Finished divergence in vector analysis.

docs/en/physics/electromagnetism/optics/polarisation.md

@@ -94,7 +94,7 @@ Using the properties of birefringence, wave plates (retarders) can be created. T
 
 ## Jones formalism of polarisation
 
-Jones formalism of polarisation with vectors and matrices cna make it easier to calculate the effect of optical elements such as linear polarizers and wave plates.
+Jones formalism of polarisation with vectors and matrices can make it easier to calculate the effects of optical elements such as linear polarizers and wave plates.
 
 > *Definition*: for an electromagnetic wave $\mathbf{E}: \mathbb{R}^2 \to \mathbb{R}^3$ with wavenumber $k \in \mathbb{R}$ and angular frequency $\omega \in \mathbb{R}$ propagating in the positive $z$-direction given by
 >

docs/en/physics/mathematical-physics/vector-analysis/curvilinear-coordinates.md

@@ -245,7 +245,7 @@ $$
     \|\mathbf{u}\| = \sqrt{u^{(i)} u_{(i)}}.
 $$
 
-We will discuss as an example the representations of the cartesian, cylindrical and spherical coordinate systems viewed from a cartesian perspective. This means that the coordinate maps are based on the cartesian interpretation of then. Every other interpretation could have been used, but our brains have a preference for cartesian it seems.z
+We will discuss as an example the representations of the cartesian, cylindrical and spherical coordinate systems viewed from a cartesian perspective. This means that the coordinate maps are based on the cartesian interpretation of them. Every other interpretation could have been used, but our brains have a preference for cartesian it seems.
 
 Let $\mathbf{x}: \mathbb{R}^3 \to \mathbb{R}^3$ map a cartesian coordinate system given by
 

