Added vector analysis to mathematical physics, finished section vectors and curves.

config/en/mkdocs.yaml

@@ -124,6 +124,14 @@ 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':
+        - '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
+        - 'Divergence': physics/mathematical-physics/vector-analysis/divergence.md
+        - 'Gradient': physics/mathematical-physics/vector-analysis/gradient.md
+        - 'Curl': physics/mathematical-physics/vector-analysis/curl.md
+        - 'Vector operators': physics/mathematical-physics/vector-analysis/vector-operators.md
     - 'Electromagnetism': 
 #      - 'Electrostatics':
 #      - 'Magnetostatics': 

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

@@ -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 $\phi \in \mathbb{R}$. 

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

@@ -0,0 +1,62 @@
+# 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. 
+
+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.
+
+## 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
+
+$$
+    \langle \mathbf{u}, \mathbf{v}, \mathbf{w} \rangle^2 = \det (J^TJ).
+$$