Updated and added some parts to differential geometry and tensors.

docs/en/mathematics/differential-geometry/curvature.md

@@ -28,7 +28,7 @@ It then follows from the definition that the curvature operator $\Omega$ can be
 
 ## Curvature tensor
 
-> *Definition 2*: the Riemann curvature tensor $\mathbf{R}: \Gamma(\mathrm{T}^*\mathrm{M}) \times \Gamma(\mathrm{TM})^3 \to \mathbb{K}$ is defined as
+> *Definition 2*: the **Riemann curvature tensor** $\mathbf{R}: \Gamma(\mathrm{T}^*\mathrm{M}) \times \Gamma(\mathrm{TM})^3 \to \mathbb{K}$ is defined as
 >
 > $$
 >   \mathbf{R}(\bm{\omega}, \mathbf{u}, \mathbf{v}, \mathbf{w}) = \mathbf{k}(\bm{\omega}, \Omega(\mathbf{v}, \mathbf{w}) \mathbf{u}), 
@@ -36,3 +36,28 @@ It then follows from the definition that the curvature operator $\Omega$ can be
 >
 > for all $\bm{\omega} \in \Gamma(\mathrm{T}^*\mathrm{M})$ and $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \Gamma(\mathrm{TM})$. 
 
+The Riemann curvature defines the curvature of the differential manifold at a certain point $x \in \mathrm{M}$. 
+
+> *Proposition 2*: let $\mathbf{R}: \Gamma(\mathrm{T}^*\mathrm{M}) \times \Gamma(\mathrm{TM})^3 \to \mathbb{K}$ be the Riemann curvature tensor, with its decomposition given by
+>
+> $$
+>   \mathbf{R} = R^i_{jkl} \partial_i \otimes dx^j \otimes dx^k \otimes dx^l, 
+> $$
+>
+> then we have that its holor is given by
+>
+> $$
+>   R^i_{jkl} = \partial_k \Gamma^i_{jl} + \Gamma^m_{jl} \Gamma^i_{mk} - \partial_k \Gamma^i_{jk} - \Gamma^m_{jk} \Gamma^i_{ml}, 
+> $$
+>
+> for all $(i,j,k,l) \in \{1, \dots, n\}^4$ with $\Gamma^i_{jk}$ denoting the linear connection symbols.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+It may then be observed that $R^i_{jkl} = - R^i_{jlk}$ such that
+
+$$
+    \mathbf{R} = \frac{1}{2} R^i_{jkl} \partial_i \otimes dx^j \otimes (dx^k \wedge dx^l).
+$$

docs/en/mathematics/differential-geometry/lengths-and-volumes.md

@@ -0,0 +1,47 @@
+# Lengths and volumes
+
+Let $\mathrm{M}$ be a differential manifold with $\dim \mathrm{M} = n \in \mathbb{N}$ used throughout the section. Let $\mathrm{TM}$ and $\mathrm{T^*M}$ denote the tangent and cotangent bundle, $V$ and $V^*$ the fiber and dual fiber bundle and $\mathscr{B}$ the tensor fiber bundle.
+
+## Riemannian geometry
+
+> *Definition 1*: the length of a vector $\mathbf{v} \in \Gamma(\mathrm{TM})$ is defined by the norm $\|\cdot\|$ induced by the inner product $\bm{g}$ such that
+>
+> $$
+>   \|\mathbf{v}\| = \sqrt{\bm{g}(\mathbf{v},\mathbf{v})}.
+> $$
+
+In the context of a smooth curve $\mathbf{v}: \mathscr{D}(\mathbf{v}) \to \Gamma(\mathrm{TM}):t \mapsto \mathbf{v}(t)$ parameterized by an open interval $\mathscr{D}(\mathbf{v}) \subset \mathbb{R}$, the length $l_{12}$ of a closed section $[t_1, t_2] \subset \mathbb{R}$ of this curve is given by
+
+$$
+\begin{align*}
+    l_{12} &= \int_{t_1}^{t_2} \|\mathbf{\dot v}(t)\| dt, \\
+           &= \int_{t_1}^{t_2} \sqrt{\bm{g}(\mathbf{\dot v},\mathbf{\dot v})} dt, \\
+           &= \int_{t_1}^{t_2} \sqrt{g_{ij} \dot v^i \dot v^j} dt,
+\end{align*}
+$$
+
+with $\mathbf{\dot v} = \dot v^i \partial_i \in \Gamma(\mathrm{TM})$. 
+
+> *Definition 2*: the volume $V$ span by the vectors $\{\mathbf{v}_i\}_{i=1}^n$ in $\Gamma(\mathrm{TM})$ is defined by
+>
+> $$
+>   V = \bm{\epsilon}(\mathbf{v}_1, \dots, \mathbf{v}_n) = \sqrt{g} \bm{\mu}(\mathbf{v}_1, \dots, \mathbf{v}_n),
+> $$
+>
+> with $\bm{\epsilon}$ the unique unit volume form. 
+
+In the context of a subspace $S \subset M$ with $\dim S = k \in \mathbb{N}[k \leq n]$, the volume $V$ is given by
+
+$$
+    V = \int_S \bm{\epsilon} = \int_S \sqrt{g} dx^1 \dots dx^k.
+$$
+
+It follows that for $k=1$ 
+
+$$
+    \int_S \bm{\epsilon} = \int_S \sqrt{\bm{g}}.
+$$
+
+## Finsler geometry
+
+Will be added later.

