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# Equations of Hamilton
## The Hamiltonian
> *Definition 1*: let $\mathcal{L}: (\mathbf{q},\mathbf{q}',t) \mapsto \mathcal{L}(\mathbf{q},\mathbf{q}',t)$ be the Lagrangian of the system, suppose that the generalized momenta $\mathbf{p}$ are defined in terms of the active variables $\mathbf{q}'$ and the passive variables $(\mathbf{q},t)$ such that
>
> $$
> \mathbf{p} = \nabla_{\mathbf{q}'}\mathcal{L}(\mathbf{q},\mathbf{q}',t),
> $$
>
> for all $t \in \mathbb{R}$.
We may now pose that there exists a function that meets the inverse, which can be obtained with Legendre transforms.
> *Theorem 1*: there exists a function $\mathcal{H}: (\mathbf{q},\mathbf{p},t) \mapsto \mathcal{H}(\mathbf{q},\mathbf{p},t)$ such that
>
> $$
> \mathbf{q}' = \nabla_{\mathbf{p}} \mathcal{H}(\mathbf{q},\mathbf{p},t),
> $$
>
> for all $t \in \mathbb{R}$. Where $\mathcal{H}$ is the Hamiltonian of the system and is related to the Lagrangian $\mathcal{L}$ by
>
> $$
> \mathcal{H}(\mathbf{q},\mathbf{p},t) = \langle \mathbf{q'}, \mathbf{p} \rangle - \mathcal{L}(\mathbf{q},\mathbf{q}',t),
> $$
>
> for all $t \in \mathbb{R}$ with $\mathcal{L}$ and $\mathcal{H}$ the Legendre transforms of each other.
??? note "*Proof*:"
Will be added later.
## The equations of Hamilton
> *Corollary 1*: the partial derivatives of $\mathcal{L}$ and $\mathcal{H}$ with respect to the passive variables are related by
>
> $$
> \begin{align*}
> \nabla_{\mathbf{q}} \mathcal{H}(\mathbf{q},\mathbf{p},t) &= - \nabla_{\mathbf{q}} \mathcal{L}(\mathbf{q},\mathbf{q}',t), \\
> \partial_t \mathcal{H}(\mathbf{q},\mathbf{p},t) &= - \partial_t \mathcal{L}(\mathbf{q},\mathbf{q}',t),
> \end{align*}
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
Obtaining the equations of Hamilton
$$
\begin{align*}
\mathbf{p}' &= -\nabla_{\mathbf{q}} \mathcal{H}(\mathbf{q},\mathbf{p},t), \\
\mathbf{q}' &= \nabla_{\mathbf{p}} \mathcal{H}(\mathbf{q},\mathbf{p},t),
\end{align*}
$$
for all $t \in \mathbb{R}$.
> *Proposition 1*: when the Hamiltonian $\mathcal{H}$ has no explicit time dependence it is a constant of motion.
??? note "*Proof*:"
Will be added later.
To put it differently; a Hamiltonian of a conservative autonomous system is conserved.
> *Theorem 2*: for conservative autonomous systems, the Hamiltonian $\mathcal{H}$ may be expressed as
>
> $$
> \mathcal{H}(\mathbf{q},\mathbf{p}) = T(\mathbf{q},\mathbf{p}) + V(\mathbf{q}),
> $$
>
> for all $t \in \mathbb{R}$ with $T: (\mathbf{q},\mathbf{p}) \mapsto T(\mathbf{q},\mathbf{p})$ and $V: \mathbf{q} \mapsto V(\mathbf{q})$ the kinetic and potential energy of the system.
??? note "*Proof*:"
Will be added later.
It may be observed that the Hamiltonian $\mathcal{H}$ and [generalised energy](/en/physics/mechanics/lagrangian-mechanics/lagrange-generalizations/#the-generalized-energy) $h$ are identical. Note however that $\mathcal{H}$ must be expressed in $(\mathbf{q},\mathbf{p},t)$ which is not the case for $h$.
> *Proposition 2*: a coordinate $q_j$ is cyclic if
>
> $$
> \partial_{q_j} \mathcal{H}(\mathbf{q},\mathbf{p},t) = 0,
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
> *Proposition 3*: the Hamiltonian is seperable if there exists two mutually independent subsystems.
??? note "*Proof*:"
Will be added later.
## Poisson brackets
> *Definition 2*: let $G: (\mathbf{q},\mathbf{p},t) \mapsto G(\mathbf{q},\mathbf{p},t)$ be an arbitrary observable, its time derivative may be given by
>
> $$
> \begin{align*}
> d_t G(\mathbf{q},\mathbf{p},t) &= \sum_{j=1}^f \Big(\partial_{q_j} G q_j' + \partial_{p_j} G p_j' \Big) + \partial_t G, \\
> &= \sum_{j=1}^f \Big(\partial_{q_j} G \partial_{p_j} \mathcal{H} - \partial_{p_j} G \partial_{q_j} \mathcal{H} \Big) + \partial_t G, \\
> &\overset{\mathrm{def}}= \{G, \mathcal{H}\} + \partial_t G.
> \end{align*}
> $$
>
> for all $t \in \mathbb{R}$ with $\mathcal{H}$ the Hamiltonian and $\{G, \mathcal{H}\}$ the Poisson bracket of $G$ and $\mathcal{H}$.
The Poisson bracket may simplify expressions; it has distinct properties that are true for any observables. The following theorem demonstrates the usefulness even more.
> *Theorem 3*: let $f: (\mathbf{q}, \mathbf{p}, t) \mapsto f(\mathbf{q}, \mathbf{p}, t)$ and $g: (\mathbf{q}, \mathbf{p}, t) \mapsto f(\mathbf{q}, \mathbf{p}, t)$ be two integrals of Hamilton's equations given by
>
> $$
> \begin{align*}
> f(\mathbf{q}, \mathbf{p}, t) = c_1, \\
> g(\mathbf{q}, \mathbf{p}, t) = c_2,
> \end{align*}
> $$
>
> for all $t \in \mathbb{R}$ with $c_{1,2} \in \mathbb{R}$. Then
>
> $$
> \{f,g\} = c_3
> $$
>
> with $c_3 \in \mathbb{R}$ for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.

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# Hamiltonian formalism of mechanics
The Hamiltonian formalism of mechanics is based on the definitions posed by [Lagrangian mechanics](/en/physics/mechanics/lagrangian-mechanics/lagrangian-formalism) and the axioms, postulates and principles posed in the [Newtonian formalism](/en/physics/mechanics/newtonian-mechanics/newtonian-formalism/).
Where the Lagrangian formalism used the [principle of virtual work](/en/physics/mechanics/lagrangian-mechanics/lagrange-equations/#principle-of-virtual-work) to derive the Lagrangian equations of motion, the Hamiltonian formalism will derive the Lagrangian equations with the stationary action principle. A derivative of Fermat's principle of least time.
In Hamilton's formulation the stationary action principle is referred to as Hamilton's principle.
## Hamilton's principle
> *Principle 1*: of all the kinematically possible motions that take a mechanical system from one given configuration to another within a time interval $T \subset \mathbb{R}$, the actual motion is the stationary point of the time integral of the Lagrangian $\mathcal{L}$ of the system. Let $S$ be the functional of the trajectories of the system, then
>
> $$
> S = \int_T \mathcal{L} dt,
> $$
>
> has stationary points.
The functional $S$ is often referred to as the action of the system. With this principle the equations of Lagrange can be derived.
> *Theorem 1*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, the equations of Lagrange are given by
>
> $$
> \partial_{q_j} \mathcal{L}(\mathbf{q}, \mathbf{q'}) - d_t \Big(\partial_{q_j'} \mathcal{L}(\mathbf{q}, \mathbf{q'}) \Big) = 0,
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Let the redefined generalized coordinates $\mathbf{q}: (t,a) \mapsto \mathbf{q}(t,a)$ be given by
$$
\mathbf{q}(t,a) = \mathbf{\hat q}(t) + a \varepsilon(t),
$$
with $\mathbf{\hat q}: t \mapsto \mathbf{\hat q}(t)$ the generalized coordinates of the system and $\varepsilon: t \mapsto \varepsilon(t)$ a smooth differentiable function.
Let $S: a \mapsto S(a)$ be the action of the system and let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian of the system, according to Hamilton's principle
$$
S(a) = \int_T \mathcal{L}(\mathbf{q}, \mathbf{q'})dt,
$$
for all $a \in \mathbb{R}$. To determine the stationary points we must have that $S'(0) = 0$. We have that $S'$ is given by
$$
\begin{align*}
S'(a) &= \int_T \partial_a \mathcal{L}(\mathbf{q}, \mathbf{q'})dt, \\
&= \int_T \sum_{j=1}^f \bigg(\partial_{q_j} \mathcal{L} \partial_a q_j + \partial_{q_j'} \mathcal{L} \partial_a q_j'\bigg)dt, \\
&= \int_T \sum_{j=1}^f \bigg(\partial_{q_j} \mathcal{L} \varepsilon_j(t) + \partial_{q_j'} \mathcal{L} \partial_a \partial_t q_j\bigg)dt. \\
\end{align*}
$$
Partial integration may be used for the second part:
$$
\begin{align*}
\int_T \partial_{q_j'} \mathcal{L} \partial_a \partial_t q_j dt &= \Big[\partial_{q_j'} \mathcal{L} \partial_a q_j \Big]_T - \int_T \partial_a q_j d_t (\partial_{q_j'} \mathcal{L})dt, \\
&= \Big[\partial_{q_j'} \mathcal{L} \varepsilon_j(t) \Big]_T - \int_T \partial_a q_j d_t (\partial_{q_j'} \mathcal{L})dt.
\end{align*}
$$
Choose $\varepsilon_j$ such that
$$
\Big[\partial_{q_j'} \mathcal{L} \varepsilon_j(t) \Big]_T = 0.
$$
Obtains
$$
\int_T \partial_{q_j'} \mathcal{L} \partial_a \partial_t q_j dt = - \int_T \partial_a q_j d_t (\partial_{q_j'} \mathcal{L})dt.
$$
The general expression of $S'$ may now be given by
$$
\begin{align*}
S'(a) &= \int_T \sum_{j=1}^f \bigg(\partial_{q_j} \mathcal{L} \varepsilon_j(t) - \partial_a q_j d_t (\partial_{q_j'} \mathcal{L})\bigg)dt, \\
&= \int_T \sum_{j=1}^f \bigg(\partial_{q_j} \mathcal{L} \varepsilon_j(t) - \varepsilon_j(t) d_t (\partial_{q_j'} \mathcal{L})\bigg)dt, \\
&= \sum_{j=1}^f \int_T \varepsilon_j(t) \Big(\partial_{q_j} \mathcal{L} - d_t (\partial_{q_j'} \mathcal{L})\Big)dt.
\end{align*}
$$
Then
$$
S'(0) = \sum_{j=1}^f \int_T \varepsilon_j(t) \Big(\partial_{q_j} \mathcal{L} - d_t (\partial_{q_j'} \mathcal{L})\Big)dt = 0,
$$
since $\varepsilon_j$ can be chosen arbitrary this implies that
$$
\partial_{q_j} \mathcal{L} - d_t (\partial_{q_j'} \mathcal{L}) = 0.
$$

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# The equations of Lagrange
## Principle of virtual work
> *Definition 1*: a virtual displacement is a displacement at a fixed moment in time that is consistent with the constraints at that moment.
The following principle addresses the problem that the constraint forces are generally unknown.
> *Principle 1*: let $\mathbf{\delta x}_i \in \mathbb{R}^m$ be a virtual displacement and let $\mathbf{F}_i: \mathbf{q} \mapsto \mathbf{F}_i(\mathbf{q})$ be the total force excluding the constraint forces. Then
>
> $$
> \sum_{i=1}^n \Big\langle \mathbf{F}_i(\mathbf{q}) - m_i \mathbf{x}_i''(\mathbf{q}), \mathbf{\delta x}_i \Big\rangle = 0,
> $$
>
> is true for sklerenomic constraints and all $t \in \mathbb{R}$.
Which implies that the constraint forces do not do any (net) virtual work.
## The equations of Lagrange
> *Theorem 1*: let $T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})$ be the kinetic energy of the system. For holonomic constraints we have that
>
> $$
> d_t \Big(\partial_{q_j'} T(\mathbf{q},\mathbf{q}') \Big) - \partial_{q_j} T(\mathbf{q},\mathbf{q}') = Q_j(\mathbf{q}),
> $$
>
> for all $t \in \mathbb{R}$. With $Q_j: \mathbf{q} \mapsto Q_j(\mathbf{q})$ the generalized forces of type I given by
>
> $$
> Q_j(\mathbf{q}) = \sum_{i=1}^n \Big\langle \mathbf{F}_i(\mathbf{q}), \partial_j \mathbf{x}_i(\mathbf{q}) \Big\rangle,
> $$
>
> for all $t \in \mathbb{R}$ with $\mathbf{F}_i: \mathbf{q} \mapsto \mathbf{F}_i(\mathbf{q})$ the total force excluding the constraint forces.
??? note "*Proof*:"
Will be added later.
Obtaining the equations of Lagrange. Note that the position of each point mass $\mathbf{x}_i$ is defined in the [Lagrangian formalism](lagrangian-formalism.md#generalizations).
### Conservative systems
For conservative systems we may express the force $\mathbf{F}_i: \mathbf{q} \mapsto \mathbf{F}_i(\mathbf{q})$ in terms of a potential energy $V: X \mapsto V(X)$ by
$$
\mathbf{F}_i(\mathbf{q}) = -\nabla_i V(X),
$$
for $X: \mathbf{q} \mapsto X(\mathbf{q}) \overset{\mathrm{def}}= \{\mathbf{x}_i(\mathbf{q})\}_{i=1}^n$.
> *Lemma 1*: for a conservative holonomic system the generalized forces of type I $Q_j: \mathbf{q} \mapsto Q_j(\mathbf{q})$ may be expressed in terms of the potential energy $V: \mathbf{q} \mapsto V(\mathbf{q})$ by
>
> $$
> Q_j(\mathbf{q}) = -\partial_{q_j} V(\mathbf{q}),
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
The equation of Lagrange may now be rewritten, which obtains the following lemma.
> *Lemma 2*: let $T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})$ and $V: \mathbf{q} \mapsto V(\mathbf{q})$ be the kinetic and potential energy of the system. The Lagrange equations for conservative systems are given by
>
> $$
> d_t \Big(\partial_{q_j'} T(\mathbf{q},\mathbf{q}')\Big) - \partial_{q_j}T(\mathbf{q},\mathbf{q}') = - \partial_{q_j} V(\mathbf{q}),
> $$
>
> for all $t \in \mathbb{R}$
??? note "*Proof*:"
Will be added later.
> *Definition 2*: let $T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})$ and $V: \mathbf{q} \mapsto V(\mathbf{q})$ be the kinetic and potential energy of the system. The Lagrangian $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ is defined as
>
> $$
> \mathcal{L}(\mathbf{q}, \mathbf{q'}) = T(\mathbf{q},\mathbf{q}') - V(\mathbf{q}),
> $$
>
> for all $t \in \mathbb{R}$.
With this definition we may write the Lagrange equations in a more formal way.
> *Theorem 2*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, the equations of Lagrange for conservative holonomic systems are given by
>
> $$
> d_t \Big(\partial_{q_j'} \mathcal{L}(\mathbf{q}, \mathbf{q'}) \Big) - \partial_{q_j} \mathcal{L}(\mathbf{q}, \mathbf{q'}) = 0,
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.

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# Lagrange generalizations
## The generalized momentum and force
> *Definition 1*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, the **generalized momentum** $p_j: (\mathbf{q}, \mathbf{q}') \mapsto p_j(\mathbf{q},\mathbf{q}')$ is defined as
>
> $$
> p_j(\mathbf{q},\mathbf{q}') = \partial_{q_j'} \mathcal{L}(\mathbf{q}, \mathbf{q'}),
> $$
>
> for all $t \in \mathbb{R}$.
The generalized momentum may also be referred to as the canonical or conjugated momentum. Recall that $j \in \mathbb{N}[j\leq f]$.
> *Definition 2*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, the **generalized force of type II** $F_j: (\mathbf{q}, \mathbf{q}') \mapsto F_j(\mathbf{q},\mathbf{q}')$ is defined as
>
> $$
> F_j(\mathbf{q},\mathbf{q}') = \partial_{q_j} \mathcal{L}(\mathbf{q}, \mathbf{q'})
> $$
>
> for all $t \in \mathbb{R}$.
We may also write $\mathbf{p} = \{p_j\}_{j=1}^f$ and $\mathbf{F} = \{F_j\}_{j=1}^f$.
## The generalized energy
> *Theorem 1*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, the generalized energy $h: (\mathbf{q}, \mathbf{q'},\mathbf{p}) \mapsto h(\mathbf{q}, \mathbf{q'},\mathbf{p})$ is given by
>
> $$
> h(\mathbf{q}, \mathbf{q'}, \mathbf{p}) = \sum_{j=1}^f \big(p_j q_j' \big) - \mathcal{L}(\mathbf{q}, \mathbf{q'}),
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
A generalization of the concept of energy.
* If the Lagrangian $\mathcal{L}: (\mathbf{q}, \mathbf{q'},t) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'},t)$ is explicitly time-dependent $\partial_t \mathcal{L}(\mathbf{q}, \mathbf{q'},t) \neq 0$ and the generalized energy $h$ is not conserved.
* If the Lagrangian $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ is not explicitly time-dependent $\partial_t \mathcal{L}(\mathbf{q}, \mathbf{q'}) = 0$ and the generalized energy $h$ is conserved.
> *Theorem 2*: for autonomous systems with only conservative forces the generalized energy $h: (\mathbf{q}, \mathbf{q'}) \mapsto h(\mathbf{q}, \mathbf{q'})$ is conserved and is given by
>
> $$
> h(\mathbf{q}, \mathbf{q'}) = T(\mathbf{q},\mathbf{q}') + V(\mathbf{q}) \overset{\mathrm{def}}= E,
> $$
>
> for all $t \in \mathbb{R}$ with $T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})$ and $V: \mathbf{q} \mapsto V(\mathbf{q})$ the kinetic and potential energy of the system and $E \in \mathbb{R}$ the total energy of the system.
??? note "*Proof*:"
Will be added later.
In this case the generalized energy $h$ is conserved and is equal to the total energy $E$ of the system.
## Conservation of generalized momentum
> *Definition 3*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, a coordinate $q_j$ is **cyclic** if
>
> $$
> \partial_{q_j} \mathcal{L}(\mathbf{q}, \mathbf{q'}) = 0,
> $$
>
> for all $t \in \mathbb{R}$.
Therefore the Lagrangian is independent of a cyclic coordinate.
> *Proposition 1*: the generalized momentum $p_j$ corresponding to a cyclic coordinate $q_j$ is conserved.
??? note "*Proof*:"
Will be added later.
## Seperable systems
> *Proposition 2*: the Lagrangian is seperable if there exists two mutually independent subsystems.
??? note "*Proof*:"
Will be added later.
Obtaining a decoupled set of partial differential equations.
## Invariances
> *Proposition 3*: the Lagrangian is invariant for Gauge transformations and therefore **not unique**.
??? note "*Proof*:"
Will be added later.
There can exist multiple Lagrangians that may lead to the same equation of motion.
According to the theorem of Noether, the invariance of a closed system with respect to continuous transformations implies that corresponding conservation laws exist.

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# Lagrangian formalism of mechanics
The Lagrangian formalism of mechanics is based on the axioms, postulates and principles posed in the [Newtonian formalism](/en/physics/mechanics/newtonian-mechanics/newtonian-formalism/).
## Configuration of a system
Considering a system of $n \in \mathbb{R}$ point masses $m_i \in \mathbb{R}$ with positions $\mathbf{x}_i \in \mathbb{R}^m$ in dimension $m \in \mathbb{N}$, for $i \in \mathbb{N}[i \leq n]$.
> *Definition 1*: the set of positions $\{\mathbf{x}_i\}_{i=1}^n$ is defined as the configuration of the system.
Obtaining a $n m$ dimensional configuration space of the system.
> *Definition 2*: let $N = nm$, the set of time dependent coordinates $\{q_i: t \mapsto q_i(t)\}_{i=1}^N$ at a time $t \in \mathbb{R}$ is a point in the $N$ dimensional configuration space of the system.
<br>
> *Definition 3*: let the generalized coordinates be a minimal set of coordinates which are sufficient to specify the configuration of a system completely and uniquely.
The minimum required number of generalized coordinates is called the number of degrees of freedom of the system.
## Classification of constraints
> *Definition 4*: geometric constraints define the range of the positions $\{\mathbf{x}_i\}_{i=1}^n$.
<br>
> *Definition 5*: holonomic constraints are defined as constraints that can be formulated as an equation of generalized coordinates and time.
Let $g: (q_1, \dots, q_N, t) \mapsto g(q_1, \dots, q_N, t) = 0$ is an example of a holonomic constraint.
> *Definition 6*: a constraint that depends on velocities is defined as a kinematic constraint.
If the kinematic constrain is integrable and can be formulated as a holonomic constraint it is referred to as a integrable kinematic constraint.
> *Definition 7*: a constraint that explicitly depends on time is defined as a rheonomic constraint. Otherwise the constraint is defined as a sklerenomic constraint.
If a system of $n$ point masses is subject to $k$ indepent holonomic constraints, then these $k$ equations can be used to eliminate $k$ of the $N$ coordinates. Therefore there remain $f \overset{\mathrm{def}}= N - k$ "independent" generalized coordinates.
## Generalizations
> *Definition 8*: the set of generalized velocities $\{q_i'\}_{i=1}^N$ at a time $t \in \mathbb{R}$ is the velocity at a point along its trajectory through configuration space.
The position of each point mass may be given by
$$
\mathbf{x}_i: \mathbf{q} \mapsto \mathbf{x}_i(\mathbf{q}),
$$
with $\mathbf{q} = \{q_i\}_{i=1}^f$ generalized coordinates.
Therefore the velocity of each point mass is given by
$$
\mathbf{x}_i'(\mathbf{q}) = \sum_{r=1}^f \partial_r \mathbf{x}_i(\mathbf{q}) q_r',
$$
for all $t \in \mathbb{R}$ (inexplicitly).
> *Theorem 1*: the total kinetic energy $T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q}')$ of the system is given by
>
> $$
> T(\mathbf{q}, \mathbf{q}') = \sum_{r,s=1}^f a_{rs}(\mathbf{q}) q_r' q_s',
> $$
>
> with
>
> $$
> a_{rs}(\mathbf{q}) = \sum_{i=1}^n \frac{1}{2} m_i \Big\langle \partial_r \mathbf{x}_i(\mathbf{q}), \partial_s \mathbf{x}_i(\mathbf{q}) \Big\rangle,
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.

