diff --git a/docs/mathematics/topology/fiber-bundles.md b/docs/mathematics/differential-geometry/topological-notions.md similarity index 95% rename from docs/mathematics/topology/fiber-bundles.md rename to docs/mathematics/differential-geometry/topological-notions.md index 1ee96e2..9d5ca02 100644 --- a/docs/mathematics/topology/fiber-bundles.md +++ b/docs/mathematics/differential-geometry/topological-notions.md @@ -1,4 +1,6 @@ -# Fiber bundles +# Topological notions + +## Fiber bundles Let $X$ be a manifold over a field $F$. @@ -54,4 +56,4 @@ with $T^{ij}_k \in \mathbb{K}$ holors of $\mathbf{T}$. Furthermore, we have a ba > > of the tensor fiber bundle $\mathscr{B}$. -Therefore, a tensor field assigns a tensor fiber (or tensor) to each point on a section of the manifold. These tensors may vary smoothly along the section of the manifold. \ No newline at end of file +Therefore, a tensor field assigns a tensor fiber (or tensor) to each point on a section of the manifold. These tensors may vary smoothly along the section of the manifold. diff --git a/docs/mathematics/linear-algebra/tensors/tensor-formalism.md b/docs/mathematics/linear-algebra/tensors/tensor-formalism.md index 5b246d0..bd62674 100644 --- a/docs/mathematics/linear-algebra/tensors/tensor-formalism.md +++ b/docs/mathematics/linear-algebra/tensors/tensor-formalism.md @@ -45,7 +45,7 @@ $$ ## Outer product -> *Definition 3*: the outer product $f \otimes g: X \times Y \to F$ of two scalar functions $f: X \to F$ and $g: Y \to F$ is defined as +> *Definition 3*: the **outer product** $f \otimes g: X \times Y \to F$ of two scalar functions $f: X \to F$ and $g: Y \to F$ is defined as > > $$ > (f \otimes g)(x,y) = f(x) g(y), @@ -133,9 +133,45 @@ By definition tensors are basis independent. Holors are basis dependent. We have from theorem 2 that the outer product of two tensors yields another tensor, with ranks adding up. +## Interior product + +> *Definition 5*: the **left interior product** $f \llcorner \alpha: Y \to F$ of a scalar function $f:X \times Y \to F$ and a scalar $\alpha \in X$ is defined as +> +> $$ +> (f \llcorner \alpha)(y) = f(\alpha,y) \qquad \forall y \in Y, +> $$ +> +> and the **right interior product** $f \lrcorner \alpha: X \to F$ for a scalar $\alpha \in Y$ is defined as +> +> $$ +> (f \lrcorner \alpha)(x) = f(x,\alpha) \qquad \forall x \in X. +> $$ + +Note that neither interior product is associative, commutatitive and only distributive in the field addition. + +With the interior product we can partially contract a tensor (i.e. reduce its basis) with a (co)vector. Consider $\mathbf{T} \in \mathscr{T}^2_0(V)$ then + +$$ + \begin{align*} + \mathbf{T} \llcorner \mathbf{\hat v} &= T^{ij} \mathbf{k}(\mathbf{\hat v}, \mathbf{e}_i) \mathbf{e}_j,\\ + &= v_i T^{ij} \mathbf{e}_j. + \end{align*} +$$ + +or similarly + +$$ + \begin{align*} + \mathbf{T} \lrcorner \mathbf{\hat v} &= T^{ij} \mathbf{e}_i \mathbf{k}(\mathbf{\hat v}, \mathbf{e}_j),\\ + &= v_j T^{ij} \mathbf{e}_i. + \end{align*} +$$ + +for all $\mathbf{\hat v} \in V^*$. + ## Inner product -> *Definition 5*: an **inner product** on $V$ is a bilinear mapping $\bm{g}: V \times V \to F$ which satisfies +> *Definition 6*: an **inner product** on $V$ is a bilinear mapping $\bm{g}: V \times V \to F$ which satisfies > > 1. for all $\mathbf{u}, \mathbf{v} \in V: \; \bm{g}(\mathbf{u}, \mathbf{v}) = \overline{\bm{g}}(\mathbf{v}, \mathbf{u}),$ > 2. for all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$ and $\lambda, \mu \in F: \;\bm{g}(\mathbf{u}, \lambda \mathbf{v} + \mu \mathbf{w}) = \lambda \bm{g}(\mathbf{u}, \mathbf{v}) + \mu \bm{g}(\mathbf{u}, \mathbf{w}),$ @@ -144,7 +180,7 @@ We have from theorem 2 that the outer product of two tensors yields another tens It may be observed that $\bm{g} \in \mathscr{T}_2^0$. Unlike the Kronecker tensor, the existence of an inner product is never implied. -> *Definition 6*: let $G$ be the Gram matrix with its components $G \overset{\text{def}}= (g_{ij})$ defined as +> *Definition 7*: let $G$ be the Gram matrix with its components $G \overset{\text{def}}= (g_{ij})$ defined as > > $$ > g_{ij} = \bm{g}(\mathbf{e}_i, \mathbf{e}_j). @@ -227,7 +263,7 @@ $$ with $u^j = g^{ij} u_i$. -> *Definition 7*: the basis $\{\mathbf{e}_i\}$ of $V$ induces a **reciprocal basis** $\{\mathbf{g}^{-1}(\mathbf{\hat e}^i)\}$ of $V$ given by +> *Definition 8*: the basis $\{\mathbf{e}_i\}$ of $V$ induces a **reciprocal basis** $\{\mathbf{g}^{-1}(\mathbf{\hat e}^i)\}$ of $V$ given by > > $$ > \mathbf{g}^{-1}(\mathbf{\hat e}^i) = g^{ij} \mathbf{e}_j. diff --git a/docs/mathematics/linear-algebra/tensors/tensor-symmetries.md b/docs/mathematics/linear-algebra/tensors/tensor-symmetries.md index 2b24de2..5b8629e 100644 --- a/docs/mathematics/linear-algebra/tensors/tensor-symmetries.md +++ b/docs/mathematics/linear-algebra/tensors/tensor-symmetries.md @@ -1,6 +1,6 @@ # Tensor symmetries -We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n$, a corresponding dual space $V^*$ with a basis $\{\mathbf{\hat e}^i\}$, a field F and a pseudo inner product $\bm{g}$ on $V$. +We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n$, a corresponding dual space $V^*$ with a basis $\{\mathbf{\hat e}^i\}$, a field $F$ and a pseudo inner product $\bm{g}$ on $V$. ## Symmetric tensors diff --git a/docs/mathematics/linear-algebra/tensors/tensor-transformations.md b/docs/mathematics/linear-algebra/tensors/tensor-transformations.md index e535f48..f33b2e9 100644 --- a/docs/mathematics/linear-algebra/tensors/tensor-transformations.md +++ b/docs/mathematics/linear-algebra/tensors/tensor-transformations.md @@ -12,7 +12,7 @@ so that $\mathbf{\hat e}^i = A^i_j \mathbf{\hat f}^j$. ## Transformation of tensors -Recall from the section of [tensor-formalism]() that a holor depends on the chosen basis, but the corresponding tensor itself does not. This implies that holors transform in a particular way under a change of basis, which is characteristic for tensors. +Recall from the section of [tensor-formalism](tensor-formalism.md) that a holor depends on the chosen basis, but the corresponding tensor itself does not. This implies that holors transform in a particular way under a change of basis, which is characteristic for tensors. > *Theorem 1*: let $\mathbf{T} \in \mathscr{T}^p_q(V)$ be a tensor with $p=q=1$ without loss of generality and $B = A^{-1}$. Then $\mathbf{T}$ may be decomposed into > @@ -26,12 +26,20 @@ Recall from the section of [tensor-formalism]() that a holor depends on the chos > with the holors related by > > $$ -> \overline T^i_j = B^i_k A^j_l T^k_l. +> \overline T^i_j = B^i_k A^l_j T^k_l. > $$ ??? note "*Proof*:" - Will be added later. + We have + + $$ + \begin{align*} + \mathbf{T} &= T^i_j B^k_i \mathbf{f}_k \otimes A^j_l \mathbf{\hat f}^l,\\ + &= B^k_i A^j_l T^i_j \mathbf{f}_k \otimes \mathbf{\hat f}^l,\\ + &= \overline T^i_j \mathbf{f}_i \otimes \mathbf{\hat f}^j. + \end{align*} + $$ The homogeneous nature of the tensor transformation