docs/en/mathematics/differential-geometry/linear-connections.md

@@ -124,7 +124,7 @@ One may interpret a geodesic as a generalization of the notion of a straight lin
 > *Proposition 2*: let $\gamma: \mathscr{D}(\gamma) \to M: t \mapsto \gamma(t)$ be a smooth curve on the manifold parameterized by an open interval $\mathscr{D}(\gamma) \subset \mathbb{R}$ and let $\mathscr{L}$ be the Lagrangian defined by
 >
 > $$
->   \mathscr{L} = \|\dot \gamma\|^2 = g_{ij} \dot \gamma^i \dot \gamma^j, 
+>   \mathscr{L} = \|\dot \gamma\|^2,
 > $$
 >
 > for all $t \in \mathscr{D}(\gamma)$. By demanding [Hamilton's principle]() we obtain the geodesic equations 

docs/en/mathematics/linear-algebra/tensors/tensor-formalism.md

@@ -135,11 +135,12 @@ We have from theorem 2 that the outer product of two tensors yields another tens
 
 ## Inner product
 
-> *Definition 5*: a **pseudo inner product** on $V$ is a nondegenerate bilinear mapping $\bm{g}: V \times V \to \mathbb{K}$ which satisfies
+> *Definition 5*: an **inner product** on $V$ is a bilinear mapping $\bm{g}: V \times V \to \mathbb{K}$ which satisfies
 >
-> 1. for all $\mathbf{u} \in V \backslash \{\mathbf{0}\} \exists \mathbf{v} \in V: \; \bm{g}(\mathbf{u},\mathbf{v}) \neq 0$,
+> 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 $\mathbf{u}, \mathbf{v} \in V: \; \bm{g}(\mathbf{u}, \mathbf{v}) = \overline{\bm{g}}(\mathbf{v}, \mathbf{u})$, 
+> 2. for all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$ and $\lambda, \mu \in \mathbb{K}: \;\bm{g}(\mathbf{u}, \lambda \mathbf{v} + \mu \mathbf{w}) = \lambda \bm{g}(\mathbf{u}, \mathbf{v}) + \mu \bm{g}(\mathbf{u}, \mathbf{w}),$
-> 3. for all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$ and $\lambda, \mu \in \mathbb{K}: \;\bm{g}(\mathbf{u}, \lambda \mathbf{v} + \mu \mathbf{w}) = \lambda \bm{g}(\mathbf{u}, \mathbf{v}) + \mu \bm{g}(\mathbf{u}, \mathbf{w}).$
+> 3. for all $\mathbf{u} \in V\backslash \{\mathbf{0}\}: \bm{g}(\mathbf{u},\mathbf{u}) > 0,$
+> 4. for $\mathbf{u} = \mathbf{0} \iff \bm{g}(\mathbf{u},\mathbf{u}) = 0.$
 
 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. 
 

docs/en/mathematics/linear-algebra/tensors/volume-forms.md

@@ -102,7 +102,7 @@ for $k \in \mathbb{N}[k < n]$.
 >   \bm{\epsilon} = \sqrt{g} \bm{\mu},
 > $$
 >
-> with $g \overset{\text{def}}{=} |\det (G)|$, the absolute value of the determinant of the [Gram matrix]().
+> with $g \overset{\text{def}}{=} \det (G)$, the determinant of the [Gram matrix]().
 
 Therefore, if we decompose the Levi-Civita tensor by