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# Energy
## Potential energy
> *Definition 1*: a force field $\mathbf{F}$ is conservative if it is [irrotational](../../mathematical-physics/vector-analysis/vector-operators/#potentials)
>
> $$
> \nabla \times \mathbf{F} = 0,
> $$
>
> obtaining a scalar potential $V$ such that
>
> $$
> \mathbf{F} = - \nabla V,
> $$
>
> referred to as the potential energy.
## Kinetic energy
> *Definition 2*: the kinetic energy $T: t \mapsto T(t)$ of a pointmass $m \in \mathbb{R}$ with position $x: t \mapsto x(t)$ subject to a force $\mathbf{F}: x \mapsto \mathbf{F}(x)$ is defined as
>
> $$
> T(t) - T(0) = \int_0^t \langle \mathbf{F}(x), dx \rangle,
> $$
>
> for all $t \in \mathbb{R}$.
<br>
> *Proposition 1*: the kinetic energy $T: t \mapsto T(t)$ of a pointmass $m \in \mathbb{R}$ with position $x: t \mapsto x(t)$ subject to a force $\mathbf{F}: x \mapsto \mathbf{F}(x)$ is given by
>
> $$
> T(t) - T(0) = \frac{1}{2} m \|x'(t)\|^2 - \frac{1}{2} m \|x'(0)\|^2,
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
## Energy conservation
> *Theorem 1*: for a pointmass $m \in \mathbb{R}$ with position $x: t \mapsto x(t)$ subject to a force $\mathbf{F}: x \mapsto \mathbf{F}(x)$ we have that
>
> $$
> T(x) + V(x) = T(0) + V(0) \overset{\mathrm{def}} = E,
> $$
>
> for all x, with $T: x \mapsto T(x)$ and $V: x \mapsto V(x)$ the kinetic and potential energy of the point mass.
??? note "*Proof*:"
Will be added later.
Obtaining conservation of energy with $E \in \mathbb{R}$ the total (constant) energy of the system.

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# Momentum
> *Definition 1*: the **momentum** $\mathbf{p}$ of a particle is defined as the product of the mass and velocity of the particle
>
> $$
> \mathbf{p} = m \mathbf{v},
> $$
>
> with $m$ the mass of the particle and $\mathbf{v}$ the velocity of the particle.
For the case that $\mathbf{v}: t \to \mathbf{v}(t) \implies \mathbf{v}'(t) = \mathbf{a}(t)$ we have the following theorem.
> *Theorem 1*: let $\mathbf{v}$, $\mathbf{a}$ be the velocity and acceleration of a particle respectively, if we have
>
> $$
> \mathbf{v}: t \to \mathbf{v}(t) \implies \forall t \in \mathbb{R}: \mathbf{v}'(t) = \mathbf{a}(t),
> $$
>
> then
>
> $$
> \mathbf{p}'(t) = \mathbf{F}(t),
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.

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# Newtonian formalism of mechanics
## Fundamental assumptions
> *Postulate 1*: there exists an absolute space in which the axioms of Euclidean geometry hold.
The properties of space are constant, immutable and entirely independent of the presence of objects and of all dynamical processes that occur within it.
> *Postulate 2*: there exists an absolute time, entirely independent.
From postulate 1 and 2 we obtain the notion that simultaneity is absolute. In the sense that incidents that occur simultaneously in one reference system, occur simultaneously in all reference systems, independent of their mutual dynamic states or relations.
The definition of a reference system will follow in the next section.
> *Principle of relativity*: all physical axioms are of identical form in all **inertial** reference systems.
It follows from the principle of relativity that the notion of absolute velocity does not exist.
> *Postulate 3*: space and time are continuous, homogeneous and isotropic.
Implying that there is no fundamental limit to the precision of measurements of spatial positions, velocities and time intervals. There are no special locations or instances in time all positions and times are equivalent. The properties of space and time are invariant under translations. And there are no special directions, all directions are equivalent. The properties of space and time are invariant under rotations and reflections.
## Galilean transformations
> *Definition 1*: a **reference system** is an abstract coordinate system whose origin, orientation, and scale are specified by a set of geometric points whose position is identified both mathematically and physically.
From the definition of a reference system and postulates 1, 2 and 3 the Galilean transformations may be posed, which may be used to transform between the coordinates of two reference systems.
> *Principle 1*: let $(\mathbf{x},t) \in \mathbb{R}^4$ be a general point in spacetime.
>
> A uniform motion with velocity $\mathbf{v}$ is given by
>
> $$
> (\mathbf{x},t) \mapsto (\mathbf{x} + \mathbf{v}t,t),
> $$
>
> for all $\mathbf{v}\in \mathbb{R}^3$.
>
> A translation by $(\mathbf{a},t)$ is given by
>
> $$
> (\mathbf{x},t) \mapsto (\mathbf{x} + \mathbf{a},t + s),
> $$
>
> for all $(\mathbf{a},t) \in \mathbb{R}^4$.
>
> A rotation by $R$ is given by
>
> $$
> (\mathbf{x},t) \mapsto (R \mathbf{x},t),
> $$
>
> for all orthogonal transformations $R: \mathbb{R}^3 \to \mathbb{R}^3$.
The Galilean transformations may form a Lie group.
## Axioms of Newton
> *Axiom 1*: in the absence of external forces, a particle moves with a constant speed along a straight line.
>
> *Axiom 2:* the net force on a particle is equal to the rate at which the particle's momentum changes with time.
>
> *Axiom 3:* if two particles exert forces onto each other, then the mutual forces have equal magnitudes but opposite directions.
From axiom 1 and the principle of relativity the definition of a inertial reference system may be posed.
> *Definition 2*: an **inertial reference system** is a reference system in which the first axiom of Newton holds.
This implies that a inertial reference system is reference system not undergoing any acceleration. Therefore we may postulate the following.
> *Postulate 4*: inertial reference systems exist.
<br>
> *Definition 3*: considering two particles $i \in \{1,2\}$ which exert forces onto each other having accelerations $\mathbf{a}_i$. Since by the 2nd and 3rd axiom we have that $\mathbf{a}_1 = - \mathbf{a}_2$ and that the ratio of their magnitudes is a constant we define the ratio of the inertial masses by
>
> $$
> \frac{m_1}{m_2} = \frac{\|\mathbf{a}_2\|}{\|\mathbf{a}_1\|}.
> $$
A particle with a mass can be considered as a point mass, which is defined below.
> *Definition 4*: a point mass is defined as a point in space and time appointed with a mass.
## Forces
> *Definition 5*: a force $\mathbf{F}$ is defined as
>
> $$
> \mathbf{F} = m \mathbf{a},
> $$
>
> with $m \in \mathbb{R}$ the inertial mass and $\mathbf{a}$ the acceleration of the particle.
Definition 5 also implies the equation of motion, for a constant force a second order ordinary differential equation of the position.
> *Proposition 1*: in the case that a force only depends on position, the equation of motion is invariant to time inversion and time translation.
??? note "*Proof*:"
Will be added later.
This implies that for a moving a particle in a force field it can not be deduced at what point in time it occured and whether it is moving forward or backward in time.
> *Definition 6*: a central force $\mathbf{F}$ representing the interaction between two point masses at positions $\mathbf{x}_1$ and $\mathbf{x}_2$ is defined as
>
> $$
> \mathbf{F} = F(\mathbf{x}_1,\mathbf{x}_2) \frac{\mathbf{x}_2 - \mathbf{x}_1}{\|\mathbf{x}_2 - \mathbf{x}_1\|} \overset{\mathrm{def}} = F(\mathbf{x}_1,\mathbf{x}_2) \mathbf{e}_r,
> $$
>
> with $F: (\mathbf{x}_1,\mathbf{x}_2) \mapsto F(\mathbf{x}_1,\mathbf{x}_2)$ the magnitude.
Which for a isotropic central force depends only on the distance between the pointmasses $\|\mathbf{x}_2 - \mathbf{x}_1\|$.
### Gravitational force of Newton
> *Postulate 5*: the force $\mathbf{F}$ between two particles described by their positions $\mathbf{x}_{1,2}: t \mapsto \mathbf{x}_{1,2}(t)$ is given by
>
> $$
> \mathbf{F} = G \frac{m_1 m_2}{\|\mathbf{x}_2 - \mathbf{x}_1\|^2} \mathbf{e}_r,
> $$
>
> with $m_{1,2} \in \mathbb{R}$ the gravitational mass of both particles and $G \in \mathbb{R}$ the gravitational constant.
According to the observation of Galilei; all object fall with equal speed (in the absence of air friction), which implies that the ratio of inertial and gravitational mass is a constant for any kind of matter.
> *Principle 2*: the inertial and gravitational mass of a particle are equal.

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# Particle systems
For a system of particles we have the mutual forces among the selected particles referred to as internal forces, otherwise external forces. If there are no external forces, the system is called closed, otherwise open.
> *Definition 1*: the internal interaction forces $\mathbf{F}_i$ in a system of $n \in \mathbb{N}$ particles with position $\mathbf{x}_i$ may be approximated by pairwise interaction forces given by
>
> $$
> \mathbf{F}_i (\mathbf{x}_i) = \sum_{j=1}^n \mathbf{F}_{ij}(\mathbf{x}_i, \mathbf{x}_j) \epsilon_{ij},
> $$
>
> for all $\mathbf{x}_i$ with $\mathbf{F}_{ij}$ the pairwise interaction force between particle $i$ and $j$.
For high density systems this approximation diverges.
## Systems with conservative internal forces
Considering a system of $n \in \mathbb{N}$ particles with position $\mathbf{x}_i$ and mass $m_i \in \mathbb{R}$ with conservative external forces $\mathbf{F}_i$. For each particle an equation of motion can be formulated using the pairwise interaction approximation (definition 1), obtaining
$$
m_i \mathbf{x}_i''(t) = \mathbf{F}_i(\mathbf{x}_i(t)) + \sum_{j=1}^n \mathbf{F}_{ij}(\mathbf{x}_i, \mathbf{x}_j) \epsilon_{ij},
$$
for all $t \in \mathbb{R}$ with $\mathbf{F}_{ij}$ the pairwise interaction force.
> *Definition 2*: the total mass $M$ of the system is defined as
>
> $$
> M = \sum_{i=1}^n m_i.
> $$
<br>
> *Definition 3*: the center of mass $\mathbf{R}: t \mapsto \mathbf{R}(t)$ of the system is defined as
>
> $$
> \mathbf{R}(t) = \frac{1}{M} \sum_{i=1}^n m_i \mathbf{x}_i(t),
> $$
>
> for all $t \in \mathbb{R}$.
<br>
> *Definition 4*: the total momentum $\mathbf{P}$ and angular momentum $\mathbf{J}$ of the system are defined as
>
> $$
> \begin{align*}
> \mathbf{P} &= \sum_{i=1}^n \mathbf{p}_i, \\
> \mathbf{J} &= \sum_{i=1}^n \mathbf{x}_i \times \mathbf{p}_i,
> \end{align*}
> $$
>
> with $\mathbf{p}_i$ the momentum of each particle.
We have for $\mathbf{P}: t \mapsto \mathbf{P}(t)$ the total momentum equivalently given by
$$
\mathbf{P}(t) = M \mathbf{R}'(t),
$$
for all $t \in \mathbb{R}$ with $\mathbf{R}: t \mapsto \mathbf{R}(t)$ the center of mass.
> *Definition 5*: the total external force $\mathbf{F}$ and torque $\mathbf{\Gamma}$ of the system are defined as
>
> $$
> \begin{align*}
> \mathbf{F} &= \sum_{i=1}^n \mathbf{F}_i, \\
> \mathbf{\Gamma} &= \sum_{i=1}^n \mathbf{x}_i \times \mathbf{F}_i,
> \end{align*}
> $$
>
> with $\mathbf{F}_i$ the conservative external force.
<br>
> *Proposition 1*: the total momentum $\mathbf{P}: t \mapsto \mathbf{P}(t)$ is related to the total external force $\mathbf{F}: t \mapsto \mathbf{F}(t)$ by
>
> $$
> \mathbf{P}'(t) = \mathbf{F}(t),
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be adder later.
> *Proposition 2*: the total angular momentum $\mathbf{J}: t \mapsto \mathbf{J}(t)$ is related to the total external torque $\mathbf{\Gamma}: t \mapsto \mathbf{\Gamma}(t)$ by
>
> $$
> \mathbf{J}'(t) = \mathbf{\Gamma}(t),
> $$
>
> for all $t \in \mathbb{R}$ if the internal forces are central forces.
??? note "*Proof*:"
Will be adder later.
### Orbital and spin angular momentum
Considering internal position vectors $\mathbf{r}_i$ relative to the center of mass $\mathbf{r}_i = \mathbf{x}_i - \mathbf{R}$. I propose that the total angular momentum $\mathbf{J}$ can be expressed as a superposition of the orbital $\mathbf{L}$ and spin $\mathbf{S}$ angular momentum components given by
$$
\mathbf{J} = \mathbf{L} + \mathbf{S}.
$$
??? note "*Proof*:"
Will be added later.
> *Definition 6*: the orbital angular momentum $\mathbf{L}$ of the system is defined as
>
> $$
> \mathbf{L} = \mathbf{R} \times \mathbf{P},
> $$
>
> with $\mathbf{R}$ the center of mass and $\mathbf{P}$ the total momentum of the system.
<br>
> *Definition 7*: the spin angular momentum $\mathbf{S}: t \mapsto \mathbf{S}(t)$ of the system is defined as
>
> $$
> \mathbf{S}(t) = \sum_{i=1}^n \mathbf{r}_i(t) \times m_i \mathbf{r}'_i(t)
> $$
>
> for all $t \in \mathbb{R}$ with $\mathbb{r}_i$ the internal position.
Analoguosly the orbital and spin torque may be defined.
> *Definition 8*: the orbital and spin torque $\mathbf{\Gamma}_{o,s}$ of the system are defined as
>
> $$
> \begin{align*}
> \mathbf{\Gamma}_o &= \mathbf{R} \times \mathbf{F}, \\
> \mathbf{\Gamma}_s &= \sum_{i=1}^n \mathbf{r}_i \times \mathbf{F}_i,
> \end{align*}
> $$
>
> with $\mathbf{R}$ the center of mass, $\mathbf{r}_i$ the internal position and $\mathbf{F}_i$ the conservative external force.
Similarly, the total torque $\mathbf{\Gamma}$ of the system is the superposition of the orbital and spin torque $\mathbf{\Gamma}_{o,s}$ given by
$$
\mathbf{\Gamma} = \mathbf{\Gamma}_o + \mathbf{\Gamma}_s.
$$
??? note "*Proof*:"
Will be added later.
> *Proposition 3*: let $\mathbf{L}: t \mapsto \mathbf{L}(t)$ be the orbital angular momentum and let $\mathbf{S}: t \mapsto \mathbf{S}(t)$ be the spin angular momentum. Then we have
>
> $$
> \begin{align*}
> \mathbf{L}'(t) &= \mathbf{\Gamma}_o(t), \\
> \mathbf{S}'(t) &= \mathbf{\Gamma}_s(t),
> \end{align*}
> $$
>
> for all $t \in \mathbb{R}$ with $\mathbf{\Gamma}_o: t \mapsto \mathbf{\Gamma}_o(t)$ and $\mathbf{\Gamma}_s: t \mapsto \mathbf{\Gamma}_s(t)$ the orbital and spin torque.
### Energy
> *Definition 9*: the total kinetic energy $T$ of the system is defined as
>
> $$
> T = \sum_{i=1}^n \frac{1}{2} m_i \|\mathbf{x}_i'\|^2,
> $$
>
> with $\mathbf{x}_i$ the position of each particle.
<br>
> *Definition 10*: the orbital and internal kinetic energy $T_{o,r}$ of the system are defined as
>
> $$
> \begin{align*}
> T_o = \frac{1}{2} M \|\mathbf{R}\|^2, \\
> T_r = \sum_{i=1}^n \frac{1}{2} m_i \|\mathbf{r}_i'\|^2,
> \end{align*}
> $$
>
> with $M$ the total mass, $\mathbf{R}$ the center of mass and $\mathbf{r}$ the internal position of each particle.
<br>
> *Proposition 4*: the total kinetic energy $T$ of the system is a superposition of the orbital and internal kinetic energy given by
>
> $$
> T = T_o + T_r.
> $$
??? note "*Proof*:"
Will be added later.
> *Proposition 5*: the dynamics of the orbital and kinetic energy $T_o: t \mapsto T_o(t)$ is decoupled
>
> $$
> T_o'(t) = \langle \mathbf{F}, \mathbf{R}'(t) \rangle,
> $$
>
> for all $t \in \mathbb{R}$ with $\mathbf{F}$ the total external force and $\mathbf{R}$ the center of mass.
>
> The dynamics of the internal kinetic energy $T_r: t \mapsto T_r(t)$ is not decoupled
>
> $$
> T_r'(t) = \sum_{i=1}^n \langle \mathbf{f}_i, \mathbf{r}_i'(t) \rangle,
> $$
>
> for all $t \in \mathbb{R}$ with $\mathbf{f}_i$ the sum of both external and internal forces for each particle.
??? note "*Proof*:"
Will be added later.

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# Rotation
Rotation is always viewed with respect to the axis of rotation, therefore in the following definitions the origin of the position is always implies to be the axis of rotation.
## Angular momentum
> *Definition 1*: the angular momentum $L$ of a point mass with position $\mathbf{r}$ and a momentum $\mathbf{p}$ is defined as
>
> $$
> \mathbf{L} = \mathbf{r} \times \mathbf{p},
> $$
>
> for all $\mathbf{r}$ and $\mathbf{p}$.
## Torque
> *Definition 2*: the torque $\mathbf{\Gamma}$ acting on a point mass with position $\mathbf{r}$ for a force $\mathbf{F}$ os defined as
>
> $$
> \mathbf{\Gamma} = \mathbf{r} \times \mathbf{F},
> $$
>
> for all $\mathbf{r}$ and $\mathbf{F}$.
The torque is related to the angular momentum by the following proposition.
> *Proposition 1*: let $\mathbf{L}: t \mapsto \mathbf{L}(t)$ be the angular momentum of a point mass, then it holds that
>
> $$
> \mathbf{L}'(t) = \mathbf{\Gamma}(t),
> $$
>
> for a constant $\mathbf{r}$ and all $t \in \mathbb{R}$ with $\mathbf{\Gamma}: t \mapsto \mathbf{\Gamma}(t)$ the torque acting on the point mass.
??? note "*Proof*:"
Let $\mathbf{L}: t \mapsto \mathbf{L}(t)$ be the angular momentum of a point mass and suppose $\mathbf{r}$ is constant, then
$$
\mathbf{L}'(t) \overset{\mathrm{def}} = d_t (\mathbf{r} \times \mathbf{p}(t)) = \mathbf{r} \times \mathbf{p}'(t),
$$
by [proposition](momentum.md) we have $\mathbf{p}'(t) = \mathbf{F}(t)$, therefore
$$
\mathbf{L}'(t) = \mathbf{r} \times \mathbf{F}(t) \overset{\mathrm{def}} = \mathbf{\Gamma}(t).
$$

