an update of various things

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 $F$. 
 

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.

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
 

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.
+    Follows directly from the definition $\bm{\epsilon} = \sqrt{g} \bm{\mu}$.

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
 

docs/physics/relativistic-mechanics/relativistic-formalism.md

@@ -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.

docs/physics/spacetime/wave-geometry.md

@@ -1 +0,0 @@
-# Wave geometry

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.

mkdocs.yml

@@ -74,8 +74,8 @@ nav:
   - 'Welcome': index.md
   - 'Mathematics':
     - mathematics/index.md
-    - 'Logic': mathematics/logic.md
+    - 'Logic': mathematics/logic.md #Extend
-    - 'Set theory':
+    - 'Set theory': # Update
       - 'Sets': mathematics/set-theory/sets.md
       - 'Relations': mathematics/set-theory/relations.md
       - 'Maps': mathematics/set-theory/maps.md
@@ -85,9 +85,11 @@ nav:
       - 'Cardinalities': mathematics/set-theory/cardinalities.md
       - 'Additional axioms': mathematics/set-theory/additional-axioms.md
     - 'Number theory': 
-      - 'Integer arithmetic': mathematics/number-theory/integer-arithmetic.md
+      - 'Integer arithmetic': mathematics/number-theory/integer-arithmetic.md #Add
-      - 'Modular arithmetic': mathematics/number-theory/modular-arithmetic.md
+      - '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':
+    #- 'Measure theory':
-      - 'Fiber bundles': mathematics/topology/fiber-bundles.md
+    #- '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
+        - 'Newtonian formalism': physics/classical-mechanics/newtonian-mechanics/newtonian-formalism.md #Update
-        - 'Momentum': physics/classical-mechanics/newtonian-mechanics/momentum.md
+        - 'Momentum': physics/classical-mechanics/newtonian-mechanics/momentum.md #Update
-        - 'Energy': physics/classical-mechanics/newtonian-mechanics/energy.md
+        - 'Energy': physics/classical-mechanics/newtonian-mechanics/energy.md #Update
-        - 'Rotation': physics/classical-mechanics/newtonian-mechanics/rotation.md
+        - 'Rotation': physics/classical-mechanics/newtonian-mechanics/rotation.md #Update
-        - 'Particle systems': physics/classical-mechanics/newtonian-mechanics/particle-systems.md
+        - 'Particle systems': physics/classical-mechanics/newtonian-mechanics/particle-systems.md #Update
       - 'Lagrangian mechanics':
-        - 'Lagrangian formalism': physics/classical-mechanics/lagrangian-mechanics/lagrangian-formalism.md
+        - 'Lagrangian formalism': physics/classical-mechanics/lagrangian-mechanics/lagrangian-formalism.md #Update
-        - 'Lagrange equations': physics/classical-mechanics/lagrangian-mechanics/lagrange-equations.md
+        - 'Lagrange equations': physics/classical-mechanics/lagrangian-mechanics/lagrange-equations.md #Update
-        - 'Lagrange generalizations': physics/classical-mechanics/lagrangian-mechanics/lagrange-generalizations.md
+        - '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':
+    #- 'Quantum mechanics':
-#    - 'Statistical 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
+    - 'Mathematical physics': #Should be removed
-#   - 'Thermodynamics':
+      - 'Error analysis': #Move to statistics
-#     - 'Classical thermodynamics'
-#     - 'Statistical thermodynamics'
-    - 'Mathematical physics': 
-      - 'Error analysis': 
         - '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