implies that a holor equation of the form $T^i_j = 0$ holds relative to any basis if it holds relative to a particular one. @@ -54,7 +62,17 @@ The homogeneous nature of the tensor transformation implies that a holor equatio ??? note "*Proof*:" - Will be added later. + We have + + $$ + \begin{align*} + \bm{\mu} &= \mu_{i_1\dots i_n} A^{i_1}_{j_1} \mathbf{\hat f}^{j_1} \otimes \dots \otimes A^{i_n}_{j_n} \mathbf{\hat f}^{j_n},\\ + &= A^{i_1}_{j_1} \cdots A^{i_n}_{j_n} \mu_{i_1\dots i_n} \mathbf{\hat f}^{j_1} \otimes \dots \otimes \mathbf{\hat f}^{j_n},\\ + &= A^{i_1}_{j_1} \cdots A^{i_n}_{j_n} [i_1,\dots,i_n] \mathbf{\hat f}^{j_1} \otimes \dots \otimes \mathbf{\hat f}^{j_n},\\ + &= \det(A) [j_1,\dots,j_n] \mathbf{\hat f}^{j_1} \otimes \dots \otimes \mathbf{\hat f}^{j_n},\\ + &= \overline \mu_{i_1 \dots i_n} \mathbf{\hat f}^{i_1} \otimes \dots \otimes \mathbf{\hat f}^{i_1}. + \end{align*} + $$ Then $\det(A)$ is the volume scaling factor of the transformation with $A$. So that if $\bm{\mu}(\mathbf{e}_1, \dots, \mathbf{e}_n) = 1$, then $\bm{\mu}(\mathbf{f}_1, \dots, \mathbf{f}_n) = \det(A).$ @@ -75,10 +93,6 @@ Then $\det(A)$ is the volume scaling factor of the transformation with $A$. So t > > then $\mu_{i_1 \dots i_n} = \overline \mu_{i_1 \dots i_n} = [i_1, \dots, i_n]$ is an invariant holor. -??? note "*Proof*:" - - Will be added later. - ## Transformation of Levi-Civita form > *Theorem 3*: let $\bm{\epsilon} \in \bigwedge_n(V)$ be the Levi-Civita tensor with @@ -94,4 +108,4 @@ Then $\det(A)$ is the volume scaling factor of the transformation with $A$. So t ??? note "*Proof*:" - Will be added later. \ No newline at end of file + Follows directly from the definition $\bm{\epsilon} = \sqrt{g} \bm{\mu}$. diff --git a/docs/mathematics/linear-algebra/tensors/volume-forms.md b/docs/mathematics/linear-algebra/tensors/volume-forms.md index 99530be..963c7d5 100644 --- a/docs/mathematics/linear-algebra/tensors/volume-forms.md +++ b/docs/mathematics/linear-algebra/tensors/volume-forms.md @@ -1,6 +1,6 @@ # Volume forms -We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n,$ a corresponding dual space $V^*$ with a basis $\{\mathbf{\hat e}^i\}_{i=1}^n$ and a pseudo inner product $\bm{g}$ on $V.$ +We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n$, a corresponding dual space $V^*$ with a basis $\{\mathbf{\hat e}^i\}_{i=1}^n$, a field $F$ and a pseudo inner product $\bm{g}$ on $V$. ## n-forms @@ -12,7 +12,7 @@ We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim > > then $\bm{\mu}$ is the **unit volume form** with respect to the basis $\{\mathbf{e}_i\}$. -Note that $\dim \bigwedge_n(V) = 1$ and consequently if $\bm{\mu}_1, \bm{\mu}_2 \in \bigwedge_n(V) \backslash \{\mathbf{0}\}$, then $\bm{\mu}_1 = \lambda \bm{\mu}_2$ with $\lambda \in \mathbb{K}$. +Note that $\dim \bigwedge_n(V) = 1$ and consequently if $\bm{\mu}_1, \bm{\mu}_2 \in \bigwedge_n(V) \backslash \{\mathbf{0}\}$, then $\bm{\mu}_1 = \lambda \bm{\mu}_2$ with $\lambda \in F$. > *Proposition 1*: the unit volume form $\bm{\mu} \in \bigwedge_n(V) \backslash \{\mathbf{0}\}$ may be given by > @@ -27,7 +27,31 @@ Note that $\dim \bigwedge_n(V) = 1$ and consequently if $\bm{\mu}_1, \bm{\mu}_2 ??? note "*Proof*:" - Will be added later. + Let $\pi = [\pi(1),\dots,\pi(n)]$ be any permutation of the set $\{1,\dots,n\}$, the unit volume form $\bm{\mu}$ is defined as + + $$ + \bm{\mu}(\mathbf{e}_{\pi(1)},\dots,\mathbf{e}_{\pi(2)}) = \mathrm{sign}(\pi), + $$ + + thus + + $$ + \bm{\mu} = \mu_{i_1\dots i_n} \mathbf{\hat e}^{i_1} \otimes \dots \otimes \mathbf{\hat e}^{i_n}. + $$ + + Furthermore $\mathscr{A}(\bm{\mu}) = \bm{\mu}$. Then + + $$ + \bm{\mu} = \mu_{i_1\dots i_n} \frac{1}{n!