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# Maxwell equations

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# Diffraction
## Huygens principle
Huygens principle will be used to derive equations for diffraction.
> *Assumption*: According to Huygens principle each point on the wavefront of an electromagnetic wave acts as a source of secondary wavelets. When summed over an extended unobstructed wavefront the secondary wavelets recreate the next wavefront. It is assumed that this principle is valid as it is consistent with the laws of reflection and refraction.
The following theorem follows from Huygens principle.
> *Law*: the net disturbance $E_P: \mathbb{R} \to \mathbb{R}$ at a perceive point $P$ for a wave travelling from source point $S$ travelling a distance $r' \in \mathbb{R}$ to an aperture opening defined for the points in $D \subseteq \mathbb{R}$ and then travelling a distance $r \in \mathbb{R}$ towards $p$ is given by
>
> $$
> E_P(t) = E_0 k e^{-i \omega t} \int_D \frac{1}{2 r r'} (1 + \cos \theta) e^{ik (r+r')} dA,
> $$
>
> for all $t \in \mathbb{R}$ with $E_0 \in \mathbb{R}$, $k \in \mathbb{R}$ the wavenumber of the light, $\omega \in \mathbb{R}$ the angular frequency of the light and $\theta \in [0, 2\pi)$ the angle between the source, aperture and perceive point.
??? note "*Proof*:"
Will be added later.
<br>
> *Law*: for two complementary apertures that when taken together form a single opaque screen. Let $E_1$ and $E_2$ be the field at point $P$ for each aperture respectively. Then the combination of these fields must give the unubstructed wave $E_0$. Therefore
>
> $$
> E_0 = E_1 + E_2.
> $$
??? note "*Proof*:"
Will be added later.
## Fraunhofer diffraction
The above law for the diffraction at a perceive point $P$ can be simplified under certain conditions such that the integral can be solved easier.
> *Corollary*: for small angles between the source, aperture and perceive point $\theta$, implying that source and perceive points are far away and the aperture opening is small then in reasonable approximation the net disturbance $E_P: \mathbb{R} \to \mathbb{R}$ at the perceive point may be given by
>
> $$
> E_P = E_0 \int_D e^{ikr}dA,
> $$
>
> with $E_0 \in \mathbb{R}$ and $k \in \mathbb{R}$ the wavenumber. Under the condition that
>
> $$
> r >> \frac{h^2}{2\lambda},
> $$
>
> with $h \in \mathbb{R}$ the height of the aperture and $\lambda \in \mathbb{R}$ the wavelength of the light.
??? note "*Proof*:"
Will be added later.
From this simplification the net disturbance caused by several apertures can be derived, given in the corollaries below.
> *Corollary*: the net disturbance $E: \mathbb{R} \to \mathbb{R}$ of the eletric field for a single slit aperture is given by
>
> $$
> E(\theta) = E_0 \text{ sinc } \beta(\theta),
> $$
>
> for all $\theta \in \mathbb{R}$ with $\beta(\theta) = \frac{kb}{2} \sin \theta$ and $E_0, k, b \in \mathbb{R}$ the magnitude of the electric field, the wavenumber and the width of the slit.
??? note "*Proof*:"
Will be added later.
<br>
> *Corollary*: the net disturbance $E: \mathbb{R} \to \mathbb{R}$ of the eletric field for a rectangular aperture is given by
>
> $$
> E(\theta, \varphi) = E_0 \text{ sinc } \alpha(\theta) \text{ sinc } \beta(\varphi),
> $$
>
> for all $(\theta, \varphi) \in \mathbb{R}^2$ with $\alpha(\theta) = \frac{ka}{2} \sin \theta$, $\beta(\varphi) = \frac{kb}{2} \sin \varphi$ and $E_0, k, a, b \in \mathbb{R}$ the magnitude of the electric field, the wavenumber, the height and the width of the rectangle.
??? note "*Proof*:"
Will be added later.
<br>
> *Corollary*: the net disturbance $E: \mathbb{R} \to \mathbb{R}$ of the eletric field for a circular aperture is given by
>
> $$
> E(\theta) = E_0 \frac{2 J_1(\sigma(\theta))}{\sigma(\theta)},
> $$
>
> for all $\theta \in \mathbb{R}^2$ with $J_1: \mathbb{R} \to \mathbb{R}$ the Bessel function of the first order, $\sigma(\theta) = \frac{kd}{2} \sin \theta$ and $E_0, k, d \in \mathbb{R}$ the magnitude of the electric field, the wavenumber and the diameter of the circle.
??? note "*Proof*:"
Will be added later.
<br>
> *Corollary*: the net disturbance $E: \mathbb{R} \to \mathbb{R}$ of the eletric field for a $N$-slit aperture with $N \in \mathbb{N}$ is given by
>
> $$
> E(\theta) = E_0 \text{ sinc } \beta(\theta) \frac{\sin N \gamma(\theta)}{N \sin \gamma(\theta)}
> $$
>
> for all $\theta \in \mathbb{R}$ with $\beta(\theta) = \frac{kb}{2} \sin \theta$, $\gamma(\theta) = \frac{kd}{2} \sin \theta$ and $E_0, k, d, b \in \mathbb{R}$ the magnitude of the electric field, the wavenumber, the distance between the slits and the width of the slits.
??? note "*Proof*:"
Will be added later.
When taking $N \to \infty$ for the $N$-slits aperture and incidence is normal principal maxima are obtained for $\gamma(\theta) = m \pi$ with $m \in \mathbb{Z}$ therefore
$$
d \sin \theta = m \lambda,
$$
with $d, \lambda \in \mathbb{R}$ the distance between the slits and the wavelength of the light.
When incidence $\theta_i \in \mathbb{R}$ is not normal the principal maxima are given by
$$
d (\sin \theta_i + \sin \theta) = m \lambda,
$$
also known as the grating equation.
??? note "*Proof*:"
Will be added later.
<br>
> *Definition*: two point sources given by the net disturbances of the eletric field $E_{1,2}: D \to \mathbb{R}$ with $D \subseteq \mathbb{R}$ such that $E_{1,2}$ are bijective can be resolved if they satisfy the Reyleigh criterion given by
>
> $$
> \min E_2^{-1}(E_{02}) \geq \min E_1^{-1}(0),
> $$
>
> $$
> \min E_1^{-1}(E_{01}) \geq \min E_2^{-1}(0),
> $$
>
> with $E_{0(1,2)} \in \mathbb{R}$ the eletric field amplitudes.
This definition will be used in the following propositions.
> *Proposition*: the chromatic resolving power $\mathcal{R}$ of a $N$-slit aperture based on the Reyleigh criterion can be determined by
>
> $$
> \mathcal{R} = N m,
> $$
>
> with $m \in \mathbb{Z}$ the order of the principal maxima and $N \in \mathbb{N}$ the number of slits.
??? note "*Proof*:"
Will be added later.
<br>
> *Proposition*: the free spectral range $\text{FSR}$ of a $N$-slit aperture can be determined by
>
> $$
> \text{FSR} = \frac{\lambda}m,
> $$
>
> with $m \in \mathbb{Z}$ the order and $\lambda \in \mathbb{R}$ the wavelength.
??? note "*Proof*:"
Will be added later.

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# Electromagnetic waves
This section is a direct follow up on the section [Maxwell equations](../maxwell-equations.md). Where the Laplacian of the electric field $\mathbf{E}: U \to \mathbb{R}^3$ and magnetic field $\mathbf{B}: U \to \mathbb{R}^3$ in vacuum ($\varepsilon = \varepsilon_0, \mu = \mu_0$) have been determined, given by
$$
\begin{align*}
&\nabla^2 \mathbf{E}(\mathbf{v}, t) = \varepsilon_0 \mu_0 \partial_t^2 \mathbf{E}(\mathbf{v}, t) \\\\
&\nabla^2 \mathbf{B}(\mathbf{v}, t) = \varepsilon_0 \mu_0 \partial_t^2 \mathbf{B}(\mathbf{v}, t)
\end{align*}
$$
for all $(\mathbf{v}, t) \in U$.
It may be observed that the eletric and magnetic field comply with the $3 + 1$ dimensional wave equation posed in the section [waves](waves.md). Obtaining the speed $v \in \mathbb{R}$ given by
$$
v = \frac{1}{\sqrt{\varepsilon_0 \mu_0}} = c,
$$
defined by $c$ the speed of light, or more generally the speed of information in the universe. Outside vacuum we have
$$
v = \frac{1}{\sqrt{\varepsilon \mu}} = \frac{c}{n},
$$
with $n = \sqrt{K_E K_B}$ the index of refraction.
> *Proposition*: let $\mathbf{E},\mathbf{B}: U \to \mathbb{R}^3$, a solution for the wave equations of the electric and magnetic field may be harmonic linearly polarized plane waves satisfying Maxwell's equations given by
>
> $$
> \begin{align*}
> \mathbf{E}(\mathbf{v}, t) &= \text{Im}\Big(\mathbf{E}_0 \exp i \big(\langle \mathbf{k}, \mathbf{v} \rangle - \omega t+ \varphi\big) \Big) \\ \\ \mathbf{B}(\mathbf{v}, t) &= \text{Im} \Big(\mathbf{B}_0 \exp i \big(\langle \mathbf{k}, \mathbf{v} \rangle - \omega t+ \varphi\big) \Big)
> \end{align*}
> $$
>
> for all $(\mathbf{v}, t) \in U$ with $\mathbf{E}_0, \mathbf{B}_0 \in \mathbb{R}^3$.
??? note "*Proof*:"
Will be added later.
The above proposition gives an example of a light wave, but note that there are much more solutions that comply to Maxwell's equations.
> *Law*: the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ for all solutions of the posed wave equations are orthogonal to the direction of propagation $\mathbf{k}$. Therefore electromagnetic waves are transverse.
??? note "*Proof*:"
Will be added later.
<br>
> *Law*: the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ in a electromagnetic wave are orthogonal to each other; $\langle \mathbf{E}, \mathbf{B} \rangle = 0$.
??? note "*Proof*:"
Will be added later.
<br>
> *Corollary*: it follows from the above law that the magnitude of the electric and magnetic fields $E, B: U \to \mathbb{R}$ in a electromagnetic wave are related by
>
> $$
> E(\mathbf{v}, t) = v B(\mathbf{v}, t)
> $$
>
> for all $(\mathbf{v}, t) \in U$ with $v = \frac{c}{n}$ the wave speed.
??? note "*Proof*:"
Will be added later.
## Energy flow
> *Law*: the energy flux density $\mathbf{S}: U \to \mathbb{R}^3$ of an electromagnetic wave is given by
>
> $$
> \mathbf{S}(\mathbf{v}, t) = \frac{1}{\mu_0} \mathbf{E}(\mathbf{v}, t) \times \mathbf{B}(\mathbf{v}, t),
> $$
>
> for all $(\mathbf{v}, t) \in U$. $\mathbf{S}$ is also called the Poynting vector.
??? note "*Proof*:"
Will be added later.
<br>
> *Definition*: the time average of the magnitude of $\mathbf{S}$ is called the irradiance.
<br>
> *Proposition*: the irradiance $I \in \mathbb{R}$ for harmonic linearly polarized plane electromagnetic waves is given by
>
> $$
> I = \frac{\varepsilon_0 c}{2} E_0^2,
> $$
>
> with $E_0$ the magnitude of $\mathbf{E}_0$.
??? note "*Proof*:"
Will be added later.

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# Geometric optics
> *Definition*: surfaces that reflect or refract rays leaving a source point $s$ to a conjugate point $p$ are defined as Cartesian surfaces.
<br>
> *Definition*: a perfect image of a point is possible with a stigmatic system. For the set of conjugated points no diffraction and abberations occur, obtaining reversible rays.
<br>
> *Assumption*: in geometric optics use will be made of the paraxial approximation that states that for small angles $\theta$
>
> $$
> \tan \theta \approx \sin \theta \approx \theta,
> $$
>
> and
>
> $$
> \cos \theta \approx 1,
> $$
>
> comes down to using the first term of the Taylor series approximation.
<br>
## Spherical surfaces
> *Law*: for a spherical reflecting interface in paraxial approximation the relation between the object and image distance $s_{o,i} \in \mathbb{R}$ and the radius $R \in \mathbb{R}$ of the interface is given by
>
> $$
> \frac{1}{s_o} + \frac{1}{s_i} = \frac{2}{R}
> $$
>
> with $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium.
??? note "*Proof*:"
Will be added later.
<br>
> *Definition*: for a object distance $s_0 \to \infty$ we let the image distance $s_i = f$ with $f \in \mathbb{R}$ the focal length defining the focal point of the spherical interface.
Then it follows from the definition that
$$
\frac{1}{s_o} + \frac{1}{s_i} = \frac{1}{f}.
$$
> *Law*: for a spherical refracting interface in paraxial approximation the relation between the object and image distance $s_{o,i} \in \mathbb{R}$ and the radius $R \in \mathbb{R}$ of the interface is given by
>
> $$
> \frac{n_i}{s_o} + \frac{n_t}{s_i} = \frac{n_t - n_i}{R}
> $$
>
> with $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium.
??? note "*Proof*:"
Will be added later.
<br>
> *Definition*: the transverse magnification $M$ for a optical system is defined as
>
> $$
> M = \frac{y'}{y}
> $$
>
> with $y, y' \in \mathbb{R}$ the object and image size.
<br>
> *Corollary*: the transverse magnification $M$ for a spherical refracting interface in paraxial approximation is by
>
> $$
> M = - \frac{n_i s_i}{n_t s_o},
> $$
>
> with $s_{o,i} \in \mathbb{R}$ the object and image distance and $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium.
??? note "*Proof*:"
Will be added later.
<br>
> *Definition*: a lens is defined by two intersecting spherical interfaces with radius $R_1, R_2 \in \mathbb{R}$ respectively.
<br>
> *Law*: for a thin lens in paraxial approximation the radii $R_1, R_2 \in \mathbb{R}$ are related to the focal length $f \in \mathbb{R}$ of the lens by
>
> $$
> \frac{1}{f} = \frac{n_t - n_i}{n_i} \bigg( \frac{1}{R_1} - \frac{1}{R_2} \bigg),
> $$
>
> with $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium.
>
> With the transverse magnification $M$ given by
>
> $$
> M = - \frac{s_i}{s_o},
> $$
>
> with the object and image distance $s_{o,i} \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
## Sign convention
Converging optics have positive focal lengths and diverging optics have negative focal lengths.
Objects are located left of the optic by a positive object distance and images are located right of the optic by a positive image distance.
## Ray tracing
> *Assumption*: using paraxial approximation and assuming that all optical elements have rotational symmetry and are aligned coaxially along a single optical axis.
A ray matrix model may be introduced where the ray is defined according to its intersection with a reference plane.
> *Definition*: a ray may be defined by its intersection with a reference plane by
>
> * the parameter $y \in \mathbb{R}$ is the perpendicular distance between the optical axis and the intersection point,
> * the angle $\theta \in [0, 2\pi)$ is the angle the ray makes with the horizontal.
<br>
> *Proposition*: for the translation of the ray between two reference planes within the same medium seperated by a horizontal distance $d \in \mathbb{R}$ the relation
>
> $$
> \begin{pmatrix} y_2 \\ \theta_2 \end{pmatrix} = \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix} \begin{pmatrix} y_1 \\ \theta_1 \end{pmatrix},
> $$
>
> holds, for $y_{1,2} \in \mathbb{R}$ and $\theta_{1,2} \in [0, 2\pi)$.
??? note "*Proof*:"
Will be added later.
<br>
> *Proposition*: for the reflection of the ray at the plane of incidence at a spherical interface of radius $R \in \mathbb{R}$ the relation
>
> $$
> \begin{pmatrix} y_2 \\ \theta_2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 2 / R & 1 \end{pmatrix} \begin{pmatrix} y_1 \\ \theta_1 \end{pmatrix},
> $$
>
> holds, for $y_{1,2} \in \mathbb{R}$ and $\theta_{1,2} \in [0, 2\pi)$.
??? note "*Proof*:"
Will be added later.
This matrix may also be given in terms of the focal length $f \in \mathbb{R}$ by
$$
\begin{pmatrix} 1 & 0 \\ f & 1 \end{pmatrix}.
$$
> *Proposition*: fir the refraction of the ray at the plane of incidence at a spherical interfance of radius $R \in \mathbb{R}$ the relation
>
> $$
> \begin{pmatrix} y_2 \\ \theta_2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ - \frac{n_t - n_i}{n_t R} & \frac{n_i}{n_t} \end{pmatrix} \begin{pmatrix} y_1 \\ \theta_1 \end{pmatrix}
> $$
>
> holds, for $y_{1,2} \in \mathbb{R}$, $\theta_{1,2} \in [0, 2\pi)$ and $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium.
??? note "*Proof*:"
Will be added later.
This matrix may also be given in terms of the focal length $f \in \mathbb{R}$ by
$$
\begin{pmatrix} 1 & 0 \\ -\frac{1}{f} & 1 \end{pmatrix}.
$$
> *Law*: the ray matrix model taken as a linear sequence of interfaces and translations can be used to model optical systems of arbitrary complexity under the posed assumptions.
??? note "*Proof*:"
Will be added later.
## Abberations
> *Definition*: an abberation is any effect that prevents a lens from forming a perfect image.
Various abberations could be
* Spherical abberation: error of the paraxial approximation.
* Chromatic abberation: error due to different index of refraction for different wavelengths of light.
* Astigmatism: deviation from the cylindrical symmetry.

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# Interference
> *Definition*: when waves are combined in phase they combine to give a larger amplitude constructive interference occurs. When waves are combined out of phase they tend to cancel, destructive interference occurs.
## Two source interference
For interference between two monochromatic electromagnetic waves given by
$$
\begin{align*}
\mathbf{E}_1(\mathbf{v}, t) = \mathbf{E}_{01} \exp i \big(\langle \mathbf{k_1}, \mathbf{v} - \mathbf{s}_1 \rangle - \omega_1 t + \varphi_1 \big), \\
\\
\mathbf{E}_2(\mathbf{v}, t) = \mathbf{E}_{02} \exp i \big(\langle \mathbf{k_2}, \mathbf{v} - \mathbf{s}_2 \rangle - \omega_2 t + \varphi_2 \big), \\
\end{align*}
$$
for all $(\mathbf{v}, t) \in U$ with $\mathbf{k}_{1,2} \in \mathbb{R}^3$ the wavenumber, $\mathbf{s}_{1,2} \in \mathbb{R}^3$ the position of the sources. Then we have the combined disturbance at $\mathbf{v}$ is given by
$$
\begin{align*}
\mathbf{E}(\mathbf{v}, t) &= \mathbf{E}_1(\mathbf{v}, t) + \mathbf{E}_2(\mathbf{v}, t), \\
&= \mathbf{E}_{01} \exp i \delta_1(\mathbf{v},t) + \mathbf{E}_{02} \exp i \delta_2(\mathbf{v},t),
\end{align*}
$$
for all $(\mathbf{v}, t) \in U$ with $\delta_i$ the phase difference for $i \in \{1,2\}$ given by
$$
\delta_i(\mathbf{v}, t) = \langle \mathbf{k_i}, \mathbf{v} - \mathbf{s}_i \rangle - \omega_i t + \varphi_i.
$$
> *Law*: the irradiance at point $\mathbf{v}$ is then given by
>
> $$
> I(\mathbf{v}, t) = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos \Big(\delta_2(\mathbf{v}, t) - \delta_1(\mathbf{v}, t) \Big),
> $$
>
> for all $(\mathbf{v}, t) \in U$ with $I_{1,2} \in \mathbb{R}$ the irradiance for each wave seperately.
??? note "*Proof*:"
Will be added later.
Let $\delta(\mathbf{v}, t) = \delta_2(\mathbf{v}, t) - \delta_1(\mathbf{v}, t)$, then we have for $\delta(\mathbf{v}, t) = 2 m \pi$ with $m \in \mathbb{Z}$ constructive interference and for $\delta(\mathbf{v}, t) = (2m + 1) \pi$ we have destructive interference.
Writing out $\delta$ for plane waves of the same angular frequency $\omega = \omega_1 = \omega_2$ and propation in the $x$-direction gives
$$
\delta(x, t) = k(x_2 - x_1) + (\varphi_2 - \varphi_1) = \frac{2\pi}{\lambda_0} n (x_2 - x_1) + (\varphi_2 - \varphi_1),
$$
for all $(x,t) \in \mathbb{R}^2$ and $n \in \mathbb{R}$ the index of refraction of the medium. The optical path difference is defined as $n (x_2 - x_1)$.
### Double slit interference
Interference is created by plane waves illuminating both slits creating disturbances at both slits that are correlated in time. Assuming the slits are points sources and the waves have the same frequency, we have a superposition point $P$ described vertically with $y \in \mathbb{R}$ and $r_{1,2} \in \mathbb{R}$ the traveling distances from the slits to this point. Obtaining a phase difference
$$
\delta = k(r_2 - r_1) + (\varphi_2 - \varphi_1),
$$
??? note "*Proof*:"
Will be added later.
<br>
If we have $L \in \mathbb{R}$ the horizontal length between the slits and the point $P$ and $d \in \mathbb{R}$ the distance between the slits and assume $L >> d$ and $\varphi_2 - \varphi_1 = 0$ then
$$
\delta(\theta) = kd \sin \theta,
$$
for all $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ with $\tan \theta = \frac{y}{L}$.
??? note "*Proof*:"
Will be added later.
## Thin film interference
Interference is created by plane waves illuminating a thin film of thickness $l \in \mathbb{R}$ and index of refraction $n_l \in \mathbb{R}$ under an angle of incidence $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ deposited on a substrate with index of refraction $n_i \in \mathbb{R}$. A phase shift is introduced between the first external and internal reflected rays obtaining a phase difference $\delta$ given by
$$
\delta(\theta) = k 2l \sqrt{n_l^2 - n_i^2 \sin^2 \theta},
$$
for all $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ with $k \in \mathbb{R}$ the wavenumber.
??? note "*Proof*:"
Will be added later.
## Michelson interferometer
Interference created by splitting and recombining plane waves that have a difference in optical path. With a setup of two mirrors displaced with lengths $L_1, L_2 \in \mathbb{R}$ from the beam splitter under an angle $\theta$ with respect to the incoming plane wave. Assuming the setup is in *one* medium with index of refraction $n \in \mathbb{R}$. Obtaining a phase difference $\delta$ given by
$$
\delta(\theta) = k 2n(L_2 - L_1) \cos \theta + \pi,
$$
for all $\delta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ with $k \in \mathbb{R}$ the wavenumber.
??? note "*Proof*:"
Will be added later.
## Fabry-perot interferometer
Interference created by a difference in optical path length with a setup consisting of two parallel flat reflective surfaces seperated by a distance $d \in \mathbb{R}$ If both surfaces have reflection and transmission amplitude ratios $r,t \in [0,1]$ then the phase difference $\delta$ between two adjecent transmitted rays under an angle $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ is given by
$$
\delta(\theta) = 2 kd \cos \theta + 2 \varphi,
$$
for all $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ with $\varphi \in [0. 2\pi)$ the phase change due to reflection dependent on the amplitude ratios.
??? note "*Proof*:"
Will be added later.
<br>
> *Definition*: The finesse $\mathcal{F}$ and the coefficient of finesse $F$ of a Fabry Perot interferometer are defined by
>
> $$
> F = \frac{4R}{(1-R)^2} \quad\text{ and }\quad \mathcal{F} = \frac{\pi \sqrt{F}}{2} = \frac{\pi \sqrt{R}}{1 - R},
> $$
>
> with $R \in [0,1]$ the reflectance. The finesse can be seen as the measure of sharpness of the interference pattern.
<br>
> *Proposition*: the transmitted irradiance $I$ of a Fabry Perot interferometer is given by
>
> $$
> I(\theta) = \frac{I_0}{1 + 4 (\mathcal{F} / \pi)^2 \sin^2 (\delta(\theta) / 2)}
> $$
>
> for all $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ with $I_0 \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
### The chromatic resolving power and free spectral range
The chromatic resolving power and free spectral range are measures that define the ability to distinguish certain features in interference or diffraction patterns.
> *Definition*: The full width at half maximum $\text{FWHM}$ for the interference pattern of the Fabry Perot interferometer is defined to be
>
> $$
> \text{FWHM} = \frac{4}{\sqrt{F}},
> $$
>
> with $F \in \mathbb{R}$.
<br>
> *Definition*: the chromatic resolving power $\mathcal{R}$ is defined by
>
> $$
> \mathcal{R} = \frac{\lambda}{\Delta \lambda},
> $$
>
> with $\lambda \in \mathbb{R}$ the base wavelength of the light and $\Delta \lambda \in \mathbb{R}$ the spectral resolution at the wavelength $\lambda$.
<br>
> *Proposition*: the chromatic resolving power $\mathcal{R}$ of a Fabry Perot interferometer based on the $\text{FWHM}$ can be determined by
>
> $$
> \mathcal{R} = \mathcal{F} m,
> $$
>
> with $m \in \mathbb{Z}$ the order of the principal maxima and $\mathcal{F} \in \mathbb{R}$ the finesse.
??? note "*Proof*:"
Will be added later.
<br>
> *Definition*: the free spectral range $\text{FSR}$ is the largest wavelength range for a given order that does not overlap the same range in an adjacent order.
<br>
> *Proposition*: the free spectral range $\text{FSR}$ of a Fabry Perot interferometer can be determined by
>
> $$
> \text{FSR} = \frac{\lambda}{m},
> $$
>
> with $m \in \mathbb{Z}$ the order and $\lambda \in \mathbb{R}$ the wavelength.
??? note "*Proof*:"
Will be added later.