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

@@ -0,0 +1,131 @@
+# The divergence of a vector field
+
+## Flux densities
+
+Considering a medium with a mass density $\rho: \mathbb{R}^4 \to \mathbb{R}$ and a velocity field $\mathbf{v}: \mathbb{R}^4 \to \mathbb{R}^3$ consisting of a orientable finite sized surface element $d\mathbf{A} \in \mathbb{R}^3$. 
+
+> *Definition*: a surface must be orientable for the surface integral to exist. It must be able to move along the surface continuously without ending up on the "other side".
+
+We then have a volume $dV \in \mathbb{R}$ defined by the parallelepiped formed by $dV = \langle d\mathbf{x}, d\mathbf{A} \rangle$ with the vector $d\mathbf{x} = \mathbf{v} dt$, for a time interval $dt \in \mathbb{R}$. The mass flux $d\Phi$ per unit of time through the surface element $d\mathbf{A}$ may then be given by
+
+$$
+    d \Phi = \rho \langle \mathbf{v}, d\mathbf{A} \rangle.
+$$
+
+The mass flux $\Phi: \mathbb{R} \to \mathbb{R}$ through a orientable finite sized surface $A \subseteq \mathbb{R}^3$ is then given by
+
+$$
+    \Phi(t) = \int_A \Big\langle \rho(\mathbf{x}, t) \mathbf{v}(\mathbf{x}, t), d\mathbf{A} \Big\rangle,
+$$
+
+for all $t \in \mathbb{R}$.
+
+> *Definition*: let $\mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3$ be the (mass) flux density given by
+>
+> $$
+>   \mathbf{\Gamma}(\mathbf{x},t) := \rho(\mathbf{x},t) \mathbf{v}(\mathbf{x},t),
+> $$
+>
+> for all $(\mathbf{x},t) \in \mathbb{R}^4$. 
+
+The (mass) flux density is a vector-valued function of position and time that expresses the rate of transport of a quantity per unit of time of area perpendicular to its direction. 
+
+The mass flux $\Phi$ through $A$ may then be given by
+
+$$
+    \Phi(t) = \int_A \Big\langle \mathbf{\Gamma}(\mathbf{x},t), d\mathbf{A} \Big\rangle,
+$$
+
+for all $t \in \mathbb{R}$.
+
+## Definition of the divergence
+
+> *Definition*: the divergence of a flux density $\mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3$ is given by
+>
+> $$
+>\nabla \cdot \mathbf{\Gamma}(\mathbf{x}, t) = \lim_{V \to 0} \frac{1}{V} \int_A \Big\langle \mathbf{\Gamma}(\mathbf{x}, t), d\mathbf{A} \Big\rangle,
+> $$
+>
+> for all $(\mathbf{x}, t) \in \mathbb{R}^4$ for a volume $V \subset \mathbb{R}^3$ with closed orientable boundary surface $A \subset V$. 
+
+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 $S: \mathbb{R}^4 \to \mathbb{R}^3$.
+
+We then have the particle flux density $\mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3$ given by 
+
+$$
+    \mathbf{\Gamma}(\mathbf{x},t) = n(\mathbf{x},t) \mathbf{v}(\mathbf{x},t),
+$$
+
+for all $(\mathbf{x},t) \in \mathbb{R}^4$. 
+
+For a volume $V \subseteq \mathbb{R}^3$ with a closed orientable boundary surface $A \subseteq \mathbb{R}^3$ we have that the amount of particles inside this volume for a specific time is given by
+
+$$
+    \int_V n(\mathbf{x}, t) dV,
+$$
+
+for all $t \in \mathbb{R}$. We have that the particle flux through $A$ is given by
+
+$$
+    \int_A \Big\langle \mathbf{\Gamma}(\mathbf{x},t), d\mathbf{A} \Big\rangle,
+$$
+
+for all $t \in \mathbb{R}$ and we have that the particle production rate in this volume $V$ is given by
+
+$$
+    \int_V S(\mathbf{x}, t)dV,
+$$
+
+for all $t \in \mathbb{R}$. We conclude that the sum of the particle flux through $A$ and the time derivative of the particles inside the volume $V$ must be equal to the production rate inside this volume $V$. Therefore we have
+
+$$
+    d_t \int_V n(\mathbf{x}, t) dV +  \int_A \Big\langle \mathbf{\Gamma}(\mathbf{x},t), d\mathbf{A} \Big\rangle = \int_V S(\mathbf{x}, t)dV,
+$$
+
+for all $t \in \mathbb{R}$. 
+
+Assuming the system is stationary the time derivative of the particles inside the volume $V$ must vanish. The divergence is then defined to be the total production for a position $\mathbf{x} \in V$. 
+
+## Divergence in curvilinear coordinates
+
+> *Theorem*: the divergence of a flux density $\mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3$ for a curvilinear coordinate system is given by
+>
+> $$
+>   \nabla \cdot \mathbf{\Gamma}(\mathbf{x}, t) = \frac{1}{\sqrt{g(\mathbf{x})}} \partial_i \Big(\Gamma^i(\mathbf{x},t) \sqrt{g(\mathbf{x})} \Big)
+> $$
+>
+> for all $\mathbf{x} \in \mathbb{R}^3$ and $i \in \{1, 2, 3\}$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+We may also give the divergence for ortho-curvilinear coordinate systems.
+
+> *Corollary*: the divergence of a flux density $\mathbf{\Gamma}: \mathbb{R}^3 \to \mathbb{R}^3$ for a ortho-curvilinear is given by
+>
+> $$
+>   \nabla \cdot \mathbf{\Gamma}(\mathbf{x}, t) = \frac{1}{h_1h_2h_3} \partial_i \Big(\Gamma^i(\mathbf{x},t)\frac{1}{h_i} h_1 h_2 h_3 \Big)
+> $$
+>
+> for all $\mathbf{x} \in \mathbb{R}^3$ and $i \in \{1, 2, 3\}$.
+
+??? note "*Proof*:"
+
+    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
+
+> *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
+>
+> $$
+>   \int_A \Big\langle \mathbf{\Gamma}(\mathbf{x}, t), d\mathbf{A} \Big\rangle = \int_V \nabla \cdot \mathbf{\Gamma}(\mathbf{x}, t) dV,
+> $$
+>
+> for all $t \in \mathbb{R}$. 
+
+??? note "*Proof*:"
+
+    Will be added later.