luc/notes
1
0
Fork 0
Code Issues Pull requests Activity

port from mathematics-physics notes

Browse source
This commit is contained in:
Luc Bijl 2025-08-26 15:48:53 +02:00
parent a4e106ce02
commit c009ea53f0
124 changed files with 13224 additions and 0 deletions
Download patch file Download diff file Expand all files Collapse all files

32
docs/physics/mathematical-physics/error-analysis/formalism.md Normal file
View file

@ -0,0 +1,32 @@
# 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.

95
docs/physics/mathematical-physics/error-analysis/maximum-error.md Normal file
View file

@ -0,0 +1,95 @@
# 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.

149
docs/physics/mathematical-physics/error-analysis/standard-error.md Normal file
View file

@ -0,0 +1,149 @@
# 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.

59
docs/physics/mathematical-physics/signal-analysis/amplitude-modulation.md Normal file
View file

@ -0,0 +1,59 @@
# 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.

84
docs/physics/mathematical-physics/signal-analysis/discrete-fourier-transform.md Normal file
View file

@ -0,0 +1,84 @@
# 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
>
> $$
> \begin{align*}
> &f_N[m] = \mathcal{DF}^{-1}[F_N[k]], \quad &\forall m \in \{0, \dots, N - 1\}, \\
> &F_N[k] = \mathcal{DF}[f[m]], \quad &\forall k \in \{0, \dots, N - 1\}.
> \end{align*}
> $$

143
docs/physics/mathematical-physics/signal-analysis/fourier-series.md Normal file
View file

@ -0,0 +1,143 @@
# 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.

136
docs/physics/mathematical-physics/signal-analysis/fourier-transform.md Normal file
View file

@ -0,0 +1,136 @@
# 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>

116
docs/physics/mathematical-physics/signal-analysis/signal-filters.md Normal file
View file

@ -0,0 +1,116 @@
# 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.

220
docs/physics/mathematical-physics/signal-analysis/signals.md Normal file
View file

@ -0,0 +1,220 @@
# 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 $k \in \mathbb{Z}$.

213
docs/physics/mathematical-physics/signal-analysis/systems.md Normal file
View file

@ -0,0 +1,213 @@
# 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.

72
docs/physics/mathematical-physics/vector-analysis/curl.md Normal file
View file

@ -0,0 +1,72 @@
# 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.

88
docs/physics/mathematical-physics/vector-analysis/curves.md Normal file
View file

@ -0,0 +1,88 @@
# 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 $\phi \in \mathbb{R}$.

300
docs/physics/mathematical-physics/vector-analysis/curvilinear-coordinates.md Normal file
View file

@ -0,0 +1,300 @@
# 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 $h_1 = 1, h_2 = r \sin \varphi, h_3 = r$.

133
docs/physics/mathematical-physics/vector-analysis/divergence.md Normal file
View file

@ -0,0 +1,133 @@
# 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.

69
docs/physics/mathematical-physics/vector-analysis/gradient.md Normal file
View file

@ -0,0 +1,69 @@
# 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.

176
docs/physics/mathematical-physics/vector-analysis/vector-operators.md Normal file
View file

@ -0,0 +1,176 @@
# 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.

64
docs/physics/mathematical-physics/vector-analysis/vectors.md Normal file
View file

@ -0,0 +1,64 @@
# 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
$$
\langle \mathbf{u}, \mathbf{v}, \mathbf{w} \rangle^2 = \det (J^TJ).
$$