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# Polarisation
If we consider an electromagnetic wave $\mathbf{E}: \mathbb{R}^2 \to \mathbb{R}^3$ with wavenumber $k \in \mathbb{R}$ and angular frequency $\omega \in \mathbb{R}$ propagating in the positve $z$-direction given by
$$
\mathbf{E}(z,t) = \exp i(kz - \omega t + \varphi_1) E_0^{(x)} \mathbf{e}_{(x)} + \exp i(kz - \omega t + \varphi_2) E_0^{(y)}\mathbf{e}_{(y)},
$$
for all $(z,t) \in \mathbb{R}^2$ with $E_0^{(x)}, E_0^{(y)} \in \mathbb{R}$ the magnitude of the wave in the $x$ and $y$ direction. We define $\Delta \varphi = \varphi_2 - \varphi_1$.
> *Definition*: the electromagnetic wave $\mathbf{E}$ is linear polarised if and only if
>
> $$
> \Delta \varphi = \pi m,
> $$
>
> for all $m \in \mathbb{Z}$.
With polarisation angle $\theta \in [0, 2\pi)$ given by
$$
\theta = \arctan \Bigg( \frac{\max E_0^{(y)}}{\max E_0^{(x)}} \Bigg).
$$
??? note "*Proof*:"
Will be added later.
> *Definition*: the electromagnetic wave $\mathbf{E}$ is left circular polarised if and only if
>
> $$
> \Delta \varphi = \frac{\pi}{2} \;\land\; E_0^{(x)} = E_0^{(y)},
> $$
>
> and right circular polarised if and only if
>
> $$
> \Delta \varphi = -\frac{\pi}{2} \;\land\; E_0^{(x)} = E_0^{(y)}.
> $$
For every state in between we have elliptical polarisation with a polarisation angle $\theta \in [0, 2\pi)$ given by
$$
\theta = \frac{1}{2} \arctan \Bigg(\frac{2 E_0^{(x)} E_0^{(y)} \cos \Delta\varphi}{ \big(E_0^{(x)} \big)^2- \big( E_0^{(y)} \big)^2} \Bigg).
$$
??? note "*Proof*:"
Will be added later.
> *Definition*: natural light is defined as light constisting of all linear polarisation states.
## Linear polarisation
> *Definition*: a linear polariser selectively removes light that is linearly polarised along a direction perpendicular to its transmission axis.
We may concretisize this definition by the following statement, considered to be Malus law.
> *Law*: for a light beam with amplitude $E_0$ incident on a linear polariser the transmitted beam has amplitude $E_0 \cos \theta$ with $\theta \in [0, 2\pi)$ the polarisation angle of the light with respect to the transmission axis. The transmitted irradiance $I: [0, 2\pi) \to \mathbb{R}$ is then given by
>
> $$
> I(\theta) = I_0 \cos^2 \theta,
> $$
>
> for all $\theta \in [0, 2\pi)$ with $I_0 \in \mathbb{R}$ the irradiance of the incident light.
??? note "*Proof*:"
Will be added later.
For natural light the average of all angles must be taken, since $\lim_{\theta \to \infty} \cos^2 \theta = \frac{1}{2}$, we have the relation $I = \frac{1}{2} I_0$ for natural light.
## Birefringence
Natural light can be polarised in several ways, some are listed below.
1. Polarisation by absorption of the other component. This can be done with a wiregrid or dichroic materials for smaller wavelengths.
2. Polarisation by scattering. Dipole radiation has distinct polarisation depending on the position.
3. Polarisation by Brewster angle, which boils down to scattering.
4. Polarisation by birefringence, the double refraction of light obtaining two orthogonal components polarised.
??? note "*Proof*:"
Will be added later.
> *Definition*: birefringence is a double refraction in a material (often crystalline) and can be derived from the Fresnel equations without assuming isotropic dielectric properties.
If isotropic dielectric properties are not assumed it implies that the refractive index may also depend on the polarisation and propgation direction of light.
Using the properties of birefringence, wave plates (retarders) can be created. They may introduce a phase difference via a speed difference in the polarisation direction.
* A half-wave plate may introduce a $\Delta \varphi = \pi$ phase difference.
* A quarter-wave plate may introduce a $\Delta \varphi = \frac{\pi}{2}$ phase difference.
## Jones formalism of polarisation
Jones formalism of polarisation with vectors and matrices can make it easier to calculate the effects of optical elements such as linear polarizers and wave plates.
> *Definition*: for an electromagnetic wave $\mathbf{E}: \mathbb{R}^2 \to \mathbb{R}^3$ with wavenumber $k \in \mathbb{R}$ and angular frequency $\omega \in \mathbb{R}$ propagating in the positive $z$-direction given by
>
> $$
> \mathbf{E}(z,t) = \mathbf{E}_0 \exp i(kz - \omega t),
> $$
>
> for all $(z,t) \in \mathbb{R}^2$. The Jones vector $\mathbf{\tilde E}$ is defined as
>
> $$
> \mathbf{\tilde E} = \mathbf{E}_0,
> $$
>
> possibly normalized with $\|\mathbf{\tilde E}\| = 1$.
For linear polarised light under an angle $\theta \in [0, 2\pi)$ the Jones vector $\mathbf{\tilde E}$ is given by
$$
\mathbf{\tilde E} = \begin{pmatrix}\cos \theta \\ \sin \theta\end{pmatrix}.
$$
For left circular polarised light the Jones vector $\mathbf{\tilde E}$ is given by
$$
\mathbf{\tilde E} = \begin{pmatrix} 1 \\ i \end{pmatrix},
$$
and for right circular polarised light
$$
\mathbf{\tilde E} = \begin{pmatrix} 1 \\ -i \end{pmatrix}.
$$
> *Definition*: Jones matrices $M_i$ with $i \in \{1, \dots, n\}$ with $n \in \mathbb{N}$ may be used to model several optical elements on an optical axis, obtaining the transmitted Jones vector $\mathbf{\tilde E}_t$ from the incident Jones vector $\mathbf{\tilde E}_i$ given by
>
> $$
> \mathbf{\tilde E}_t = M_n \cdots M_1 \mathbf{\tilde E}_i.
> $$
The Jones matrices for several optical elements are now given.
> *Proposition*: the Jones matrix $M$ of a linear polariser is given by
>
> $$
> M = \begin{pmatrix} \cos^2 \theta & \frac{1}{2} \sin 2\theta \\ \frac{1}{2} \sin 2\theta & \sin^2 \theta \end{pmatrix},
> $$
>
> with $\theta \in [0, 2\pi)$ the transmission axis of the linear polariser.
??? note "*Proof*:"
Will be added later.
<br>
> *Proposition*: the Jones matrix $M$ of a half-wave plate is given by
>
> $$
> M = \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix},
> $$
>
> with $\theta \in [0, 2\pi)$ the fast axis of the half-wave plate.
??? note "*Proof*:"
Will be added later.
<br>
> *Proposition*: the Jones matrix $M$ of a quarter-wave plate is given by
>
> $$
> M = \begin{pmatrix} \cos^2 \theta + \sin^2 \theta & (1 - i) \sin \theta \cos \theta \\ (1 - i) \sin \theta \cos \theta & i(\cos^2 \theta + \sin^2 \theta) \end{pmatrix},
> $$
>
> with $\theta \in [0, 2\pi)$ the fast axis of the quarter-wave plate.
??? note "*Proof*:"
Will be added later.
<br>

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# Reflection and refraction
> *Definition*: light rays are perpendicular to electromagnetic wave fronts.
Reflection and refraction occur whenever light rays enter into a new medium with index of refraction $n \in \mathbb{R}$. Reflection may be informally defined as the change of direction of the rays that stay within the initial medium. Refraction may be informally defined as the change of direction of the rays that transport to the other medium.
> *Law*: the law of reflection states that the angle of reflection of a light ray equals the angle of incidence.
??? note "*Proof*:"
Will be added later.
<br>
> *Law*: the law of refraction states that the angle of refraction $\theta_t \in [0, 2\pi)$ is related to the angle of incidence $\theta_i \in [0, 2\pi)$ by
>
> $$
> n_i \sin \theta_i = n_t \sin \theta_t,
> $$
>
> with $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium.
??? note "*Proof*:"
Will be added later.
## Fresnel equations
In this section the fractions of reflected and transmitted power for specific electromagnetic waves will be derived.
> *Lemma*: for the electric field perpendicular to the plane of incidence (s-polarisation) the Fresnel amplitude ratios for reflection $r_s \in [0,1]$ and transmission $t_s \in [0,1]$ are given by
>
> $$
> \begin{align*}
> r_s &= \frac{n_i \cos \theta_i - n_t \cos \theta_t}{n_i \cos \theta_i + n_t \cos \theta_t}, \\
> \\
> t_s &= \frac{2 n_i \cos \theta_i}{n_i \cos \theta_i + n_t \cos \theta_t},
> \end{align*}
> $$
>
> with $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium and $\theta_{i,t} \in [0, 2\pi)$ the angle of incidence and refraction.
??? note "*Proof*:"
Will be added later.
<br>
> *Lemma*: for the electric field parallel to the plane of incidence (p-polarisation) the Fresnel amplitude ratios for reflection $r_p \in [0,1]$ and transmission $t_p \in [0,1]$ are given by
>
> $$
> \begin{align*}
> r_p &= \frac{n_i \cos \theta_t - n_t \cos \theta_i}{n_i \cos \theta_t + n_t \cos \theta_i}, \\
> \\
> t_p &= \frac{2 n_i \cos \theta_i}{n_i \cos \theta_t + n_t \cos \theta_i},
> \end{align*}
> $$
>
> with $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium and $\theta_{i,t} \in [0, 2\pi)$ the angle of incidence and refraction.
??? note "*Proof*:"
Will be added later.
<br>
> *Law*: the fraction of the incident power that is reflected is called the reflectivity $R \in [0,1]$ and is given by
>
> $$
> R = r^2,
> $$
>
> with $r \in [0, 1]$ the Fresnel amplitude ratio for reflection.
??? note "*Proof*:"
Will be added later.
<br>
> *Law*: the fraction of the incident power that is transmitted is called the transmissivity $T \in [0,1]$ and is given by
>
> $$
> T = \bigg(\frac{n_t \cos \theta_t}{n_i \cos \theta_i}\bigg) t^2
> $$
>
> with $t \in [0, 1]$ the Fresnel amplitude ratio for transmission, $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium and $\theta_{i,t} \in [0, 2\pi)$ the angle of incidence and refraction.
??? note "*Proof*:"
Will be added later.
<br>
## Limiting cases
> *Corollary*: we have $r_p = 0$ for an incident angle given by
>
> $$
> \tan \theta_b = \frac{n_t}{n_i},
> $$
>
> with $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium. The angle $\theta_b$ is called the Brewster angle.
??? note "*Proof*:"
Will be added later.
Therefore we have for the Brewster angle the reflectivity equal to zero for p-polarisation. Such relation does not exist for s-polarisation.
> *Corollary*: we have $r_s = 1$ or total reflection for $n_i > n_t$ and an incident angle given by
>
> $$
> \sin \theta_i > \frac{n_t}{n_i},
> $$
>
> with $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium. With
>
> $$
> \sin \theta_c = \frac{n_t}{n_i},
> $$
>
> the critical angle.
??? note "*Proof*:"
Will be added later.
## Phase changes on reflection
> *Proposition*: a reflected light ray may obtain a phase shift if
>
> 1. for all incident angles and $n_i < n_t$ the reflected light ray is phase shifted by $\pi$,
> 2. for incident angles $\theta_i > \theta_c$ and $n_i > n_t$ the reflected light ray is not phase shifted,
> 3. the transmitted light ray is not phase shifted.
??? note "*Proof*:"
Will be added later.
For incident angles $\theta_i < \theta_c$ and $n_i > n_t$ the phase shifts are complex.
## Dispersion
Will be added later.
## Scattering
Will be added later.

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# Waves
> *Definition*: a wave is a propagating disturbance transporting energy and momentum. A $1 + 1$ dimensional wave $\Psi: \mathbb{R}^2 \to \mathbb{R}$ travelling can be defined by a linear combination of a right and left travelling function $f,g: \mathbb{R} \to \mathbb{R}$ obtaining
>
> $$
> \Psi(x,t) = f(x - vt) + g(x + vt),
> $$
>
> for all $(x,t) \in \mathbb{R}^2$ and $v \in \mathbb{R}$ the speed of the wave. Satisfies the $1 + 1$ dimensional wave equation
>
> $$
> \partial_x^2 \Psi(x,t) = \frac{1}{v^2} \partial_t^2 \Psi(x,t).
> $$
The derivation of the wave equation can be obtained in section...
> *Theorem*: a right travelling harmonic wave $\Psi: \mathbb{R}^2 \to \mathbb{R}$ with $\lambda, T, A, \varphi \in \mathbb{R}$ the wavelength, period, amplitude and phase of the wave is given by
>
> $$
> \begin{align*}
> \Psi(x,t) &= A \sin \big(k(x-vt) + \varphi\big), \\
> &= A \sin(kx-\omega t + \varphi), \\
> &= A \sin \Big(2\pi \Big(\frac{x}{\lambda} - \frac{t}{T} \Big) + \varphi \Big),
> \end{align*}
> $$
>
> for all $(x,t) \in \mathbb{R}^2$. With $k = \frac{2\pi}{\lambda}$ the wavenumber, $\omega = \frac{2\pi}{T}$ the angular frequency and $v = \frac{\lambda}{T}$ the wave speed.
??? note "*Proof*:"
Will be added later.
A right travelling harmonic wave $\Psi: \mathbb{R}^2 \to \mathbb{R}$ can also be represented in the complex plane given by
$$
\Psi(x,t) = \text{Im} \big(A \exp i(kx - \omega t + \varphi )\big),
$$
for all $(x,t) \in \mathbb{R}^2$.
> *Theorem*: let $\Psi: \mathbb{R}^4 \to \mathbb{R}$ be a $3 + 1$ dimensional wave then it satisfies the $3 + 1$ dimensional wave equation given by
>
> $$
> \nabla^2 \Psi(\mathbf{x},t) = \frac{1}{v^2} \partial_t^2 \Psi(\mathbf{x},t),
> $$
>
> for all $(\mathbf{x},t) \in \mathbb{R}^4$.
??? note "*Proof*:"
Will be added later.
We may formulate various solutions $\Psi: \mathbb{R}^4 \to \mathbb{R}$ for this wave equation.
The first solution may be the plane wave that follows cartesian symmetry and can therefore best be described in a cartesian coordinate system $\mathbf{v}(x,y,z)$. The solution is given by
$$
\Psi(\mathbf{v}, t) = \text{Im}\big(A \exp i(\langle \mathbf{k}, \mathbf{v} \rangle - \omega t + \varphi) \big),
$$
for all $(\mathbf{v}, t) \in \mathbb{R}^4$ with $\mathbf{k} \in \mathbb{R}^3$ the wavevector.
The second solution may be the cylindrical wave that follows cylindrical symmetry and can therefore best be described in a cylindrical coordinate system $\mathbf{v}(r,\theta,z)$. The solution is given by
$$
\Psi(\mathbf{v}, t) = \text{Im}\Bigg(\frac{A}{\sqrt{\|\mathbf{v}\|}} \exp i(k \|\mathbf{v} \| - \omega t + \varphi) \Bigg),
$$
for all $(\mathbf{v}, t) \in \mathbb{R}^4$.
The third solution may be the spherical wave that follows spherical symmetry and can therefore best be described in a spherical coordinate system $\mathbf{v}(r, \theta, \varphi)$. The solution is given by
$$
\Psi(\mathbf{v}, t) = \text{Im}\Bigg(\frac{A}{\|\mathbf{v}\|} \exp i(k\|\mathbf{v}\| - \omega t + \varphi) \Bigg)
$$
for all $(\mathbf{v}, t) \in \mathbb{R}^4$.
> *Principle*: the principle of superposition is valid for waves, since the solution space of the wave equation is linear.
From this principle we obtain the property of constructive and destructive interference of waves.

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# Physics

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# Formalism of error analysis
When measuring a physical quantity with an experiment it is key to know how accurate the physical quantity has been determined, or equivalently, what the uncertainty is in the measured value.
## Measurement errors
Experimental uncertainties that cause a difference between the measured value and the real value of a physical quantity can be grouped into two categories; the **random error** and the **systematic error**.
Systematic errors always give an error in the same direction when the experiment is repeated. Whereas random errors have no preferential direction when the experiment is repeated.
## Confidence intervals
The uncertainty in the measured value may be expressed in a **confidence interval**. We will distinguish between two kinds of confidence intervals, the **maximum error** or 100% confidence interval and the **standard error** or 68% confidence interval. The percentage corresponding to this confidence interval is the probability that the real value lies within this interval.
### The maximum error
When a measurement is performed in which all systematic errors have been eliminated and no random errors are observed the maximum error should be used. Additionaly, the maximum error should be used for experiments where only a single measurement has been performed.
When the maximum error is used it is self-evident that multiple measurements of the same quantity are consistent if their confidence intervals overlap.
### The standard error
The standard error should be used whenever random errors in the measurements are present and when more than one measurement is performed. The standard error may then be determined from the spread in the results.
## Conventions
The following conventions are in use to denote uncertainties.
1. Uncertainties in the measurement results will be denoted with one significant figure, rounding is necessary. For intermediate results, two significant figures can be taken into account.
2. The least significant figure in a result has to have the same position as that of the uncertainty.
3. Units have to be mentioned and both the results and the uncertainty should obviously have the same unit.
4. Uncertainties are always positive.

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# The maximum error
## Determining the transformed maximum error
In this section a method will be postulated and derived under certain assumptions to determine the maximum error, after a transformation with a map $f$.
> *Definition 1*: let $f: \mathbb{R}^n \to \mathbb{R} :(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \mapsto f(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \overset{.} = y \pm \Delta_y$ be a function that maps independent measurements with a corresponding maximum error to a new quantity $y$ with maximum error $\Delta_y$ for $n \in \mathbb{N}$.
In assumption that the maximum errors of the independent measurements are small the following may be posed.
> *Postulate 1*: let $f:(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \mapsto f(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \overset{.} = y \pm \Delta_y$, the maximum error $\Delta_y$ may be given by
>
> $$
> \Delta_y = \sum_{i=1}^n | \partial_i f(x_1, \dots, x_n) | \Delta_{x_i},
> $$
>
> and $y = f(x_1, \dots, x_n)$ correspondingly for all $(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \in \mathbb{R}^n$.
??? note "*Derivation*:"
Will be added later.
With this general expression the following properties may be derived.
### Properties
The sum of the independently measured quantities is posed in the following corollary.
> *Corollary 1*: let $f:(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \mapsto f(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \overset{.} = y \pm \Delta_y$ with $y$ given by
>
> $$
> y = f(x_1, \dots, x_n) = x_1 + \dots x_n,
> $$
>
> then the maximum error $\Delta_y$ may be given by
>
> $$
> \Delta_y = \Delta_{x_1} + \dots + \Delta_{x_n},
> $$
>
> for all $(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \in \mathbb{R}^n$.
??? note "*Proof*:"
Will be added later.
The multiplication of a constant with the independently measured quantities is posed in the following corollary.
> *Corollary 2*: let $f:(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \mapsto f(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \overset{.} = y \pm \Delta_y$ with $y$ given by
>
> $$
> y = f(x_1, \dots, x_n) = \lambda(x_1 + \dots x_n),
> $$
>
> for $\lambda \in \mathbb{R}$ then the maximum error $\Delta_y$ may be given by
>
> $$
> \Delta_y = |\lambda| (\Delta_{x_1} + \dots + \Delta_{x_n}),
> $$
>
> for all $(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \in \mathbb{R}^n$.
??? note "*Proof*:"
Will be added later.
The product of two independently measured quantities is posed in the following corollary.
> *Corollary 3*: let $f: (x_1 \pm \Delta_{x_1}, x_2 \pm \Delta_{x_2}) \mapsto f(x_1 \pm \Delta_{x_1}, x_2 \pm \Delta_{x_2}) \overset{.} = y \pm \Delta_y$ with $y$ given by
>
> $$
> y = f(x_1, x_2) = x_1 x_2,
> $$
>
> then the maximum error $\Delta_y$ may be given by
>
> $$
> \Delta_y = \frac{\Delta_{x_1}}{|x_1|} + \frac{\Delta_{x_2}}{|x_2|},
> $$
>
> for all $(x_1 \pm \Delta_{x_1}, x_2 \pm \Delta_{x_2}) \in \mathbb{R}^2$.
??? note "*Proof*:"
Will be added later.
## Combining measurements
If by a measurement series $m \in \mathbb{N}$ values $\{y_1 \pm \Delta_{y_1}, \dots, y_m \pm \Delta_{y_m}\}$ have been found for the same quantity then
$$
[y \pm \Delta_y] = \bigcap_{i \in \mathbb{N}[i \leq m]} [y_i \pm \Delta_{y_i}],
$$
the overlap of all the intervals with $[y \pm \Delta_y]$ denoting the interval in which the real value exists.

