Finished gradient in vector analysis.

docs/en/physics/mathematical-physics/vector-analysis/divergence.md

@@ -50,7 +50,7 @@ for all $t \in \mathbb{R}$.
 
 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 hte divergence can be interpreted with the species 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 $S: \mathbb{R}^4 \to \mathbb{R}^3$.
+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 
 
@@ -116,7 +116,9 @@ We may also give the divergence for ortho-curvilinear coordinate systems.
 
     Will be added later.
 
-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 given below.s
+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
 >

docs/en/physics/mathematical-physics/vector-analysis/gradient.md

@@ -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.

docs/en/physics/mathematical-physics/vector-analysis/vectors.md

@@ -11,7 +11,9 @@ The defining property of axiomatic geometry is that it can be introduced without
 
 The fifth postulate as formulated below is only valid for Euclidean geometry; flat space informally. 
 
-5. 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.
+<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