} \mathbf{\hat e}^{i_1} \wedge \dots \wedge \mathbf{\hat e}^{i_n}, + $$ + + and going back to the definition only requires us to consider + + $$ + \bm{\mu} = \mathbf{\hat e}^{1} \wedge \dots \wedge \mathbf{\hat e}^{n}, + $$ + + such that $\mu_{i_1\dots i_n} = [i_1,\dots,i_n]$. The normalisation of the unit volume form $\bm{\mu}$ requires a basis. Consequently, the identification $\mu_{i_1 \dots i_n} = [i_1, \dots, i_n]$ holds only relative to the basis. @@ -47,7 +71,7 @@ $$ by convention, to resolve ambiguity with respect to the meaning of $\mu_{i_1 \dots i_n}$ without using another symbol or extra accents. -Using theorem 2 in the section of [tensor symmetries]() we may state the following. +Using theorem 2 in the section of [tensor symmetries](tensor-symmetries.md) we may state the following. > *Proposition 2*: let $(V, \bm{\mu})$ be a vector space with an oriented volume form, then we have > @@ -59,7 +83,15 @@ Using theorem 2 in the section of [tensor symmetries]() we may state the followi ??? note "*Proof*:" - Will be added later. + We have + + $$ + \begin{align*} + \bm{\mu}(\mathbf{v}_1,\dots,\mathbf{v}_n) &= \mu_{i_1\dots i_n} \mathbf{k}(\mathbf{\hat e}^{i_1},\mathbf{v}_1) \cdots \mathbf{k}(\mathbf{\hat e}^{i_n},\mathbf{v}_n),\\ + &= [i_1,\dots,i_n] \mathbf{k}(\mathbf{\hat e}^{i_1},\mathbf{v}_1) \cdots \mathbf{k}(\mathbf{\hat e}^{i_n},\mathbf{v}_n),\\ + &= \det\big(\mathbf{k}(\mathbf{\hat e}^i,\mathbf{v}_j)\big). + \end{align*} + $$ Which reveals the role of the Kronecker tensor and thus the role of the dual space in the definition of $\bm{\mu}$. We may also conclude that an oriented volume $\bm{\mu} \in \bigwedge_n(V)$ on a vector space $V$ does not require an inner product. @@ -73,9 +105,9 @@ From proposition 2 it may also be observed that within a geometrical context the > \bm{\mu} \lrcorner \mathbf{u}_1 \lrcorner \dots \lrcorner \mathbf{u}_k(\mathbf{v}_{k+1}, \dots, \mathbf{v}_n) = \bm{\mu}(\mathbf{u}_1, \dots, \mathbf{u}_k, \mathbf{v}_{k+1}, \dots, \mathbf{v}_n), > $$ > -> for all $\mathbf{v}_{k+1}, \dots, \mathbf{v}_n \in V$ with $\lrcorner$ the insert operator. +> for all $\mathbf{v}_{k+1}, \dots, \mathbf{v}_n \in V$. -It follows that $(n-k)$-form $\bm{\mu} \lrcorner \mathbf{u}_1 \lrcorner \dots \lrcorner \mathbf{u}_k \in \bigwedge_{n-k}(V)$ can be written as +It follows that the $(n-k)$-form $\bm{\mu} \lrcorner \mathbf{u}_1 \lrcorner \dots \lrcorner \mathbf{u}_k \in \bigwedge_{n-k}(V)$ can be written as $$ \begin{align*} @@ -120,7 +152,16 @@ then we have $\epsilon_{i_1 \dots i_n} = \sqrt{g} \mu_{i_1 \dots i_n}$ and $\eps ??? note "*Proof*:" - Will be added later. + The reciprocal Levi-Civita tensor may be written as + + $$ + \begin{align*} + \mathbf{g}(\bm{\epsilon}) &= \sqrt{g} \mathbf{g}(\mathbf{\hat e}^1) \wedge \dots \wedge \mathbf{g}(\mathbf{\hat e}_n),\\ + &= \sqrt{g} g^{1i_1} \mathbf{e}_{i_1} \wedge \dots \wedge g^{ni_n} \mathbf{e}_{i_n},\\ + &= \sqrt{g} \det (G^{-1}) \mathbf{e}_1 \wedge \dots \wedge \mathbf{e}_n,\\ + &= \frac{1}{\sqrt{g}} \mathbf{e}_1 \wedge \dots \wedge \mathbf{e}_n. + \end{align*} + $$ We may decompose the reciprocal Levi-Civita tensor by @@ -128,4 +169,4 @@ $$ \mathbf{g}(\bm{\epsilon}) = \epsilon^{i_1 \dots i_n} \mathbf{e}_{i_1} \otimes \cdots \otimes \mathbf{e}_{i_n} = \epsilon^{|i_1 \dots