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# Standard error
## The spread in the mean
> *Definition 1*: for a series of $N \in \mathbb{N}$ independent measurements $\{x_1, \dots, x_N\}$ of the same quantity, the mean $\bar x$ of the measurements is defined as
>
> $$
> \bar x = \frac{1}{N} \sum_{i=1}^N x_i,
> $$
>
> for all $x_i \in \mathbb{R}$.
??? note "*Derivation from the expectation value*:"
Will be added later.
Which is closely related to the expectation value defined in [probability theory](), the difference is the experimental notion of a finite amount of measurements. Similarly, the mean should be an approximation of the true value.
> *Definition 2*: for a series of $N \in \mathbb{N}$ independent measurements $\{x_1, \dots, x_N\}$ of the same quantity, the spread $S$ in the measurements is defined as
>
> $$
> S = \sqrt{\frac{1}{N - 1} \sum_{i=1}^N (\bar x - x_i)^2},
> $$
>
> for all $x_i \in \mathbb{R}$.
??? note "*Derivation from the variance*:"
Will be added later.
Which is closely related to the variance defined in [probability theory](), the difference is once again the experimental notion of a finite amount of measurements.
With the spread $S$ in the measurements the spread in the mean $S_{\bar x}$ may be determined.
> *Theorem 1*: for a series of $N \in \mathbb{N}$ independent measurements $\{x_1, \dots, x_N\}$ of the same quantity, the spread in the mean $S_{\bar x}$ is given by
>
> $$
> S_{\bar x} = \sqrt{\frac{1}{N(N-1)} \sum_{i=1}^N (\bar x - x_i)^2},
> $$
>
> for all $x_i \in \mathbb{R}$ with $\bar x$ the mean.
??? note "*Proof*:"
Will be added later.
## Determining the transformed spread
In this section a method will be postulated and derived under certain assumptions to determine the spread in the transformed means with a map $f$.
> *Definition 3*: let $f: \mathbb{R}^n \to \mathbb{R} :(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \mapsto f(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \overset{.} = \bar y \pm S_{\bar y}$ be a function that maps the mean for each independent measurement series with a corresponding spread to a new mean quantity $\bar y$ with a spread $S_{\bar y}$ for $n \in \mathbb{N}$.
In assumption that the spread in the mean for each independent measurement series is small, the following may be posed.
> *Postulate 1*: let $f: (\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \mapsto f(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \overset{.} = \bar y \pm S_{\bar y}$, the spread $S_{\bar y}$ may be given by
>
> $$
> S_{\bar y} = \sqrt{\sum_{i=1}^n \Big(\partial_i f(\bar x_1, \dots, \bar x_n) S_{\bar x_i} \Big)^2},
> $$
>
> and $\bar y = f(\bar x_1, \dots, \bar x_n)$ correspondingly for all $(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \in \mathbb{R}^n$.
??? note "*Derivation*:"
Will be added later.
With this general expression the following properties may be derived.
### Properties
The sum of the independently measured quantities is posed in the following corollary.
> *Corollary 1*: let $f: (\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \mapsto f(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \overset{.} = \bar y \pm S_{\bar y}$ with $\bar y$ given by
>
> $$
> \bar y = f(\bar x_1, \dots, \bar x_n) = \bar x_1 + \dots \bar x_n,
> $$
>
> then the spread $S_{\bar y}$ may be given by
>
> $$
> S_{\bar y} = \sqrt{S_{\bar x_1}^2 + \dots + S_{\bar x_n}^2},
> $$
>
> for all $(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \in \mathbb{R}^n$.
??? note "*Proof*:"
Will be added later.
The multiplication of a constant with the independently measured quantities is posed in the following corollary.
> *Corollary 2*: let $f: (\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \mapsto f(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \overset{.} = \bar y \pm S_{\bar y}$ with $\bar y$ given by
>
> $$
> \bar y = f(\bar x_1, \dots, \bar x_n) = \lambda(\bar x_1 + \dots \bar x_n),
> $$
>
> for $\lambda \in \mathbb{R}$ then the spread $S_{\bar y}$ may be given by
>
> $$
> S_{\bar y} = |\lambda| (S_{\bar x_1} + \dots + S_{\bar x_n}),
> $$
>
> for all $(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \in \mathbb{R}^n$.
??? note "*Proof*:"
Will be added later.
The product of two independently measured quantities is posed in the following corollary.
> *Corollary 3*: let $f: (\bar x_1 \pm S_{\bar x_1}, \bar x_2 \pm S_{\bar x_2}) \mapsto f(\bar x_1 \pm S_{\bar x_1}, \bar x_2 \pm S_{\bar x_2}) \overset{.} = \bar y \pm S_{\bar y}$ with $\bar y$ given by
>
> $$
> \bar y = f(\bar x_1, \bar x_2) = \bar x_1 \bar x_2,
> $$
>
> then the spread $S_{\bar y}$ may be given by
>
> $$
> S_{\bar y} = \sqrt{\bigg(\frac{S_{\bar x_1}}{\bar x_1}\bigg)^2 + \bigg(\frac{S_{\bar x_2}}{\bar x_2} \bigg)^2},
> $$
>
> for all $(\bar x_1 \pm S_{\bar x_1}, x_2 \pm S_{\bar x_2}) \in \mathbb{R}^2$.
??? note "*Proof*:"
Will be added later.
## Combining measurements
If by a measurement series $m \in \mathbb{N}$ values $\{\bar y_1 \pm S_{\bar y_1}, \dots, \bar y_m \pm S_{\bar y_m}\}$ have been found for the same quantity then $\bar y$ is given by
$$
\bar y = \frac{\sum_{i=1}^m (1 / S_{\bar y_i})^2 \bar y_i}{\sum_{i=1}^m (1 / S_{\bar y_i})^2},
$$
with its corresponding spread $S_{\bar y}$ given by
$$
S_{\bar y} = \frac{1}{\sqrt{\sum_{i=1}^m (1 / S_{\bar y_i})^2}},
$$
for all $\{\bar y_1 \pm S_{\bar y_1}, \dots, \bar y_m \pm S_{\bar y_m}\} \in \mathbb{R}^m$.
??? note "*Proof*:"
Will be added later.

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# Amplitude modulation
> *Theorem*: a multiplication of two harmonic functions results in a sum of harmonics withh the sum and difference of the original frequencies. This is called *heterodyne*.
??? note "*Proof*:"
Will be added later.
For example if we have a harmonic signal $m: \mathbb{R} \to \mathbb{R}$ with $\omega, A \in \mathbb{R}$ given by
$$
m(t) = A \cos \omega t,
$$
for all $t \in \mathbb{R}$ and a harmonic carrier signal $c: \mathbb{R} \to \mathbb{R}$ with $\omega_c \in \mathbb{R}$ given by
$$
c(t) = \cos \omega_c t.
$$
for all $t \in \mathbb{R}$. Then the multiplication of both is given by
$$
m(t)c(t) = A \cos (\omega t) \cos (\omega_c t) = \frac{A}{2} \bigg(\cos t(\omega + \omega)c + \cos t(\omega - \omega_c) \bigg),
$$
obtaining heterodyne.
> *Definition*: amplitude modulation makes use of a harmonic carrier signal $c: \mathbb{R} \to \mathbb{R}$ with a reasonable angular frequency $\omega_c \in \mathbb{R}$ given by
>
> $$
> c(t) = \cos \omega_c t
> $$
>
> for all $t \in \mathbb{R}$ to modulate a signal $m: \mathbb{R} \to \mathbb{R}$.
<br>
> *Theorem*: For the case that the carrier signal is not additionaly transmitted we obtain
>
> $$
> m(t) c(t) \overset{\mathcal{F}}\longleftrightarrow \frac{1}{2} \big(M(\omega + \omega_c) + M(\omega - \omega_c) \big),
> $$
>
> for all $t, \omega \in \mathbb{R}$.
>
> For the case that the carrier signal is additionaly transmitted we obtain
>
> $$
> m(t) (1 + c(t)) \overset{\mathcal{F}}\longleftrightarrow \frac{1}{2} \Big(M(\omega + \omega_c) + M(\omega - \omega_c) + \pi \big(\delta(\omega + \omega_c) + \delta(\omega - \omega_c) \big) \Big)
> $$
>
> for all $t, \omega \in \mathbb{R}$.
>
> Therefore multiple bandlimited signals can be transmitted simultaneously in frequency bands.
??? note "*Proof*:"
Will be added later.

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# The discrete Fourier transform
> *Theorem*: sampling a signal with the impulse train makes the spectrum of the signal periodic.
??? note "*Proof*:"
Will be added later.
A bandlimited signal implies that its frequency components are zero outside the bandwidth frequency interval.
> *Theorem*: if a signal has a bandwidth $\omega_b \in \mathbb{R}$ then it can be completely determined from its samples at a sampling frequency $\omega_s \in \mathbb{R}$ given by
>
> $$
> \omega_s > 2 \omega_b.
> $$
??? note "*Proof*:"
Will be added later.
When the sampling frequency does not comply to this statement, the reconstruction of the spectrum will exhibit imperfections known as aliasing. The critical value of the sampling frequency is known as the *Nyquist* frequency.
## The discrete time Fourier transform
> *Theorem*: let $f: \mathbb{R} \to \mathbb{C}$ be a signal with its sampled signal $f_s(t) = f(t) \delta_{T_s}(t)$ for all $t \in \mathbb{R}$ with sampling period $T_s \in \mathbb{R}$. Then the discrete time Fourier transform $F: \mathbb{R} \to \mathbb{C}$ of $f_s$ is given by
>
> $$
> F(\Omega) = \sum_{m = -\infty}^\infty f[m] e^{-im\Omega},
> $$
>
> for all $\Omega \in \mathbb{R}$. With $\Omega = \omega T_s$ the dimensionless frequency and $F_s(\omega) := F(\Omega)$.
??? note "*Proof*:"
Will be added later.
## The discrete Fourier transform
> *Theorem*: let $f: \mathbb{R} \to \mathbb{C}$ be a signal and $f_N: \mathbb{R} \to \mathbb{C}$ the truncated signal of $f$ by $N \in \mathbb{N}$ given by
>
> $$
> f_N[m] = \begin{cases} f[m] &\text{ if } m \in \{0, \dots, N - 1\}, \\ 0 &\text{ if } m \notin \{0, \dots, N - 1\}, \end{cases}
> $$
>
> sampled by $T_s \in \mathbb{R}$. Its discrete Fourier transform $F_N: \mathbb{R} \to \mathbb{C}$ is given by
>
> $$
> F_N[k] = \sum_{m=0}^{N-1} f[m] \exp \bigg(-2\pi i \frac{km}{N} \bigg)
> $$
>
> for all $k \in \{0, \dots, N-1\}$.
??? note "*Proof*:"
Will be added later.
We have that $F_N[k] = F_N(k\Delta \omega)$ with $\Delta \omega = \frac{2\pi}{N T_s}$ the angular frequency resolution.
> *Theorem*: let $F_N: \mathbb{R} \to \mathbb{C}$ be a spectrum of a signal truncated by $N \in \mathbb{N}$ then its inverse discrete Fourier transform $f_N: \mathbb{R} \to \mathbb{C}$ is given by
>
> $$
> f[m] = \frac{1}{N} \sum_{k=0}^{N-1} F_N[k] \exp \bigg(2\pi i \frac{km}{N} \bigg)
> $$
>
> for all $m \in \{0, \dots, N - 1\}$.
??? note "*Proof*:"
Will be added later.
> *Definition*: therefore $f_N$ and $F_N$ with $N \in \mathbb{N}$ form a discrete Fourier transform pair denoted by
>
> $$
> f_N \overset{\mathcal{DF}}\longleftrightarrow F_N,
> $$
>
> therefore we have
>
> $$
> \begin{align*}
> &f_N[m] = \mathcal{DF}^{-1}[F_N[k]], \quad &\forall m \in \{0, \dots, N - 1\}, \\
> &F_N[k] = \mathcal{DF}[f[m]], \quad &\forall k \in \{0, \dots, N - 1\}.
> \end{align*}
> $$

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# Fourier series
> *Theorem*: the "Fourier" inner product of two functions $g, f: \mathbb{C} \to \mathbb{C}$ is defined as
>
> $$
> \langle f, g \rangle = \int_a^b f(t) \overline g(t) dt
> $$
>
> with $f, g$ members of the square integrable functions $L^2[a,b]$ with $a,b \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
The overline generally implies the complex conjugate.
> *Corollary*: the "Fourier" norm of a square integrable function $f: \mathbb{C} \to \mathbb{C}$ is defined as
>
> $$
> \|f\| = \sqrt{\langle f, f \rangle}.
> $$
??? note "*Proof*:"
Will be added later.
<br>
> *Proposition*: let $f: \mathbb{R} \to \mathbb{R}$ be a periodic function with period $T_0 \in \mathbb{R}$ then the autocorrelation of $f$ will create peaks for $t = zT_0$ for all $t \in \mathbb{R}$ and $z \in \mathbb{Z}$.
??? note "*Proof*:"
Will be added later.
<br>
> *Definition*: two functions $f, g: \mathbb{C} \to \mathbb{C}$ are orthogonal if and only if
>
> $$
> \langle f, g \rangle = 0
> $$
## Approximating functions
> *Lemma*: a function $f: \mathbb{R} \to \mathbb{C}$ can be approximated with a linear combination of orthogonal functions $b_k: \mathbb{R} \to \mathbb{C}$ given by
>
> $$
> \phi_n(t) = \sum_{k=0}^n c_k b_k(t),
> $$
>
> for all $t \in \mathbb{R}$ with $n \in \mathbb{N}$ the order. The coefficients $c_k \in \mathbb{C}$ that minimise $\|f - \phi_n\|$ may be determined by
>
> $$
> c_k = \frac{\langle f, b_k \rangle}{\langle b_k, b_k \rangle}.
> $$
??? note "*Proof*:"
Will be added later.
The orthogonal functions $b_k: \mathbb{R} \to \mathbb{C}$ have not yet been specified. There are many possible choices (Legendre polynomials, Bessel functions, spherical harmonics etc.) for these functions, for the Fourier series specifically we make use trigonometric or more generally imaginary exponential functions.
> *Lemma*: in the special case that $b_k: \mathbb{R} \to \mathbb{C}$ is given by
>
> $$
> b_k(t) = \exp(i k \omega_0 t),
> $$
>
> for all $t \in \mathbb{R}$ with $k \in \mathbb{Z}$ and $\omega_0 \in \mathbb{R}$ the angular frequency. A periodic function $f: \mathbb{R} \to \mathbb{C}$ with period $T_0 = \frac{2\pi}{\omega_0}$ may be approximated by
>
> $$
> \phi_n(t) = \sum_{k = 0}^n c_k e^{i k \omega_0 t},
> $$
>
> for all $t \in \mathbb{R}$. With the coefficients $c_k \in \mathbb{C}$ given by
>
> $$
> c_k = \frac{1}{T_0} \int_0^{T_0} f(t) e^{-i k \omega_0 t}dt.
> $$
??? note "*Proof*:"
Will be added later.
<br>
> *Lemma*: For a periodic function $f: \mathbb{R} \to \mathbb{C}$ and its approximation $\phi_n$ given in the above lemma we have
>
> $$
> \lim_{n \to \infty} \|f - \phi_n \| = 0,
> $$
>
> implies that the resulting series approximation converges to $f$. Similarly the series approximation converges also pointwise
>
> $$
> \lim_{n \to \infty} |f(t) - \phi_n(t)| = 0,
> $$
>
> for all $t \in D$ with $D \subseteq \mathbb{R}$ the interval where $f$ is continuous.
??? note "*Proof*:"
Will be added later.
## The Fourier series
With the above lemmas we may state the following theorems.
> *Theorem*: the classical Fourier series of a periodic function $f: \mathbb{R} \to \mathbb{C}$ with period $T_0 = \frac{2\pi}{\omega_0}$ may be given by
>
> $$
> f(t) = \sum_{k = -\infty}^\infty c_k e^{i k \omega_0 t},
> $$
>
> for all $t \in \mathbb{R}$. With the coefficients $c_k \in \mathbb{C}$ given by
>
> $$
> c_k = \frac{1}{T_0} \int_0^{T_0} f(t) e^{-i k \omega_0 t}dt.
> $$
??? note "*Proof*:"
Will be added later.
Expanding the Fourier series such that it can also approximate aperiodic functions obtains.
> *Theorem*: the Fourier series of an aperiodic function $f: \mathbb{R} \to \mathbb{C}$ may be given
>
> $$
> f(t) = \frac{1}{2\pi} \int_{-\infty}^\infty F(\omega) e^{i \omega t} d\omega,
> $$
>
> for all $t \in \mathbb{R}$. The expansion coefficient $F: \mathbb{R} \to \mathbb{C}$ is given by
>
> $$
> F(\omega) = \int_{-\infty}^\infty f(t) e^{-i \omega t}dt
> $$
>
> for all $\omega \in \mathbb{R}$. Is called the Fourier transform of $f$ and represents the continuous frequency spectrum of $f$.
??? note "*Proof*:"
Will be added later.

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# Fourier transformations
## Definition of the Fourier transform
> *Definition*: let $f, F: \mathbb{R} \to \mathbb{C}$, the Fourier transform of $f$ is given by
>
> $$
> F(\omega) = \int_{-\infty}^\infty f(t) e^{-i \omega t}dt,
> $$
>
> for all $\omega \in \mathbb{R}$. The inverse Fourier transform of $F$ is given by
>
> $$
> f(t) = \frac{1}{2\pi} \int_{-\infty}^\infty F(\omega) e^{i \omega t} d\omega,
> $$
>
> for all $t \in \mathbb{R}$. Therefore $f$ and $F$ form a Fourier transform pair denoted by
>
> $$
> f \overset{\mathcal{F}}\longleftrightarrow F,
> $$
>
> therefore we have
>
> $$
> \begin{align*}
> &f(t) = \mathcal{F}^{-1}[F(\omega)], \quad &\forall t \in \mathbb{R}, \\
> &F(\omega) = \mathcal{F}[f(t)], \quad &\forall \omega \in \mathbb{R}.
> \end{align*}
> $$
## Properties of the Fourier transform
> *Proposition*: let $f, g, F, G: \mathbb{R} \to \mathbb{C}$, we have linearity given by
>
> $$
> af(t) + bg(t) \overset{\mathcal{F}}\longleftrightarrow aF(\omega) + bG(\omega),
> $$
>
> with $a,b \in \mathbb{C}$.
??? note "*Proof*:"
Will be added later.
<br>
> *Proposition*: let $f,F: \mathbb{R} \to \mathbb{C}$, we have time shifting given by
>
> $$
> f(t - t_0) \overset{\mathcal{F}}\longleftrightarrow F(\omega) e^{-i\omega t_0},
> $$
>
> with $t_0 \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
<br>
> *Proposition*: let $f,F: \mathbb{R} \to \mathbb{C}$, we have frequency shifting given by
>
> $$
> e^{i \omega_0 t} f(t) \overset{\mathcal{F}}\longleftrightarrow F(\omega - \omega_0)
> $$
>
> with $\omega_0 \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
<br>
> *Proposition*: let $f,F: \mathbb{R} \to \mathbb{C}$, we have time or frequency scaling given by
>
> $$
> f(t/a) \overset{\mathcal{F}}\longleftrightarrow |a| F(a\omega)
> $$
>
> with $a \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
<br>
> *Proposition*: let $f, g, F, G: \mathbb{R} \to \mathbb{C}$, we have time convolution given by
>
> $$
> f(t) * g(t) \overset{\mathcal{F}}\longleftrightarrow F(\omega) G(\omega).
> $$
??? note "*Proof*:"
Will be added later.
<br>
> *Proposition*: let $f, g, F, G: \mathbb{R} \to \mathbb{C}$, we have frequency convolution given by
>
> $$
> f(t) g(t) \overset{\mathcal{F}}\longleftrightarrow \frac{1}{2\pi} F(\omega) * G(\omega).
> $$
??? note "*Proof*:"
Will be added later.
<br>
> *Proposition*: let $f,F: \mathbb{R} \to \mathbb{C}$ be differentiable, we have time differentation given by
>
> $$
> f'(t) \overset{\mathcal{F}}\longleftrightarrow i \omega F(\omega).
> $$
??? note "*Proof*:"
Will be added later.
<br>
> *Proposition*: let $f,F: \mathbb{R} \to \mathbb{C}$ be differentiable, we have time integration given by
>
> $$
> \int_{-\infty}^t f(u)du \overset{\mathcal{F}}\longleftrightarrow \frac{1}{i\omega} F(\omega) + \pi F(0)\delta(\omega).
> $$
??? note "*Proof*:"
Will be added later.
<br>

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# Signal filters
## The impedance
> *Proposition*: considering an ideal resistor $R \in \mathbb{R}$. When a voltage difference $v_i - v_o$ is applied to both ends a current $I: \mathbb{R} \to \mathbb{R}$ will be induced given by
>
> $$
> v_i(t) - v_o(t) = R I(t),
> $$
>
> for all $t \in \mathbb{R}$. By taking the Fourier transform of both sides of the expression we find
>
> $$
> V_i(\omega) - V_o(\omega) = R I(\omega),
> $$
>
> for all $\omega \in \mathbb{R}$ with $V_{i,o}: \mathbb{R} \to \mathbb{C}$ the fourier transforms of the voltage difference.
??? note "*Proof*:"
Will be added later.
<br>
> *Proposition*: considering a load coil with inductance $L \in \mathbb{R}$. When a voltage difference $v_i - v_o$ is applied to both ends a current $I: \mathbb{R} \to \mathbb{R}$ will be induced given by
>
> $$
> v_i(t) - v_o(t) = L I'(t),
> $$
>
> for all $t \in \mathbb{R}$. By taking the Fourier transform of both sides of the expression we find
>
> $$
> V_i(\omega) - V_o(\omega) = i \omega L I(\omega),
> $$
>
> for all $\omega \in \mathbb{R}$ with $V_{i,o}: \mathbb{R} \to \mathbb{C}$ the fourier transforms of the voltage difference.
??? note "*Proof*:"
Will be added later.
<br>
> *Proposition*: considering a capacitor with capacity $C \in \mathbb{R}$. When a voltage difference $v_i - v_o$ is applied to both ends a current $I: \mathbb{R} \to \mathbb{R}$ will be induced given by
>
> $$
> v_i(t) - v_o(t) = \frac{1}{C} \int_{-\infty}^t I(t)dt,
> $$
>
> for all $t \in \mathbb{R}$. By taking the Fourier transform of both sides of the expression we find
>
> $$
> V_i(\omega) - V_o(\omega) = \bigg(\frac{1}{i \omega C} + \frac{\pi \delta(\omega)}{C} \bigg) I(\omega),
> $$
>
> for all $\omega \in \mathbb{R}$ with $V_{i,o}: \mathbb{R} \to \mathbb{C}$ the fourier transforms of the voltage difference.
??? note "*Proof*:"
Will be added later.
<br>
> *Definition*: the complex impedance $Z: \mathbb{R} \to \mathbb{C}$ is defined as
>
> $$
> V_i(\omega) - V_o(\omega) = Z(\omega) I(\omega)
> $$
>
> for all $\omega \in \mathbb{R}$.
Therefore the complex impedance for the ideal resistor is given by $Z(\omega) = R$ and for the load coil $Z(\omega) = i \omega L$ for all $\omega \in \mathbb{R}$.
> *Proposition*: the impedance elements $Z_i$ for $i \in \{1, \dots, n\}$ with $n \in \mathbb{N}$ in series can be summed to obtain $Z$
>
> $$
> Z = Z_1 + \dots + Z_n.
> $$
??? note "*Proof*:"
Will be added later.
<br>
> *Proposition*: the impedance elements $Z_i$ for $i \in \{1, \dots, n\}$ with $n \in \mathbb{N}$ in parallel can be inversely summed to obtain $Z$
>
> $$
> \frac{1}{Z} = \frac{1}{Z_1} + \dots + \frac{1}{Z_n}.
> $$
??? note "*Proof*:"
Will be added later.
<br>
## The transfer function
> *Definition*: the relation between the input and output voltage in the frequency domain $V_{i,o}: \mathbb{R} \to \mathbb{C}$ can be written as
>
> $$
> V_o(\omega) = H(\omega) V_i(\omega),
> $$
>
> for all $\omega \in \mathbb{R}$ with $H: \mathbb{R} \to \mathbb{C}$ the transfer function.
The transfer function may be interpreted as a frequency filter of the signal.
Some ideal filters are given in the list below
* a *low-pass* filter removes all frequency components $\omega > \omega_c$ with $\omega_c \in \mathbb{R}$ the cut-off frequency,
* a *high-pass* filter removes all frequency components $\omega < \omega_c$,
* a *band-pass* filter removes all frequency componets outside a particular frequency range,
* a *band-stop* filter removes all frequency compnents inside a particular frequency range.

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# Signals
## Definitions
> *Definition*: a signal is a function of space and time.
>
> * Output can be analog or quantised.
> * Input can be continuous or discrete.
<br>
> *Definition*: a signal can be sampled at particular moments $k T_s$ in time, with $k \in \mathbb{Z}$ and $T_s \in \mathbb{R}$ the sampling period. For a signal $f: \mathbb{R} \to \mathbb{R}$ sampled with a sampling period $T_s$ may be denoted by
>
> $$
> f[k] = f(kT_s), \qquad \forall k \in \mathbb{Z}.
> $$
<br>
> *Definition*: signal transformations on a function $x: \mathbb{R} \to \mathbb{R}$ obtaining the function $y: \mathbb{R} \to \mathbb{R}$ are given by
>
> | Signal transformation | Time | Amplitude |
> | :-: | :-: | :-: |
> | Reversal | $y(t) = x(-t)$ | $y(t) = -x(t)$ |
> | Scaling | $y(t) = x(at)$ | $y(t) = ax(t)$ |
> | Shifting | $y(t) = x(t - b)$ | $y(t) = x(t) + b$ |
>
> for all $t \in \mathbb{R}$.
For sampled signals similar definitions hold.
### Symmetry
> *Definition*: consider a signal $f: \mathbb{R} \to \mathbb{R}$ which is defined in an interval which is symmetric around $t = 0$, we define.
>
> * $f$ is *even* if $f(t) = f(-t)$, $\forall t \in \mathbb{R}$.
> * $f$ is *odd* if $f(t) = -f(-t)$, $\forall t \in \mathbb{R}$.
For sampled signals similar definitions hold.
> *Theorem*: every signal can be decomposed into symmetric parts.
??? note "*Proof*:"
Will be added later.
### Periodicity
> *Definition*: a signal $f: \mathbb{R} \to \mathbb{R}$ is defined to be periodic in $T$ if and only if
>
> $$
> f(t + T) = f(t), \qquad \forall t \in \mathbb{R}.
> $$
For sampled signals similar definitions hold.
> *Theorem*: a summation of two periodic signals with periods $T_1, T_2 \in \mathbb{R}$ respectively is periodic if and only if
>
> $$
> \frac{T_1}{T_2} \in \mathbb{Q}.
> $$
??? note "*Proof*:"
Will be added later.
### Signals
> *Definition*: the Heaviside step signal $u: \mathbb{R} \to \mathbb{R}$ is defined by
>
> $$
> u(t) = \begin{cases} 1 &\text{ if } t > 0,\\ 0 &\text{ if } t < 0,\end{cases}
> $$
>
> for all $t \in \mathbb{R}$.
For a sampled function the Heaviside step signal is given by
$$
u[k] = \begin{cases} 1 \text{ if } k \geq 0, \\ 0 \text{ if } k < 0, \end{cases}
$$
for all $k \in \mathbb{Z}$.
> *Definition*: the rectangular signal $\text{rect}: \mathbb{R} \to \mathbb{R}$ is defined by
>
> $$
> \text{rect} (t) = \begin{cases} 1 &\text{ if } |t| < \frac{1}{2}, \\ 0 &\text{ if } |t| > \frac{1}{2},\end{cases}
> $$
>
> for all $t \in \mathbb{R}$.
The rect signal can be normalised obtaining the scaled rectangular signal $D: \mathbb{R} \to \mathbb{R}$ defined by
$$
D(t, \varepsilon) = \begin{cases} \frac{1}{\varepsilon} &\text{ if } |t| < \frac{\varepsilon}{2},\\ 0 &\text{ if } |t| > \frac{\varepsilon}{2},\end{cases}
$$
for all $t \in \mathbb{R}$.
The following signal has been derived from the scaled rectangular signal $D: \mathbb{R} \to \mathbb{R}$ used on a signal $f: \mathbb{R} \to \mathbb{R}$ for
$$
\lim_{\varepsilon \;\downarrow\; 0} \int_{-\infty}^{\infty} f(t) D(t, \varepsilon)dt = \lim_{\varepsilon \;\downarrow\; 0} \frac{1}{\varepsilon} \int_{-\frac{\varepsilon}{2}}^{\frac{\varepsilon}{2}} f(t) dt = f(0),
$$
using the mean [value theorem for integrals](../../../mathematics/calculus/integration.md#the-mean-value-theorem-for-integrals).
> *Definition*: the Dirac signal $\delta$ is a generalized signal defined by the properties
>
> $$
> \begin{align*}
> \delta(t - t_0) = 0 \quad \text{ for } t \neq t_0,& \\
> \int_{-\infty}^\infty f(t) \delta(t - t_0) dt = f(t_0),&
> \end{align*}
> $$
>
> for a signal $f: \mathbb{R} \to \mathbb{R}$ continuous in $t_0$.
For sampled signals the $\delta$ signal is given by
$$
\delta[k] = \begin{cases} 1 &\text{ if } k = 0, \\ 0 &\text{ if } k \neq 0.\end{cases}
$$
## Signal sampling
We already established that a signal $f: \mathbb{R} \to \mathbb{R}$ can be sampled with a sampling period $T_s \in \mathbb{R}$ obtaining $f[k] = f(kT_s)$ for all $k \in \mathbb{Z}$. We can also define a *time-continuous* signal $f_s: \mathbb{R} \to \mathbb{R}$ that represents the sampled signal using the Dirac signal, obtaining
$$
f_s(t) = f(t) \sum_{k = - \infty}^\infty \delta(t - k T_s), \qquad \forall t \in \mathbb{R}.
$$
> *Definition*: the sampling signal or impulse train $\delta_{T_s}: \mathbb{R} \to \mathbb{R}$ is defined as
>
> $$
> \delta_{T_s}(t) = \sum_{k = - \infty}^\infty \delta(t - k T_s)
> $$
>
> for all $t \in \mathbb{R}$ with a sampling period $T_s \in \mathbb{R}$.
Then integration works out since we have
$$
\int_{-\infty}^\infty f(t) \delta_{T_s}(t) dt = \sum_{k = -\infty}^\infty \int_{-\infty}^\infty f(t) \delta(t - k T_s) dt = \sum_{k = -\infty}^\infty f [k],
$$
by definition.
## Convolutions
> *Definition*: let $f,g: \mathbb{R} \to \mathbb{R}$ be two continuous signals, the convolution product is defined as
>
> $$
> f(t) * g(t) = \int_{-\infty}^\infty f(u)g(t-u)du
> $$
>
> for all $t \in \mathbb{R}$.
<br>
> *Proposition*: the convolution product is commutative, distributive and associative.
??? note "*Proof*:"
Will be added later.
> *Theorem*: let $f: \mathbb{R} \to \mathbb{R}$ be a signal then we have for the convolution product between $f$ and the Dirac signal $\delta$ and some $t_0 \in \mathbb{R}$
>
> $$
> f(t) * \delta(t - t_0) = f(t - t_0)
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
let $f: \mathbb{R} \to \mathbb{R}$ be a signal and $t_0 \in \mathbb{R}$, using the definition of the Dirac signal
$$
f(t) * \delta(t - t_0) = \int_{-\infty}^\infty f(u) \delta(t - t_0 - u)du = f(t - t_0),
$$
for all $t \in \mathbb{R}$.
In particular $f(t) * \delta(t) = f(t)$ for all $t \in \mathbb{R}$; $\delta$ is the unity of the convolution.
The average value of a signal $f: \mathbb{R} \to \mathbb{R}$ for an interval $\varepsilon \in \mathbb{R}$ may be given by
$$
f(t) * D(t, \varepsilon) = \frac{1}{\varepsilon} \int_{t - \frac{\varepsilon}{2}}^{t + \frac{\varepsilon}{2}} f(u)du.
$$
For sampled/discrete signals we have a similar definition for the convolution product, given by
$$
f[k] * g[k] = \sum_{m = -\infty}^{\infty} f[m]g[k - m],
$$
for all $k \in \mathbb{Z}$.
## Correlations
> *Definition*: let $f,g: \mathbb{R} \to \mathbb{R}$ be two continuous signals, the cross-correlation is defined as
>
> $$
> f(t) \star g(t) = \int_{-\infty}^\infty f(u) g(t + u)du
> $$
>
> for all $t \in \mathbb{R}$.
Especially the auto-correlation of a continuous signal $f: \mathbb{R} \to \mathbb{R}$ given by $f(t) \star f(t)$ for all $t \in \mathbb{R}$ is useful, as it can detect periodicity. This is proved in the section [Fourier series](fourier-series.md).
For sampled/discrete signals a similar definition exists given by
$$
f[k] \star g[k] = \sum_{m = -\infty}^{\infty} f[m]g[k + m],
$$
for all $k \in \mathbb{Z}$.

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# Systems
> *Definition*: a system transforms signals.
## Operators
> *Definition*: let $x,y: \mathbb{R} \to \mathbb{R}$ be the input and output signal related to an operator $T$ by
>
> $$
> y(t) = T[x(t)]
> $$
>
> for all $t \in \mathbb{R}$.
For example for a time shift of the signal $S_{t_0}: y(t) = x(t - t_0)$ we have $y(t) = S_{t_0}[x(t)]$ for all $t \in \mathbb{R}$. For an amplifier of the signal $P: y(t) = k(t) x(t)$ we have $y(t) = P[x(t)]$ for all $t \in \mathbb{R}$.
> *Definition*: for systems $T_i$ for $i \in \{1, \dots, n\}$ with $n \in \mathbb{N}$ in parallel we define operator addition by
>
> $$
> T = T_1 + \dots + T_n,
> $$
>
> such that for $x,y: \mathbb{R} \to \mathbb{R}$ the input and output signal obtains
>
> $$
> y(t) = T[x(t)] = (T_1 + \dots + T_n)[x(t)] = T_1[x(t)] + \dots + T_n[x(t)],
> $$
>
> for all $t \in \mathbb{R}$.
<br>
> *Definition*: for systems $T_i$ for $i \in \{1, \dots, n\}$ with $n \in \mathbb{N}$ in series we define operator multiplication by
>
> $$
> T = T_n \cdots T_1,
> $$
>
> such that for $x,y: \mathbb{R} \to \mathbb{R}$ the input and output signal obtains
>
> $$
> y(t) = T[x(t)] =T_n \cdots T_1 [x(t)] = T_n[T_{n-1}\cdots T_1[x(t)]],
> $$
>
> for all $t \in \mathbb{R}$.
It may be observed that the operator product is not commutative.
## Properties of systems.
> *Definition*: a system $T$ with inputs $x_{1,2}: \mathbb{R} \to \mathbb{R}$ is linear if and only if
>
> $$
> T[a x_1(t) + b x_2(t)] = a T_1[x_1(t)] + b T_2[x_2(t)]
> $$
>
> for all $t \in \mathbb{R}$ with $a,b \in \mathbb{C}$.
<br>
> *Definition*: a system $T$ is time invariant if and only if for all $t \in \mathbb{R}$ a shift in the input $x: \mathbb{R} \to \mathbb{R}$ results only in a shift in the output $y: \mathbb{R} \to \mathbb{R}$
>
> $$
> y(t) = T[x(t)] \iff y(t - t_0) = T[x(t - t_0)],
> $$
>
> for all $t_0 \in \mathbb{R}$.
<br>
> *Definition*: a system $T$ is invertible if distinct input $x: \mathbb{R} \to \mathbb{R}$ results in distinct output $y: \mathbb{R} \to \mathbb{R}$; the system is injective. The inverse of $T$ is defined such that
>
> $$
> T^{-1}[y(t)] = T^{-1}[T[x(t)]] = x(t)
> $$
>
> for all $t \in \mathbb{R}$.
<br>
> *Definition*: a system $T$ is memoryless if the image of the output $y(t_0)$ with $y: \mathbb{R} \to \mathbb{R}$ depends only on the input $x(t_0)$ with $x: \mathbb{R} \to \mathbb{R}$ for all $t_0 \in \mathbb{R}$.
<br>
> *Definition*: a system $T$ is causal if the image of the output $y(t_0)$ with $y: \mathbb{R} \to \mathbb{R}$ depends only on images of the input $x(t)$ for $t \leq t_0$ with $x: \mathbb{R} \to \mathbb{R}$ for all $t_0 \in \mathbb{R}$.
It is commenly accepted that all physical systems are causal since by definition, a cause precedes its effect. But do not be fooled.
> *Definition*: a system $T$ is bounded-input $\implies$ bounded-output (BIBO) -stable if and only if for all $t \in \mathbb{R}$ the output $y: \mathbb{R} \to \mathbb{R}$ is bounded for bounded input $x: \mathbb{R} \to \mathbb{R}$. Then
>
> $$
> |x(t)| \leq M \implies |y(t)| \leq P,
> $$
>
> for all $M, P \in \mathbb{R}$.
## Linear time invariant systems
Linear time invariant systems are described by linear operators whose action on a system does not expicitly depend on time; time invariance.
> *Definition*: consider a LTI-system $T$ given by
>
> $$
> y(t) = T[x(t)],
> $$
>
> for all $t \in \mathbb{R}$. The impulse response $h: \mathbb{R} \to \mathbb{R}$ of this systems is defined as
>
> $$
> h(t) = T[\delta(t)]
> $$
>
> for all $t \in \mathbb{R}$ with $\delta$ the Dirac delta function.
It may be literally interpreted as the effect of an impulse at $t = 0$ on the system.
> *Theorem*: for a LTI-system $T$ with $x,y,h: \mathbb{R} \to \mathbb{R}$ the input, output and impulse response of the system we have
>
> $$
> y(t) = h(t) * x(t),
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
Therefore the system $T$ is completely characterized by the impulse response of $T$.
> *Theorem*: for two LTI-systems in parallel given by $T = T_1 + T_2$ with $x,y,h_1,h_2: \mathbb{R} \to \mathbb{R}$ the input, output and impulse response of both systems we have
>
> $$
> y(t) = (h_1(t) + h_2(t)) * x(t),
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
<br>
> *Theorem*: for two LTI-systems in series given by $T = T_2 T_1$ with $x,y,h_1,h_2: \mathbb{R} \to \mathbb{R}$ the input, output and impulse response of both systems we have
>
> $$
> y(t) = (h_2(t) * h_1(t)) * x(t),
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
From the definition of convolutions we have $h_2 * h_1 = h_1 * h_2$ therefore the product of LTI-systems *is* commutative.
For a causal system there is no effect before its cause, a causal LTI system therefore must have an impulse response $h: \mathbb{R} \to \mathbb{R}$ that must be zero for all $t \in \mathbb{R}^-$.
> *Theorem*: for a LTI-system and its impulse response $h: \mathbb{R} \to \mathbb{R}$ we have
>
> $$
> h(t) \overset{\mathcal{F}}\longleftrightarrow H(\omega),
> $$
>
> for all $t, \omega \in \mathbb{R}$ with $H: \mathbb{R} \to \mathbb{C}$ the transfer function.
??? note "*Proof*:"
Will be added later.
<br>
> *Theorem*: for a LTI system $T$ with $x,y,h: \mathbb{R} \to \mathbb{R}$ the input, output and its impulse if the inverse system $T^{-1}$ exists it has an impulse response $h^{-1}: \mathbb{R} \to \mathbb{R}$ such that
>
> $$
> x(t) = h^{-1}(t) * y(t),
> $$
>
> for all $t \in \mathbb{R}$ if and only if
>
> $$
> h^{-1} * h(t) = \delta(t),
> $$
>
> for all $t \in \mathbb{R}$. The transfer function of $T^{-1}$ is then given by
>
> $$
> H^{-1}(\omega) = \frac{1}{H(\omega)},
> $$
>
> for all $\omega \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
<br>
Therefore a LTI-system is invertible if and only if $H(\omega) \neq 0$ for all $\omega \in \mathbb{R}$.
> *Theorem*: the low pass filter $H: \mathbb{R} \to \mathbb{C}$ given by the transfer function
>
> $$
> H(\omega) = \text{rect} \frac{\omega}{2\omega_b},
> $$
>
> for all $\omega \in \mathbb{R}$ with $\omega_b \in \mathbb{R}$ is not causal. Therefore assumed to be not physically realisable.
??? note "*Proof*:"
Will be added later.

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# The curl of a vector field
> *Definition*: the Levi-Civita permutation symbol is defined as
>
> $$
> e_{ijk} = \begin{cases} 0 &\text{ if $i,j,k$ are identical}, \\ 1 &\text{ if the permutation $(i,j,k)$ is even}, \\ -1 &\text{ if the permutation $(i,j,k)$ is odd}.\end{cases}
> $$
>
>
The curl of a vector field may describe the circulation of a vector field and is defined below.
> *Definition*: derivation and definition is missing for now.
Note that the "cross product " between the nabla operator and the vector field $\mathbf{v}$ does not imply anything and is only there for notational sake. An alternative to this notation is using $\text{rot } \mathbf{v}$ to denote the curl or rotation.
> *Theorem*: the curl of a vector field $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ for a curvilinear coordinate system is defined as
>
> $$
> \nabla \times \mathbf{v}(\mathbf{x}) = \frac{1}{\sqrt{g(\mathbf{x})}} e^{ijk} \partial_i \big(v_j(\mathbf{x}) \big) \mathbf{a}_k(\mathbf{x}),
> $$
>
> for all $\mathbf{x} \in \mathbb{R}^3$.
??? note "*Proof*:"
Will be added later.
The curl of a vector field for a ortho-curvilinear coordinate system may also be derived and can be found below.
> *Corollary*: the curl of a vector field $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ for a ortho-curvilinear coordinate system is defined as
>
> $$
> \nabla \times \mathbf{v}(\mathbf{x}) = \frac{1}{h_1 h_2 h_3} e^{ijk} \partial_i \big(h_j v_{(j)}(\mathbf{x}) \big) h_k \mathbf{e}_{(k)},
> $$
>
> for all $\mathbf{x} \in \mathbb{R}^3$.
??? note "*Proof*:"
Will be added later.
Please note that the scaling factors may also depend on the position $\mathbf{x} \in \mathbb{R}^3$ depending on the coordinate system.
> *Proposition*: let $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ be a vector field and $f: \mathbb{R}^3 \to \mathbb{R}$ a scalar field then we have
>
> $$
> \begin{align*}
> \nabla \cdot \big(\nabla \times \mathbf{v}(\mathbf{x}) \big) &= 0, \\
> \nabla \times \nabla f(\mathbf{x}) &= \mathbf{0},
> \end{align*}
> $$
>
> for all $\mathbf{x} \in \mathbb{R}^3$.
??? note "*Proof*:"
Will be added later.
Similarly to the [divergence theorem](divergence.md#divergence-in-curvilinear-coordinates) for the divergence, the curl is related to Kelvin-Stokes theorem given below.
> *Theorem*: let $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ be a smooth vector field and $A \subset \mathbb{R}^3$ a closed surface with boundary curve $C \subset \mathbb{R}^3$ piecewise smooth we have that
>
> $$
> \oint_C \big\langle \mathbf{v}(\mathbf{x}), d\mathbf{x} \big\rangle = \int_A \big\langle \nabla \times \mathbf{v}(\mathbf{x}), d\mathbf{A} \big\rangle,
> $$
>
> is true.
??? note "*Proof*:"
Will be added later.

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# Curves
> *Definition*: a curve is a continuous vector-valued function of one real-valued parameter.
>
> * A closed curve $\mathbf{c}: \mathbb{R} \to \mathbb{R}^3$ is defined by $\mathbf{c}(a) = \mathbf{c}(b)$ with $a \in \mathbb{R}$ the begin point and $b \in \mathbb{R}$ the end point.
> * A simple curve has no crossings.
<br>
> *Definition*: let $\mathbf{c}: \mathbb{R} \to \mathbb{R}^3$ be a curve, the derivative of $\mathbf{c}$ is defined as the velocity of the curve $\mathbf{c}'$. The length of the velocity is defined as the speed of the curve $\|\mathbf{c}'\|$.
<br>
> *Proposition*: let $\mathbf{c}: \mathbb{R} \to \mathbb{R}^3$ be a curve, the velocity of the curve $\mathbf{c}'$ is tangential to the curve.
??? note "*Proof*:"
Will be added later.
<br>
> *Definition*: let $\mathbf{c}: \mathbb{R} \to \mathbb{R}^3$ be a differentiable curve, the infinitesimal arc length $ds: \mathbb{R} \to \mathbb{R}$ of the curve is defined as
>
> $$
> ds(t) := \|d \mathbf{c}(t)\| = \|\mathbf{c}'(t)\|dt
> $$
>
> for all $t \in \mathbb{R}$.
<br>
> *Theorem*: let $\mathbf{c}: \mathbb{R} \to \mathbb{R}^3$ be a differentiable curve, the arc length $s: \mathbb{R} \to \mathbb{R}$ of a section that start at $t_0 \in \mathbb{R}$ is given by
>
> $$
> s(t) = \int_{t_0}^t \|\mathbf{c}'(u)\|du,
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
## Arc length parameterization
To obtain a speed of unity everywhere on the curve, or differently put equidistant arc lengths between each time step an arc length parameterization can be performed. It can be performed in 3 steps:
1. For a given curve determine the arc length function for a given start point.
2. Find the inverse of the arc length function if it exists.
3. Adopt the arc length as variable of the curve.
Obtaining a speed of unity on the entire defined curve.
For example consider a curve $\mathbf{c}: \mathbb{R} \to \mathbb{R}^3$ given in Cartesian coordinates by
$$
\mathbf{c}(\phi) = \begin{pmatrix} r \cos \phi \\ r \sin \phi \\ \rho r \phi\end{pmatrix},
$$
for all $\phi \in \mathbb{R}$ with $r, \rho \in \mathbb{R}^+$.
Determining the arc length function $s: \mathbb{R} \to \mathbb{R}$ of the curve
$$
\begin{align*}
s(\phi) &= \int_0^\phi \|\mathbf{c}'(u)\|du, \\
&= \int_0^\phi r \sqrt{1 + \rho^2}du, \\
&= \phi r \sqrt{1 + \rho^2},
\end{align*}
$$
for all $\phi \in \mathbb{R}$. It may be observed that $s$ is a bijective mapping.
The inverse of the arc length function $s^{-1}: \mathbb{R} \to \mathbb{R}$ is then given by
$$
s^{-1}(\phi) = \frac{\phi}{r\sqrt{a + \rho^2}},
$$
for all $\phi \in \mathbb{R}$.
The arc length parameterization $\mathbf{c}_s: \mathbb{R} \to \mathbb{R}^3$ of $\mathbf{c}$ is then given by
$$
\mathbf{c}_s(\phi) = \mathbf{c}(s^{-1}(\phi)) = \begin{pmatrix} r \cos (\phi / r\sqrt{a + \rho^2}) \\ r \sin (\phi / r\sqrt{a + \rho^2}) \\ \rho \phi / \sqrt{a + \rho^2}\end{pmatrix},
$$
for all $\phi \in \mathbb{R}$.