i_n|} \mathbf{e}_{i_1} \wedge \cdots \wedge \mathbf{e}_{i_n}, $$ -then we have $\epsilon^{i_1 \dots i_n} = \frac{1}{\sqrt{g}} \mu^{i_1 \dots i_n}$ and $\epsilon^{|i_1 \dots i_n|} = \frac{1}{\sqrt{g}}$. \ No newline at end of file +then we have $\epsilon^{i_1 \dots i_n} = \frac{1}{\sqrt{g}} \mu^{i_1 \dots i_n}$ and $\epsilon^{|i_1 \dots i_n|} = \frac{1}{\sqrt{g}}$. diff --git a/docs/physics/relativistic-mechanics/relativistic-formalism.md b/docs/physics/relativistic-mechanics/relativistic-formalism.md deleted file mode 100644 index 1815e5b..0000000 --- a/docs/physics/relativistic-mechanics/relativistic-formalism.md +++ /dev/null @@ -1,140 +0,0 @@ -# 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. \ No newline at end of file diff --git a/docs/physics/spacetime/wave-geometry.md b/docs/physics/spacetime/wave-geometry.md deleted file mode 100644 index 4a263d7..0000000 --- a/docs/physics/spacetime/wave-geometry.md +++ /dev/null @@ -1 +0,0 @@ -# Wave geometry \ No newline at end of file diff --git a/docs/physics/thermodynamics/equilibrium-formalism.md b/docs/physics/thermodynamics/equilibrium-formalism.md new file mode 100644 index 0000000..56c4a47 --- /dev/null +++ b/docs/physics/thermodynamics/equilibrium-formalism.md @@ -0,0 +1,9 @@ +# Formalism of equilibrium thermodynamics + +Thermodynamics, the study of the behaviour of physical systems in terms of their macroscopic observables. A physical theory that perhaps is closest to our macroscopic perception of the physical world. + +## References + +* A.M. Steane, *Thermodynamics*, Oxford University Press, 2017. +* Dep. of Physics Oxford University, *Basics of Thermodynamics*. +* D.A. Lavis and R. Frigg, *The Fundamentals of Thermodynamics*, Springer, 2025. diff --git a/docs/physics/thermodynamics/non-equilibrium-formalism.md b/docs/physics/thermodynamics/non-equilibrium-formalism.md new file mode 100644 index 0000000..e69de29 diff --git a/mkdocs.yml b/mkdocs.yml index a2e946e..f4e96a6 100755 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -74,8 +74,8 @@ nav: - 'Welcome': index.md - 'Mathematics': - mathematics/index.md - - 'Logic': mathematics/logic.md - - 'Set theory': + - 'Logic': mathematics/logic.md #Extend + - 'Set theory': # Update - 'Sets': mathematics/set-theory/sets.md - 'Relations': mathematics/set-theory/relations.md - 'Maps': mathematics/set-theory/maps.md @@ -84,10 +84,12 @@ nav: - 'Recursion and induction': mathematics/set-theory/recursion-induction.md - 'Cardinalities': mathematics/set-theory/cardinalities.md - 'Additional axioms': mathematics/set-theory/additional-axioms.md - - 'Number theory': - - 'Integer arithmetic': mathematics/number-theory/integer-arithmetic.md - - 'Modular arithmetic': mathematics/number-theory/modular-arithmetic.md + - 'Number theory': + - 'Integer arithmetic': mathematics/number-theory/integer-arithmetic.md #Add + - 'Modular arithmetic': mathematics/number-theory/modular-arithmetic.md #Add - 'Complex numbers': mathematics/number-theory/complex-numbers.md + #- 'Group theory': + #- 'Ring theory': - 'Linear algebra': - 'Systems of linear equations': mathematics/linear-algebra/systems-of-linear-equations.md - 'Matrices': @@ -104,8 +106,10 @@ nav: - 'Tensors': - 'Tensor formalism': mathematics/linear-algebra/tensors/tensor-formalism.md - 'Tensor symmetries': mathematics/linear-algebra/tensors/tensor-symmetries.md - - 'Volume forms': mathematics/linear-algebra/tensors/volume-forms.md + - 'Volume forms': mathematics/linear-algebra/tensors/volume-forms.md #Update - 'Tensor transformations': mathematics/linear-algebra/tensors/tensor-transformations.md + #- 'Analysis': + #- 'Complex analysis': - 'Functional analysis': - 'Metric spaces': - 'Metric spaces': mathematics/functional-analysis/metric-spaces/metric-spaces.md @@ -133,8 +137,9 @@ nav: - 'Laguerre polynomials': mathematics/functional-analysis/inner-product-spaces/polynomials/laguerre-polynomials.md - 'Representations of functionals': mathematics/functional-analysis/inner-product-spaces/representations-of-functionals.md - 'Operator classes': mathematics/functional-analysis/inner-product-spaces/operator-classes.md - - 'Topology': - - 'Fiber bundles': mathematics/topology/fiber-bundles.md + #- 'Measure theory': + #- 'Probability theory': + #- 'Statistics': - 'Calculus': - 'Limits': mathematics/calculus/limits.md - 'Continuity': mathematics/calculus/continuity.md @@ -156,13 +161,17 @@ nav: - 'Taylor polynomials': mathematics/multivariable-calculus/taylor-polynomials.md - 'Extrema': mathematics/multivariable-calculus/extrema.md - 'Integration': mathematics/multivariable-calculus/integration.md + #- 'Vector calculus': + #- 'Ordinary difference equations': - 'Ordinary differential equations': - 'First order differential equations': mathematics/ordinary-differential-equations/first-order-ode.md - 'Second order differential equations': mathematics/ordinary-differential-equations/second-order-ode.md - 'Systems of linear differential equations': mathematics/ordinary-differential-equations/systems-of-linear-ode.md - 'The Laplace transform': mathematics/ordinary-differential-equations/laplace-transform.md + #- 'Partial differential equations': - 'Differential geometry': - 'Differential manifolds': mathematics/differential-geometry/differential-manifolds.md + - 'Topological notions': mathematics/differential-geometry/topological-notions.md - 'Tangent spaces': mathematics/differential-geometry/tangent-spaces.md - 'Transformations': mathematics/differential-geometry/transformations.md - 'Lengths and volumes': mathematics/differential-geometry/lengths-and-volumes.md @@ -175,25 +184,28 @@ nav: - physics/index.md - 'Classical Mechanics': - 'Newtonian mechanics': - - 'Newtonian formalism': physics/classical-mechanics/newtonian-mechanics/newtonian-formalism.md - - 'Momentum': physics/classical-mechanics/newtonian-mechanics/momentum.md - - 'Energy': physics/classical-mechanics/newtonian-mechanics/energy.md - - 'Rotation': physics/classical-mechanics/newtonian-mechanics/rotation.md - - 'Particle systems': physics/classical-mechanics/newtonian-mechanics/particle-systems.md + - 'Newtonian formalism': physics/classical-mechanics/newtonian-mechanics/newtonian-formalism.md #Update + - 'Momentum': physics/classical-mechanics/newtonian-mechanics/momentum.md #Update + - 'Energy': physics/classical-mechanics/newtonian-mechanics/energy.md #Update + - 'Rotation': physics/classical-mechanics/newtonian-mechanics/rotation.md #Update + - 'Particle systems': physics/classical-mechanics/newtonian-mechanics/particle-systems.md #Update - 'Lagrangian mechanics': - - 'Lagrangian formalism': physics/classical-mechanics/lagrangian-mechanics/lagrangian-formalism.md - - 'Lagrange equations': physics/classical-mechanics/lagrangian-mechanics/lagrange-equations.md - - 'Lagrange generalizations': physics/classical-mechanics/lagrangian-mechanics/lagrange-generalizations.md + - 'Lagrangian formalism': physics/classical-mechanics/lagrangian-mechanics/lagrangian-formalism.md #Update + - 'Lagrange equations': physics/classical-mechanics/lagrangian-mechanics/lagrange-equations.md #Update + - 'Lagrange generalizations': physics/classical-mechanics/lagrangian-mechanics/lagrange-generalizations.md #Update - 'Hamiltonian mechanics': - 'Hamiltonian formalism': physics/classical-mechanics/hamiltonian-mechanics/hamiltonian-formalism.md - - 'Equations of Hamilton': physics/classical-mechanics/hamiltonian-mechanics/equations-of-hamilton.md - - + - 'Equations of Hamilton': physics/classical-mechanics/hamiltonian-mechanics/equations-of-hamilton.md #Update + #- 'Field theory': + - 'Thermodynamics': + - 'Equilibrium formalism': physics/thermodynamics/equilibrium-formalism.md + - 'Non-equilibrium formalism': physics/thermodynamics/non-equilibrium-formalism.md + #- 'Statistical mechanics': - 'Electromagnetism': - 'Electrostatics': physics/electromagnetism/electrostatics.md - 'Magnetostatics': physics/electromagnetism/magnetostatics.md - 'Electromagnetic dynamics': physics/electromagnetism/electromagnetic-dynamics.md - - 'Optics': + - 'Optics': #Update - 'Waves': physics/electromagnetism/optics/waves.md - 'Electromagnetic waves': physics/electromagnetism/optics/electromagnetic-waves.md - 'Reflection and refraction': physics/electromagnetism/optics/reflection-and-refraction.md @@ -201,25 +213,20 @@ nav: - 'Interference': physics/electromagnetism/optics/interference.md - 'Diffraction': physics/electromagnetism/optics/diffraction.md - 'Polarisation': physics/electromagnetism/optics/polarisation.md -# - 'Quantum mechanics': -# - 'Statistical mechanics': + #- 'Quantum mechanics': - 'Spacetime': - 'Special formalism': physics/spacetime/special-formalism.md - 'General formalism': physics/spacetime/general-formalism.md - 'Schwarzschild geometry': physics/spacetime/schwarzschild-geometry.md - 'Kerr geometry': physics/spacetime/kerr-geometry.md - 'Isotropic geometry': physics/spacetime/isotropic-geometry.md -# - 'Relativistic mechanics': -# - 'Relativistic formalism': physics/relativistic-mechanics/relativistic-formalism.md -# - 'Thermodynamics': -# - 'Classical thermodynamics' -# - 'Statistical thermodynamics' - - 'Mathematical physics': - - 'Error analysis': + ## + - 'Mathematical physics': #Should be removed + - 'Error analysis': #Move to statistics - 'Error analysis formalism': physics/mathematical-physics/error-analysis/formalism.md - 'Maximum error': physics/mathematical-physics/error-analysis/maximum-error.md - 'Standard error': physics/mathematical-physics/error-analysis/standard-error.md - - 'Signal analysis': + - 'Signal analysis': #Partly move to functional analysis, partly to ?? - 'Signals': physics/mathematical-physics/signal-analysis/signals.md - 'Fourier series': physics/mathematical-physics/signal-analysis/fourier-series.md - 'Fourier transform': physics/mathematical-physics/signal-analysis/fourier-transform.md @@ -227,7 +234,7 @@ nav: - 'Amplitude modulation': physics/mathematical-physics/signal-analysis/amplitude-modulation.md - 'Signal filters': physics/mathematical-physics/signal-analysis/signal-filters.md - 'Systems': physics/mathematical-physics/signal-analysis/systems.md - - 'Vector analysis': + - 'Vector analysis': #Update and move to vector calculus - 'Vectors': physics/mathematical-physics/vector-analysis/vectors.md - 'Curves': physics/mathematical-physics/vector-analysis/curves.md - 'Curvilinear coordinates': physics/mathematical-physics/vector-analysis/curvilinear-coordinates.md