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# Curvilinear coordinate systems
In this section curvilinear coordinate systems will be presented, these are coordinate systems that are based on a set of basis vectors that are neither orthognal nor normalized.
> *Principle*: space can be equipped with a smooth and continuous coordinate net.
## Covariant basis
> *Definition*: consider a coordinate system $(x_1, x_2, x_3)$ that is mapped by $\mathbf{x}: \mathbb{R}^3 \to \mathbb{R}^3$ with respect to a reference coordinate system. Producing a position vector for every combination of coordinate values.
>
> * For two coordinates fixed, a coordinate curve is obtained.
> * For one coordinate fixed, a coordinate surface is obtained.
We will now use this coordinate system described as $\mathbf{x}$ to formulate a set of basis vectors.
> *Definition*: for a valid coordinate system $\mathbf{x}$ a set of linearly independent covariant (local) basis vectors can be described by
>
> $$
> \mathbf{a}_i(x_1, x_2, x_3) := \partial_i \mathbf{x}(x_1, x_2, x_3),
> $$
>
> for all $(x_1, x_2, x_3) \in \mathbb{R}^3$ and $i \in \{1, 2, 3\}$.
Obtaining basis vectors that are tangential to the corresponding coordinate curves. Therefore any vector $\mathbf{u} \in \mathbb{R}^3$ can be written in terms of its components with respect to this basis
$$
\mathbf{u} = \sum_{i=1}^3 u_i \mathbf{a}_i
$$
with $u_{1,2,3} \in \mathbb{R}$ the components.
> *Definition*: the Einstein summation convention omits the summation symbol and is defined by
>
> $$
> \mathbf{u} = \sum_{i=1}^3 u_i \mathbf{a}_i = u^i \mathbf{a}_i,
> $$
>
> with $u^{1,2,3} \in \mathbb{R}$ the contravariant components. The definition states that
>
> 1. When an index appears twice in a product, one as a subscript and once as a superscript, summation over that index is implied.
> 2. A superscript that appears in denominator counts as a subscript.
This convention makes writing summation a lot easier, though one may see it as a little unorthodox.
## The metric tensor
> *Definition*: for two vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^3$ that are represented in terms of a covariant basis, the scalar product is given by
>
> $$
> \langle \mathbf{u}, \mathbf{v} \rangle = u^i v^j \langle \mathbf{a}_i, \mathbf{a}_j \rangle = u^i v^j g_{ij},
> $$
>
> with $g_{ij}$ the components of a structure that is called the metric tensor given by
>
> $$
> (g_{ij}) := \begin{pmatrix} \langle \mathbf{a}_1, \mathbf{a}_1 \rangle & \langle \mathbf{a}_1, \mathbf{a}_2 \rangle & \langle \mathbf{a}_1, \mathbf{a}_3 \rangle \\ \langle \mathbf{a}_2, \mathbf{a}_1 \rangle & \langle \mathbf{a}_2, \mathbf{a}_2 \rangle & \langle \mathbf{a}_2, \mathbf{a}_3 \rangle \\ \langle \mathbf{a}_3, \mathbf{a}_1 \rangle & \langle \mathbf{a}_3, \mathbf{a}_2 \rangle & \langle \mathbf{a}_3, \mathbf{a}_3 \rangle \end{pmatrix}.
> $$
For the special case of an orthogonal set of basis vectors, all of-diagonal elements are zero and we have a metric tensor $g_{ij}$ given by
$$
(g_{ij}) = \begin{pmatrix} \langle \mathbf{a}_1, \mathbf{a}_1 \rangle & & \\ & \langle \mathbf{a}_2, \mathbf{a}_2 \rangle & \\ & & \langle \mathbf{a}_3, \mathbf{a}_3 \rangle\end{pmatrix} = \begin{pmatrix} h_1^2 & & \\ & h_2^2 & \\ & & h_3^2\end{pmatrix},
$$
with $h_i = \sqrt{\langle \mathbf{a}_i, \mathbf{a}_i \rangle} = \|\mathbf{a}_i\|$ the scale factors for $i \in \{1, 2, 3\}$.
> *Theorem*: the determinant of the metric tensor $g := \det(g_{ij})$ can be written as the square of the scalar triple product of the covariant basis vectors
>
> $$
> g = \langle \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \rangle^2.
> $$
??? note "*Proof*:"
Will be added later.
<br>
> *Corollary*: consider a covariant basis and the infinitesimal coordinate transformations $(dx_1, dx_2, dx_3)$ spanned by the covariant basis then the volume defined by these infinitesimal coordinate transformations is given by
>
> $$
> \begin{align*}
> dV &= \langle dx_1 \mathbf{a}_2, dx_2 \mathbf{a}_1, dx_3 \mathbf{a}_3 \rangle, \\
> &= \sqrt{g} dx_1 dx_2 dx_3,
> \end{align*}
> $$
>
> by definition of the scalar triple product. For a function $f: \mathbb{R}^3 \to \mathbb{R}$ its integral in the domain $D \subseteq \mathbb{R}^3$ with $D = [a_1, b_1] \times [a_2, b_2] \times [a_3, b_3]$ and $a_i, b_i \in \mathbb{R}$ for $i \in \{1, 2, 3\}$ closed may be given by
>
> $$
> \int_D f(x_1, x_2, x_3)dV = \int_{a_1}^{b_1} \int_{a_2}^{b_2} \int_{a_3}^{b_3} f(x_1, x_2, x_3) \sqrt{g} dx_1 dx_2 dx_3.
> $$
??? note "*Proof*:"
Will be added later.
## Contravariant basis
The covariant basis vectors have been constructed as tangential vectors of the coordinate curves. An alternative basis can be constructed from vectors that are perpendicular to coordinate surfaces.
> *Definition*: for a valid set of covariant basis vectors the contravariant basis vectors may be defined, given by
>
> $$
> \begin{align*}
> \mathbf{a}^1 &:= \frac{1}{\sqrt{g}} (\mathbf{a}_2 \times \mathbf{a}_3), \\
> \mathbf{a}^2 &:= \frac{1}{\sqrt{g}} (\mathbf{a}_3 \times \mathbf{a}_1), \\
> \mathbf{a}^3 &:= \frac{1}{\sqrt{g}} (\mathbf{a}_1 \times \mathbf{a}_2)
> \end{align*}
> $$
From this definition it follows that $\langle \mathbf{a}^i, \mathbf{a}_j \rangle = \delta_j^i$, with $\delta_j^i$ the Kronecker delta defined by
> *Definition*: the Kronecker delta $\delta_{ij}$ is defined as
>
> $$
> \delta_{ij} = \begin{cases} 1 &\text{ if } i = j, \\ 0 &\text{ if } i \neq j.\end{cases}
> $$
A metric tensor for contravariant basis vectors may be defined. With which the relations between covariant and contravariant quantities can be found.
> *Definition*: the components of the metric tensor for contravariant basis vectors are defined as
>
> $$
> g^{ij} := \langle \mathbf{a}^i, \mathbf{a}^j \rangle,
> $$
>
> therefore the metric tensor for contravariant basis vectors is given by
>
> $$
> (g^{ij}) = \begin{pmatrix} \langle \mathbf{a}^1, \mathbf{a}^1 \rangle & \langle \mathbf{a}^1, \mathbf{a}^2 \rangle & \langle \mathbf{a}^1, \mathbf{a}^3 \rangle \\ \langle \mathbf{a}^2, \mathbf{a}^1 \rangle & \langle \mathbf{a}^2, \mathbf{a}^2 \rangle & \langle \mathbf{a}^2, \mathbf{a}^3 \rangle \\ \langle \mathbf{a}^3, \mathbf{a}^1 \rangle & \langle \mathbf{a}^3, \mathbf{a}^2 \rangle & \langle \mathbf{a}^3, \mathbf{a}^3 \rangle \end{pmatrix}.
> $$
These relations are stated in the proposition below.
> *Proposition*: considering the two ways of representing the vector $\mathbf{u} \in \mathbb{R}^3$ given by
>
> $$
> \mathbf{u} = u^i \mathbf{a}_i = u_i \mathbf{a}^i.
> $$
>
> From the definitions given above the relations between the covariant and contravariant quantities of the vector $\mathbf{u}$ have been found to be
>
> $$
> u_i = g_{ij} u^j, \qquad \mathbf{a}_i = g_{ij} \mathbf{a}^j,
> $$
>
> $$
> u^i = g^{ij} u_j, \qquad \mathbf{a}^i = g^{ij} \mathbf{a}_j.
> $$
??? note "*Proof*:"
Will be added later.
By combining the expressions for the components a relation can be established between $g_{ij}$ and $g^{ij}$.
> *Theorem*: the components of the metric tensor for covariant and contravariant basis vectors are related by
>
> $$
> g_{ij} g^{jk} = \delta_i^k.
> $$
??? note "*Proof*:"
Will be added later.
This is the index notation for $(g_{ij})(g^{ij}) = I$, with $I$ the identity matrix, therefore we have
$$
(g^{ij}) = (g_{ij})^{-1},
$$
concluding that both matrices are nonsingular.
> *Corollary*: let $\mathbf{u} \in \mathbb{R}^3$ be a vector, for orthogonal basis vectors it follows that the covariant and contravariant basis vectors are proportional by
>
> $$
> \mathbf{a}^i = \frac{1}{h_i^2} \mathbf{a}_i,
> $$
>
> and for the components of $\mathbf{u}$ we have
>
> $$
> u^i = \frac{1}{h_i^2} u_i,
> $$
>
> for all $i \in \{1, 2, 3\}$.
??? note "*Proof*:"
Will be added later.
Therefore it also follows that for the special case of orthogonal basis vectors the metric tensor for contrariant basis vectors $(g^{ij})$ is given by
$$
(g^{ij}) = \begin{pmatrix} \langle \mathbf{a}^1, \mathbf{a}^1 \rangle & & \\ & \langle \mathbf{a}^2, \mathbf{a}^2 \rangle & \\ & & \langle \mathbf{a}^3, \mathbf{a}^3 \rangle\end{pmatrix} = \begin{pmatrix} \frac{1}{h_1^2} & & \\ & \frac{1}{h_2^2} & \\ & & \frac{1}{h_3^2}\end{pmatrix},
$$
with $h_i = \sqrt{\langle \mathbf{a}_i, \mathbf{a}_i \rangle} = \|\mathbf{a}_i\|$ the scale factors for $i \in \{1, 2, 3\}$.
## Phyiscal components
A third representation of vectors uses physical components and normalized basis vectors.
> *Definition*: from the above corollary the physical component representation for a vector $\mathbf{u} \in \mathbb{R}^3$ can be defined as
>
> $$
> \mathbf{e}_{(i)} := h_i \mathbf{a}^i = \frac{1}{h_i} \mathbf{a}_i,
> $$
>
> $$
> u_{(i)} := h_i u^i = \frac{1}{h_i} u_i,
> $$
>
> for all $i \in \{1, 2, 3\}$.
Contributing to the physical component representation given by
$$
\mathbf{u} = u^{(i)} \mathbf{e}_{(i)},
$$
for $i \in \{1, 2, 3\}$.
> *Proposition*: obtaining the properties
>
> $$
> \langle \mathbf{e}_{(i)}, \mathbf{e}_{(i)} \rangle = \frac{1}{h_i^2} \langle \mathbf{a}_i, \mathbf{a}_i \rangle = 1,
> $$
>
> and for vectors $\mathbf{u}, \mathbf{v} \in \mathbb{R}^3$ we have
>
> $$
> \langle \mathbf{u}, \mathbf{v} \rangle = u^{(i)} v_{(i)}.
> $$
??? note "*Proof*:"
Will be added later.
In particular the length of a vector $\mathbf{u} \in \mathbb{R}^3$ can then be determined by
$$
\|\mathbf{u}\| = \sqrt{u^{(i)} u_{(i)}}.
$$
We will discuss as an example the representations of the cartesian, cylindrical and spherical coordinate systems viewed from a cartesian perspective. This means that the coordinate maps are based on the cartesian interpretation of them. Every other interpretation could have been used, but our brains have a preference for cartesian it seems.
Let $\mathbf{x}: \mathbb{R}^3 \to \mathbb{R}^3$ map a cartesian coordinate system given by
$$
\mathbf{x}(x,y,z) = \begin{pmatrix} x \\ y \\ z\end{pmatrix},
$$
then we have the covariant basis vectors given by
$$
\mathbf{a}_i(x,y,z) = \partial_i \mathbf{x}(x,y,z),
$$
obtaining $\mathbf{a}_1 = \begin{pmatrix} 1 \\ 0 \\ 0\end{pmatrix}, \mathbf{a}_2 = \begin{pmatrix} 0 \\ 1 \\ 0\end{pmatrix}, \mathbf{a}_3 = \begin{pmatrix} 0 \\ 0 \\ 1\end{pmatrix}$.
It may be observed that this set of basis vectors is orthogonal. Therefore the scaling factors are given by $h_1 = 1, h_2 = 1, h_3 = 1$ as to be expected for the reference.
<br>
Let $\mathbf{x}: \mathbb{R}^3 \to \mathbb{R}^3$ map a cylindrical coordinate system given by
$$
\mathbf{x}(r,\theta,z) = \begin{pmatrix} r \cos \theta \\ r \sin \theta \\ z\end{pmatrix},
$$
then we have the covariant basis vectors given by
$$
\mathbf{a}_i(r,\theta,z) = \partial_i \mathbf{x}(r,\theta,z),
$$
obtaining $\mathbf{a}_1(\theta) = \begin{pmatrix} \cos \theta \\ \sin \theta \\ 0\end{pmatrix}, \mathbf{a}_2(r, \theta) = \begin{pmatrix} -r\sin \theta \\ r \cos \theta \\ 0\end{pmatrix}, \mathbf{a}_3 = \begin{pmatrix} 0 \\ 0 \\ 1\end{pmatrix}$.
It may be observed that this set of basis vectors is orthogonal. Therefore the scaling factors are given by $h_1 = 1, h_2 = r, h_3 = 1$.
<br>
Let $\mathbf{x}: \mathbb{R}^3 \to \mathbb{R}^3$ map a spherical coordinate system given by
$$
\mathbf{x}(r,\theta,\varphi) = \begin{pmatrix}r \cos \theta \sin \varphi \\ r \sin \theta \sin \varphi \\ r \cos \varphi\end{pmatrix},
$$
using the mathematical convention, then we have the covariant basis vectors given by
$$
\mathbf{a}_i(r,\theta,\varphi) = \partial_i \mathbf{x}(r,\theta,\varphi),
$$
obtaining $\mathbf{a}_1(\theta, \varphi) = \begin{pmatrix} \cos \theta \sin \varphi \\ \sin \theta \sin \varphi\\ \cos \varphi\end{pmatrix}, \mathbf{a}_2(r, \theta, \varphi) = \begin{pmatrix} -r\sin \theta \sin \varphi \\ r \cos \theta \sin \varphi \\ 0\end{pmatrix}, \mathbf{a}_3 = \begin{pmatrix} r \cos \theta \cos \varphi \\ r \sin \theta \cos \varphi \\ - r \sin \varphi\end{pmatrix}$.
It may be observed that this set of basis vectors is orthogonal. Therefore the scaling factors are given by $h_1 = 1, h_2 = r \sin \varphi, h_3 = r$.

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# The divergence of a vector field
## Flux densities
Considering a medium with a mass density $\rho: \mathbb{R}^4 \to \mathbb{R}$ and a velocity field $\mathbf{v}: \mathbb{R}^4 \to \mathbb{R}^3$ consisting of a orientable finite sized surface element $d\mathbf{A} \in \mathbb{R}^3$.
> *Definition*: a surface must be orientable for the surface integral to exist. It must be able to move along the surface continuously without ending up on the "other side".
We then have a volume $dV \in \mathbb{R}$ defined by the parallelepiped formed by $dV = \langle d\mathbf{x}, d\mathbf{A} \rangle$ with the vector $d\mathbf{x} = \mathbf{v} dt$, for a time interval $dt \in \mathbb{R}$. The mass flux $d\Phi$ per unit of time through the surface element $d\mathbf{A}$ may then be given by
$$
d \Phi = \rho \langle \mathbf{v}, d\mathbf{A} \rangle.
$$
The mass flux $\Phi: \mathbb{R} \to \mathbb{R}$ through a orientable finite sized surface $A \subseteq \mathbb{R}^3$ is then given by
$$
\Phi(t) = \int_A \Big\langle \rho(\mathbf{x}, t) \mathbf{v}(\mathbf{x}, t), d\mathbf{A} \Big\rangle,
$$
for all $t \in \mathbb{R}$.
> *Definition*: let $\mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3$ be the (mass) flux density given by
>
> $$
> \mathbf{\Gamma}(\mathbf{x},t) := \rho(\mathbf{x},t) \mathbf{v}(\mathbf{x},t),
> $$
>
> for all $(\mathbf{x},t) \in \mathbb{R}^4$.
The (mass) flux density is a vector-valued function of position and time that expresses the rate of transport of a quantity per unit of time of area perpendicular to its direction.
The mass flux $\Phi$ through $A$ may then be given by
$$
\Phi(t) = \int_A \Big\langle \mathbf{\Gamma}(\mathbf{x},t), d\mathbf{A} \Big\rangle,
$$
for all $t \in \mathbb{R}$.
## Definition of the divergence
> *Definition*: the divergence of a flux density $\mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3$ is given by
>
> $$
>\nabla \cdot \mathbf{\Gamma}(\mathbf{x}, t) = \lim_{V \to 0} \frac{1}{V} \int_A \Big\langle \mathbf{\Gamma}(\mathbf{x}, t), d\mathbf{A} \Big\rangle,
> $$
>
> for all $(\mathbf{x}, t) \in \mathbb{R}^4$ for a volume $V \subset \mathbb{R}^3$ with closed orientable boundary surface $A \subset V$.
Note that this "dot product" between the nabla operator and the flux density $\mathbf{\Gamma}$ does not imply anything and is only there for notational sake. An alternative to this notation is using $\text{div } \mathbf{\Gamma}$ to denote the divergence.
The definition of the divergence can be interpreted with the particle mass balance for a medium with a particle density $n: \mathbb{R}^4 \to \mathbb{R}$ and a velocity field $\mathbf{v}: \mathbb{R}^4 \to \mathbb{R}^3$. Furthermore we have that the particles are produced at a rate $S: \mathbb{R}^4 \to \mathbb{R}^3$.
We then have the particle flux density $\mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3$ given by
$$
\mathbf{\Gamma}(\mathbf{x},t) = n(\mathbf{x},t) \mathbf{v}(\mathbf{x},t),
$$
for all $(\mathbf{x},t) \in \mathbb{R}^4$.
For a volume $V \subseteq \mathbb{R}^3$ with a closed orientable boundary surface $A \subseteq \mathbb{R}^3$ we have that the amount of particles inside this volume for a specific time is given by
$$
\int_V n(\mathbf{x}, t) dV,
$$
for all $t \in \mathbb{R}$. We have that the particle flux through $A$ is given by
$$
\int_A \Big\langle \mathbf{\Gamma}(\mathbf{x},t), d\mathbf{A} \Big\rangle,
$$
for all $t \in \mathbb{R}$ and we have that the particle production rate in this volume $V$ is given by
$$
\int_V S(\mathbf{x}, t)dV,
$$
for all $t \in \mathbb{R}$. We conclude that the sum of the particle flux through $A$ and the time derivative of the particles inside the volume $V$ must be equal to the production rate inside this volume $V$. Therefore we have
$$
d_t \int_V n(\mathbf{x}, t) dV + \int_A \Big\langle \mathbf{\Gamma}(\mathbf{x},t), d\mathbf{A} \Big\rangle = \int_V S(\mathbf{x}, t)dV,
$$
for all $t \in \mathbb{R}$.
Assuming the system is stationary the time derivative of the particles inside the volume $V$ must vanish. The divergence is then defined to be the total production for a position $\mathbf{x} \in V$.
## Divergence in curvilinear coordinates
> *Theorem*: the divergence of a flux density $\mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3$ for a curvilinear coordinate system is given by
>
> $$
> \nabla \cdot \mathbf{\Gamma}(\mathbf{x}, t) = \frac{1}{\sqrt{g(\mathbf{x})}} \partial_i \Big(\Gamma^i(\mathbf{x},t) \sqrt{g(\mathbf{x})} \Big)
> $$
>
> for all $\mathbf{x} \in \mathbb{R}^3$ and $i \in \{1, 2, 3\}$.
??? note "*Proof*:"
Will be added later.
We may also give the divergence for ortho-curvilinear coordinate systems.
> *Corollary*: the divergence of a flux density $\mathbf{\Gamma}: \mathbb{R}^3 \to \mathbb{R}^3$ for a ortho-curvilinear coordinate system is given by
>
> $$
> \nabla \cdot \mathbf{\Gamma}(\mathbf{x}, t) = \frac{1}{h_1h_2h_3} \partial_i \Big(\Gamma^{(i)}(\mathbf{x},t)\frac{1}{h_i} h_1 h_2 h_3 \Big)
> $$
>
> for all $\mathbf{x} \in \mathbb{R}^3$ and $i \in \{1, 2, 3\}$.
??? note "*Proof*:"
Will be added later.
Please note that the scaling factors may also depend on the position $\mathbf{x} \in \mathbb{R}^3$ depending on the coordinate system.
It has been found that the volume integral over the divergence of a vector field is equal to the integral of the vector field itself over the surface that bounds the volume. It is known as the divergence theorem and is given below.
> *Theorem*: for a volume $V \subset \mathbb{R}^3$ with a closed and orientable boundary surface $A \subset V$ with a continuously differentiable flux density $\mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3$ we have that
>
> $$
> \int_A \Big\langle \mathbf{\Gamma}(\mathbf{x}, t), d\mathbf{A} \Big\rangle = \int_V \nabla \cdot \mathbf{\Gamma}(\mathbf{x}, t) dV,
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.

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# The gradient of a scalar field
Considering a scalar field $f: \mathbb{R}^3 \to \mathbb{R}$, if the field is continuously differentiable we have
$$
df(\mathbf{x}) = \partial_i f(\mathbf{x}) dx_i,
$$
for all $x \in \mathbb{R}^4$ and $i \in \{1,2,3\}$. We may rewrite this in terms of a displacement vector $d\mathbf{x} = \mathbf{a}_i dx^i$ into
$$
\begin{align*}
df &= \partial_i f(\mathbf{x}) \delta^i_j dx^j, \\
&= \partial_i f(\mathbf{x}) \langle \mathbf{a}^i, \mathbf{a}_j \rangle dx^j, \\
&= \partial_i f(\mathbf{x})\langle \mathbf{a}^i, d\mathbf{x} \rangle.
\end{align*}
$$
> *Definition*: the gradient of a scalar field $f: \mathbb{R}^3 \to \mathbb{R}$ for a curvilinear coordinate system is defined as
>
> $$
> \nabla f(\mathbf{x}) := \partial_i f(\mathbf{x}) \mathbf{a}^i,
> $$
>
> for all $\mathbf{x} \in \mathbb{R}^3$.
Note that in the differentation section of multivariable calculus the definition of the gradient explicitly for Cartesian coordinate systems was given. This definition is rather general for all coordinate systems, although is limited to only 3 dimensions.
> *Proposition*: let $f: \mathbb{R}^3 \to \mathbb{R}$ be a scalar field, the gradient of $f$ points in the direction for which $f$ has the greatest increase.
??? note "*Proof*:"
Will be added later.
The following definition introduces the material derivative, it may appear to be a little unorthodox.
> *Definition*: let $f: \mathbb{R}^4 \to \mathbb{R}$ be a scalar field and $\mathbf{x}: \mathbb{R} \to \mathbb{R}^3$ a vector field. The material derivative of $f$ is defined as
>
> $$
> f'(\mathbf{x}(t), t) := \big\langle \nabla f(\mathbf{x}, t), \mathbf{x}'(t) \big\rangle + \partial_t f(\mathbf{x}, t),
> $$
>
> for all $t \in \mathbb{R}$. Note that the gradient in the scalar product is only taken for $\mathbf{x}$.
The following definition introduces the directional derivative.
> *Definition*: let $f: \mathbb{R}^3 \to \mathbb{R}$ be a scalar field and $\mathbf{v} \in \mathbb{R}^3$ a normalised vector such that $\|\mathbf{v}\| = 1$. The directional derivative of $f$ in the direction of $\mathbf{v}$ is defined as
>
> $$
> D_{\mathbf{v}} f(\mathbf{x}) := \big\langle \mathbf{v}, \nabla f(\mathbf{x}) \big\rangle,
> $$
>
> for all $\mathbf{x} \in \mathbb{R}^3$.
Consider a vector field $\mathbf{u}: \mathbb{R}^3 \to \mathbb{R}^3$, the integral of $\mathbf{u}$ along a curve $C \subset \mathbb{R}^3$ is given by
$$
\int_C \big\langle \mathbf{u}(\mathbf{x}), d\mathbf{x} \big\rangle.
$$
> *Theorem*: let $f: \mathbb{R}^3 \to \mathbb{R}$ be a scalar field and consider a curve $C \subset \mathbb{R}^3$ then we have
>
> $$
> \int_C \big\langle \nabla f(\mathbf{x}), d\mathbf{x} \big\rangle = \big[f(\mathbf{x}) \big]_C.
> $$
??? note "*Proof*:"
Will be added later.

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# Vector operators
## Properties of the gradient, divergence and curl
> *Proposition*: let $a,b \in \mathbb{R}$, $f,g: \mathbb{R}^3 \to \mathbb{R}$ be differentiable scalar fields and $\mathbf{u}, \mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ be differentiable vector fields. Then we have the following identities:
>
> **Linearity:**
>
> $$
> \begin{align*}
> \nabla (af + bg) &= a \nabla f + b \nabla g, \\
> \nabla \cdot (a\mathbf{u} + b \mathbf{v}) &= a (\nabla \cdot \mathbf{u}) + b (\nabla \cdot \mathbf{v}), \\
> \nabla \times (a\mathbf{u} + b \mathbf{v}) &= a (\nabla \times \mathbf{u}) + b (\nabla \times\mathbf{v}).
> \end{align*}
> $$
>
> **Multiplication rules:**
>
> $$
> \begin{align*}
> \nabla (fg) &= f \nabla g+ g \nabla f, \\
> \nabla \cdot (f \mathbf{u}) &= f (\nabla \cdot \mathbf{u}) + \langle \nabla f, \mathbf{u} \rangle, \\
> \nabla \cdot (\mathbf{u} \times \mathbf{v}) &= \langle \nabla \times \mathbf{u}, \mathbf{v} \rangle - \langle \mathbf{u}, \nabla \times \mathbf{v} \rangle, \\
> \nabla \times (f\mathbf{u}) &= f (\nabla \times \mathbf{u}) + \nabla f \times \mathbf{u}.
> \end{align*}
> $$
??? note "*Proof*:"
Will be added later.
## The laplacian
> *Definition*: the laplacian of a differentiable scalar field $f: \mathbb{R}^3 \to \mathbb{R}$ is defined as
>
> $$
> \nabla^2 f(\mathbf{x}) := \nabla \cdot \nabla f(\mathbf{x}),
> $$
>
> for all $\mathbf{x} \in \mathbb{R}^3$.
The notation may be unorthodox for some. An alternative notatation for the laplacian is $\Delta f$, though generally deprecated.
We can also rewrite the laplacian for curvilinear coordinate systems as has been done below.
> *Theorem*: the laplacian of a differentiable scalar field $f: \mathbb{R}^3 \to \mathbb{R}$ for a curvilinear coordinate system is given by
>
> $$
> \nabla^2 f(\mathbf{x}) = \frac{1}{g(\mathbf{x})} \partial_i \Big(\sqrt{g(\mathbf{x})} g^{ij}(\mathbf{x}) \partial_j f(\mathbf{x}) \Big),
> $$
>
> for all $\mathbf{x} \in \mathbb{R}^3$.
??? note "*Proof*:"
Will be added later.
The laplacian for a ortho-curvilinear coordinate system may also be derived and can be found below.
> *Corollary*: the laplacian of a differentiable scalar field $f: \mathbb{R}^3 \to \mathbb{R}$ for a ortho-curvilinear coordinate system is given by
>
> $$
> \nabla^2 f(\mathbf{x}) = \frac{1}{h_1 h_2 h_3} \bigg(\partial_1 \Big(\frac{h_2 h_3}{h_1} \partial_1 f(\mathbf{x}) \Big) + \partial_2 \Big(\frac{h_1 h_3}{h_2} \partial_2 f(\mathbf{x}) \Big) + \partial_3 \Big(\frac{h_1 h_2}{h_3} \partial_3 f(\mathbf{x}) \Big) \bigg),
> $$
>
> for all $\mathbf{x} \in \mathbb{R}^3$.
??? note "*Proof*:"
Will be added later.
Please note that the scaling factors may also depend on the position $\mathbf{x} \in \mathbb{R}^3$ depending on the coordinate system.
> *Proposition*: the laplacian of a differentiable vector field $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ is given by
>
> $$
> \nabla^2 \mathbf{v}(\mathbf{x}) = \nabla \big(\nabla \cdot \mathbf{v}(\mathbf{x})\big) - \nabla \times \big(\nabla \times \mathbf{v}(\mathbf{x})\big),
> $$
>
> for all $\mathbf{x} \in \mathbb{R}^3$.
??? note "*Proof*:"
Will be added much later.
## Potentials
> *Definition*: a vector field $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ is irrotational or curl free if
>
> $$
> \nabla \times \mathbf{v}(\mathbf{x}) = \mathbf{0},
> $$
>
> for all $\mathbf{x} \in \mathbb{R}^3$.
If $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ is the gradient of some scalar field $\Phi: \mathbb{R}^3 \to \mathbb{R}$ it is irrotational since
$$
\nabla \times\big (\nabla \Phi(\mathbf{x})\big) = \mathbf{0},
$$
for all $\mathbf{x} \in \mathbb{R}^3$.
> *Proposition*: an irrotational vector field $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ has a scalar potential $\Phi: \mathbb{R}^3 \to \mathbb{R}$ such that
>
> $$
> \mathbf{v}(\mathbf{x}) = \nabla \Phi(\mathbf{x}),
> $$
>
> for all $\mathbf{x} \in \mathbb{R}^3$.
??? note "*Proof*:"
Will be added later.
In physics the scalar potential is generally given by the negative of the gradient, both are correct but one is more stupid than the other.
> *Definition*: a vector field $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ is solenoidal or divergence-free if
>
> $$
> \nabla \cdot \mathbf{v}(\mathbf{x}) = 0,
> $$
>
> for all $\mathbf{x} \in \mathbb{R}^3$.
If $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ is the curl of some vector field $\mathbf{u}: \mathbb{R}^3 \to \mathbb{R}^3$ it is solenoidal since
$$
\nabla \cdot \big(\nabla \times \mathbf{u}(\mathbf{x}) \big) = 0,
$$
for all $\mathbf{x} \in \mathbb{R}^3$.
> *Proposition*: a solenoidal vector field $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ has a vector potential $\mathbf{u}: \mathbb{R}^3 \to \mathbb{R}^3$ such that
>
> $$
> \mathbf{v}(\mathbf{x}) = \nabla \times \mathbf{u}(\mathbf{x}),
> $$
>
> for all $\mathbf{x} \in \mathbb{R}^3$.
??? note "*Proof*:"
Will be added later.
The theorem below is the Helmholtz decomposition theorem and states that every vector field can be written in terms of two potentials.
> *Theorem*: every vector field $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ can be written in terms of a scalar $\Phi: \mathbb{R}^3 \to \mathbb{R}$ and a vector $\mathbf{u}: \mathbb{R}^3 \to \mathbb{R}^3$ potential as
>
> $$
> \mathbf{v}(\mathbf{x}) = \nabla \Phi(\mathbf{x}) + \nabla \times \mathbf{u}(\mathbf{x}),
> $$
>
> for all $\mathbf{x} \in \mathbb{R}^3$.
??? note "*Proof*:"
Will be added later.
It then follows that the scalar and vector potentials can be determined for a volume $V \subset \mathbb{R}^3$ with a boundary surface $A \subset \mathbb{R}^3$ that encloses the domain $V$.
> *Corollary*: the scalar $\Phi: \mathbb{R}^3 \to \mathbb{R}$ and vector $\mathbf{u}: \mathbb{R}^3 \to \mathbb{R}^3$ potentials for a volume $V \subset \mathbb{R}^3$ with a boundary surface $A \subset \mathbb{R}^3$ that encloses the domain $V$ are given by
>
> $$
> \begin{align*}
> \Phi(\mathbf{x}) &= \frac{1}{4\pi} \int_V \frac{\nabla \cdot \mathbf{v}(\mathbf{r})}{\|\mathbf{x} - \mathbf{r}\|}dV - \frac{1}{4\pi} \oint_A \bigg\langle \frac{1}{\|\mathbf{x} - \mathbf{r}\|} \mathbf{v}(\mathbf{r}), d\mathbf{A} \bigg\rangle, \\
> \\
> \mathbf{u}(\mathbf{x}) &= \frac{1}{4\pi} \int_V \frac{\nabla \times \mathbf{v}(\mathbf{r})}{\|\mathbf{x} - \mathbf{r}\|}dV - \frac{1}{4\pi} \oint_A \frac{1}{\|\mathbf{x} - \mathbf{r}\|} \mathbf{v}(\mathbf{r}) \times d\mathbf{A},
> \end{align*}
> $$
>
> for all $\mathbf{x} \in \mathbb{R}^3$.
??? note "*Proof*:"
Will be added later.

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# Vectors and geometry
## Axiomatic geometry
The defining property of axiomatic geometry is that it can be introduced without any reference to a coordinate system. The 5 postulates of classical geometry are listed below.
1. A straight line segment can be drawn between any pair of two points.
2. A straight line segment can be extended indefinitely into a straight line.
3. A line segment is the radius of a circle with one of the end points as its center.
4. All right angles are congruent.
The fifth postulate as formulated below is only valid for Euclidean geometry; flat space informally.
<ol start="5">
<li>Given in a plane, a line and a point not on that line there is only one line through that point that does not intersect with the other line.</li>
</ol>
## Vectors
Referring to linear algebra section [vector spaces](../../../mathematics/linear-algebra/vector-spaces.md) for the axioms of the Euclidean vector space and its vector definitions. Some vector products in 3 dimensional Euclidean space are defined below
> *Definition*: the Euclidean scalar product of $\mathbf{u}, \mathbf{v} \in \mathbb{R}^3$ is given by
>
> $$
> \langle \mathbf{u}, \mathbf{v} \rangle := \|\mathbf{u}\| \|\mathbf{v}\| \cos \varphi,
> $$
>
> with $\|\mathbf{u}\|$ and $\|\mathbf{v}\|$ the length of $\mathbf{u}$ and $\mathbf{v}$ and the $\varphi$ the angle between $\mathbf{u}$ and $\mathbf{v}$.
It follows than that for $\mathbf{v} = \mathbf{u}$ we have
$$
\|\mathbf{u}\| = \langle \mathbf{u}, \mathbf{u} \rangle.
$$
> *Definition*: the Euclidean cross product of $\mathbf{u}, \mathbf{v} \in \mathbb{R}^3$ is given by
>
> $$
> \|\mathbf{u} \times \mathbf{v}\| := \|\mathbf{u}\| \|\mathbf{v}\| \sin \varphi,
> $$
>
> with $\|\mathbf{u}\|$ and $\|\mathbf{v}\|$ the length of $\mathbf{u}$ and $\mathbf{v}$ and the $\varphi$ the angle between $\mathbf{u}$ and $\mathbf{v}$. Defining the area of a parallelogram span by $\mathbf{u}$ and $\mathbf{v}$. The normal direction of the surface is obtained by not taking the length of the cross product.
The scalar and cross product can be combined obtaining a parallelepiped spanned by three 3-dimensional vectors.
> *Definition*: the Euclidean scalar triple of $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{R}^3$ is given by
>
> $$
> \langle \mathbf{u}, \mathbf{v}, \mathbf{w} \rangle := \langle \mathbf{u}, \mathbf{v} \times \mathbf{w} \rangle,
> $$
>
> defining the volume of a parallelepiped spanned by $\mathbf{u}, \mathbf{v}$ and $\mathbf{w}$.
Let $J$ be a $3 \times 3$ matrix given by $J = (\mathbf{u}^T, \mathbf{v}^T, \mathbf{w}^T)$, the Euclidean scalar product may also be defined as
$$
\langle \mathbf{u}, \mathbf{v}, \mathbf{w} \rangle = \det (J),
$$
with $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{R}^3$. We also have that
$$
\langle \mathbf{u}, \mathbf{v}, \mathbf{w} \rangle^2 = \det (J^TJ).
$$

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# Kerr geometry

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# Relativistic formalism of mechanics
From now on, we refer to space and time as spacetime.
## Fundamental assumptions
> *Postulate 1*: spacetime is continuous.
Implying that there is no fundamental limit to the precision of measurements of spatial positions, velocities and time intervals.
> *Postulate 2*: there exists a [neighbourhood]() in spacetime in which the axioms of [Euclidean]() geometry hold.
A reformulation of the postulate in the Newtonian formalism compatible with the new formulation.
> *Postulate 3*: all physical axioms have the same form in all inertial frames.
This principle is dependent on the definition of an inertial frame, which in my view is not optimal. It will have to be improved.
> *Principle 1*: spacetime is not instantaneous.
Implying that there exists a maximum speed with which information can travel.
> *Axiom 1*: spacetime is represented by a torsion-free pseudo Riemannian manifold $M$ with 3 spacial dimensions and 1 time dimension.
Torsion-free means that $\mathbf{T} = \mathbf{0}$, the [torsion tensor]() is always zero.
## Lorentz transformations
Will be added later.
## Results from the fundamental assumptions
> *Theorem 1*: let $\bm{g} \in \Gamma(\mathrm{TM})$ be the pseudo Riemannian inner product on $\mathrm{TM}$, then it follows that from [Hamilton's principle]() that the covariant derivative is equal to zero:
>
> $$
> \forall i \in \{1, 2, 3, 4\}: D_i \bm{g} = \mathbf{0},
> $$
>
> which is called *metric compatibility*.
??? note "*Proof*:"
Will be added later.
A linear connection $\nabla$ on a torsion-free pseudo Riemannian manifold with metric compatibility is called the **Levi-Civita connection** with its linear connection symbols denoted as the **Christoffel symbols**.
> *Theorem 2*: the Christoffel symbols $\Gamma_{ij}^k$ (of a Levi-Civita connection) are covariantly symmetric
>
> $$
> \Gamma_{ij}^k = \Gamma_{ji}^k,
> $$
>
> for all $(i,j,k) \in \{1,2,3,4\}^3$, and may be given by
>
> $$
> \Gamma_{ij}^k = \frac{1}{2} g^{kl} (\partial_i g_{ij} + \partial_j g_{il} - \partial_l g_{ij}),
> $$
>
> for all $\bm{g} = g_{ij} dx^i \otimes dx^j \in \Gamma(\mathrm{TM})$.
??? note "*Proof*:"
Will be added later.
Similarly, we have the following.
> *Proposition 1*: let $\mathbf{R}: \Gamma(\mathrm{T^*M}) \times \Gamma(\mathrm{TM})^3 \to F$ be the Riemann curvature tensor on a manifold $M$ over a field $F$, defined under the Levi-Civita connection. Then it may be decomposed by
>
> $$
> \mathbf{R} = \frac{1}{8} R^i_{jkl} (\partial_i \wedge dx^j) \vee (dx^k \wedge dx^l).
> $$
??? note "*Proof*:"
Will be added later.
Such that $R^i_{jkl}$ has a dimension of
$$
\frac{4^2 (4^2 - 1)}{12} = 20.
$$
## Curvature
> *Definition 1*: let $\mathbf{W}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to F$ denote the **Ricci tensor** which is defined as
>
> $$
> \begin{align*}
> \mathbf{W} &= \frac{1}{2} R_{ijk}^k dx^i \vee dx^j,\\
> &= \frac{1}{2} W_{ij} dx^i \vee dx^j,
> \end{align*}
> $$
>
> with $R_{ijk}^k$ the contracted Riemann holor and let $W$ be the **Ricci scalar** be defined as $W = W_{ij} g^{ij}$ with $g^{ij}$ the dual metric holor.
The Ricci tensor and scalar are normally denoted by the symbol $R$ but this would impose confusion with the curvature tensor, therefore it has been chosen to assign symbol $W$ to the Ricci tensor and scalar.
The **Ricci tensor** is a contraction (simplification) of the Riemann curvature tensor. It provides a way to summarize the curvature of a manifold by focusing on how volumes change in the presence of curvature. The **Ricci scalar** summarizes the curvature information contained in the **Ricci tensor**.
> *Definition 2*: let $\mathbf{G}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to F$ denote the **Einstein tensor** which is defined as
>
> $$
> \mathbf{G} = \mathbf{W} - \frac{1}{2} W \bm{g},
> $$
>
> with $\mathbf{W}$ the Ricci tensor, $\bm{g}$ the metric tensor and $W$ the Ricci scalar.
The **Einstein tensor** encapsulates the curvature of the manifold while satisfying the posed conditions (Lovelock's theorem). Such as the following proposition.
> *Proposition 2*: the Einstein tensor $\mathbf{G}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to F$ has the following properties
>
> 1. $\mathbf{G} = G_{|ij|} dx^i \vee dx^j$,
> 2. $D_i \mathbf{G} = 0$.
??? note "*Proof*:"
Will be added later.
## Energy and momentum
> *Definition 3*: let $\mathbf{T}: \Gamma(\mathrm{T^*M}) \times \Gamma(\mathrm{T^*M}) \to F$ denote the **energy momentum tensor** which is defined by the following properties,
>
> 1. $\mathbf{T} = T^{|ij|} \partial_i \vee \partial_j \in \bigvee^2(\mathrm{TM})$,
> 2. $D_i \mathbf{T} = 0$.
Property 1. is a result of the zero torsion axiom and property 2. is the demand of conservation of energy and momentum.
The **energy momentum tensor** describes the matter distribution at each event in spacetime. It acts as a *source* term.
## Einstein field equations
> *Axiom 2*: the Einstein tensor $\mathbf{G}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to F$ relates to the energy momentum tensor $\mathbf{T}: \Gamma(\mathrm{T^*M}) \times \Gamma(\mathrm{T^*M}) \to F$ by
>
> $$
> \mathbf{G} + \Lambda \bm{g} = \kappa \mathbf{T},
> $$
>
> with $\kappa = \frac{8 \pi G}{c^4}$ and $\Lambda, G$ the cosmological and gravitational constants respectively.
This equation (these equations) relate the geometry of spacetime to the distribution of matter within it. For a given $\mathbf{T}$ the system of equations can solve for $\bm{g}$ and vice versa.

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# Schwarzschild geometry
## Spherical symmetry
A metric that is time-reversal and time-translation invariant is said to be **static**.
> *Lemma 1*: a static, spherically symmetric metric tensor $\bm{g}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to F$ must be of the form
>
> $$
> \bm{g} = A(r) dr \otimes dr + r^2 (\sin^2 (\varphi) d\theta \otimes d\theta + d\varphi \otimes d\varphi) - B(r) dt \otimes dt,
> $$
>
> for all $(r, \theta, \varphi, t) \in \mathbb{R}^4$ with $A,B: r \mapsto A(r),B(r)$.
??? note "*Proof*:"
Will be added later.
Reducing the determination of the metric to only two functions $A$ and $B$.
## Exterior solution
Outside of the mass distribution the energy-momentum tensor vanishes, so we can impose $\mathbf{W} = \mathbf{0}$. Then, by imposing the weak field limit we have the following.
> *Principle 1*: a metric outside a static, spherically symmetric mass distribution is described by the **Schwarzschild metric**
>
> $$
> \bm{g} = \Big(1 - \frac{2 G M}{c^2 r}\Big)^{-1} dr \otimes dr + r^2 (\sin^2 (\varphi) d\theta \otimes d\theta + d\varphi \otimes d\varphi) - c^2 \Big(1 - \frac{2 G M}{c^2 r} \Big) dt \otimes dt,
> $$
>
> for all $(r, \theta, \varphi, t) \in \mathbb{R}^4$ with $G$ the gravitational constant and $M$ the mass of the spherically symmetric mass distribution.
??? note "*Derivation*:"
Will be added later.
Notice that for $r_s = \frac{2 G M}{c^2}$ the metric with these coordinates is not defined. This radius is called the **Schwarzschild radius**.
> *Theorem 1 (Birkhoff's theorem)*: the Schwarzschild metric is the only spherically symmetric solution, outside a spherical mass distribution.
??? note "*Proof*:"
Will be added later.
Note that static is automatically implied by spherical symmetry. An important consequence of the theorem is that a purely radially pulsating star cannot emit gravitational radiation, because outside of this star such gravitational radiation would amount to a time-dependent spherically symmetric spacetime geometry in (approximate) vacuum, which, according to the Birkhoff’s theorem, cannot be consistent with Einstein’s field equations.

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# Wave geometry