Finished vector operators.

docs/en/physics/index.md

@@ -12,4 +12,7 @@ Welcome to the physics page. Some special physical environments that will be use
 * *Corollaries*: a result in which the proof relies heavily on a given theorem or law.
 * *Proofs*: a convincing argument that a certain physical statement is necessarily true. A proof generally uses deductive reasoning and logic but also contains some amount of ordinary language.
 
-The physics sections of this wiki are based on various books and lectures. A comprehensive list of references can be found below.
+The physics sections of this wiki are based on various books and lectures. A comprehensive list of references can be found below.
+
+* The sections of signal and vector analysis in mathematical physics are based on the lectures and lecture notes of Jan van Dijk.
+* The section of optics in electromagnetism is based on the lectures of Jürgen Kohlhepp and the book Principles of Physical Optics by Chuck Bennet. 

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

@@ -65,7 +65,7 @@ Similarly to the [divergence theorem](divergence.md#divergence-in-curvilinear-co
 >   \oint_C \big\langle \mathbf{v}(\mathbf{x}), d\mathbf{x} \big\rangle = \int_A \big\langle \nabla \times \mathbf{v}(\mathbf{x}), d\mathbf{A} \big\rangle,
 > $$
 >
-> is valid. 
+> is true. 
 
 ??? note "*Proof*:"
 

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

@@ -1,2 +1,176 @@
 # Vector operators
 
+## Properties of the gradient, divergence and curl
+
+> *Proposition*: let $a,b \in \mathbb{R}$, $f,g: \mathbb{R}^3 \to \mathbb{R}$ be differentiable scalar fields and $\mathbf{u}, \mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ be differentiable vector fields. Then we have the following identities:
+>
+> **Linearity:**
+>
+> $$
+> \begin{align*}
+>   \nabla (af + bg) &= a \nabla f + b \nabla g, \\
+>   \nabla \cdot (a\mathbf{u} + b \mathbf{v}) &= a (\nabla \cdot \mathbf{u}) + b (\nabla \cdot \mathbf{v}), \\
+>   \nabla \times (a\mathbf{u} + b \mathbf{v}) &=  a (\nabla \times \mathbf{u}) + b (\nabla \times\mathbf{v}).
+> \end{align*}
+> $$
+>
+> **Multiplication rules:**
+>
+> $$
+> \begin{align*}
+>   \nabla (fg) &= f \nabla g+ g \nabla f, \\
+>   \nabla \cdot (f \mathbf{u}) &= f (\nabla \cdot \mathbf{u}) + \langle \nabla f, \mathbf{u} \rangle, \\
+>   \nabla \cdot (\mathbf{u} \times \mathbf{v}) &= \langle \nabla \times \mathbf{u}, \mathbf{v} \rangle - \langle \mathbf{u}, \nabla \times \mathbf{v} \rangle, \\
+>   \nabla \times (f\mathbf{u}) &= f (\nabla \times \mathbf{u}) + \nabla f \times \mathbf{u}.
+> \end{align*}
+> $$
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+## The laplacian
+
+> *Definition*: the laplacian of a differentiable scalar field $f: \mathbb{R}^3 \to \mathbb{R}$ is defined as
+>
+> $$
+>   \nabla^2 f(\mathbf{x}) := \nabla \cdot \nabla f(\mathbf{x}),
+> $$
+>
+> for all $\mathbf{x} \in \mathbb{R}^3$. 
+
+The notation may be unorthodox for some. An alternative notatation for the laplacian is $\Delta f$, though generally deprecated.
+
+We can also rewrite the laplacian for curvilinear coordinate systems as has been done below.
+
+> *Theorem*: the laplacian of a differentiable scalar field $f: \mathbb{R}^3 \to \mathbb{R}$ for a curvilinear coordinate system is given by
+>
+> $$
+>   \nabla^2 f(\mathbf{x}) = \frac{1}{g(\mathbf{x})} \partial_i \Big(\sqrt{g(\mathbf{x})} g^{ij}(\mathbf{x}) \partial_j f(\mathbf{x}) \Big),
+> $$
+>
+> for all $\mathbf{x} \in \mathbb{R}^3$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+The laplacian for a ortho-curvilinear coordinate system may also be derived and can be found below.
+
+> *Corollary*: the laplacian of a differentiable scalar field $f: \mathbb{R}^3 \to \mathbb{R}$ for a ortho-curvilinear coordinate system is given by
+>
+> $$
+>   \nabla^2 f(\mathbf{x}) = \frac{1}{h_1 h_2 h_3} \bigg(\partial_1 \Big(\frac{h_2 h_3}{h_1} \partial_1 f(\mathbf{x}) \Big) + \partial_2 \Big(\frac{h_1 h_3}{h_2} \partial_2 f(\mathbf{x}) \Big) + \partial_3 \Big(\frac{h_1 h_2}{h_3} \partial_3 f(\mathbf{x}) \Big) \bigg),
+> $$
+>
+> for all $\mathbf{x} \in \mathbb{R}^3$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+Please note that the scaling factors may also depend on the position $\mathbf{x} \in \mathbb{R}^3$ depending on the coordinate system. 
+
+> *Proposition*: the laplacian of a differentiable vector field $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ is given by
+>
+> $$
+>   \nabla^2 \mathbf{v}(\mathbf{x}) = \nabla \big(\nabla \cdot \mathbf{v}(\mathbf{x})\big) - \nabla \times \big(\nabla \times \mathbf{v}(\mathbf{x})\big),
+> $$
+>
+> for all $\mathbf{x} \in \mathbb{R}^3$. 
+
+??? note "*Proof*:"
+
+    Will be added much later.
+
+## Potentials
+
+> *Definition*: a vector field $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ is irrotational or curl free if 
+>
+> $$
+>   \nabla \times \mathbf{v}(\mathbf{x}) = \mathbf{0},
+> $$
+>
+> for all $\mathbf{x} \in \mathbb{R}^3$. 
+
+If $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ is the gradient of some scalar field $\Phi: \mathbb{R}^3 \to \mathbb{R}$ it is irrotational since 
+
+$$
+    \nabla \times\big (\nabla \Phi(\mathbf{x})\big) = \mathbf{0},
+$$
+
+for all $\mathbf{x} \in \mathbb{R}^3$. 
+
+> *Proposition*: an irrotational vector field $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ has a scalar potential $\Phi: \mathbb{R}^3 \to \mathbb{R}$ such that
+>
+> $$
+>   \mathbf{v}(\mathbf{x}) = \nabla \Phi(\mathbf{x}),
+> $$
+>
+> for all $\mathbf{x} \in \mathbb{R}^3$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+In physics the scalar potential is generally given by the negative of the gradient, both are correct but one is more stupid than the other.
+
+> *Definition*: a vector field $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ is solenoidal or divergence-free if 
+>
+> $$
+>   \nabla \cdot \mathbf{v}(\mathbf{x}) = 0,
+> $$
+>
+> for all $\mathbf{x} \in \mathbb{R}^3$. 
+
+If $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ is the curl of some vector field $\mathbf{u}: \mathbb{R}^3 \to \mathbb{R}^3$ it is solenoidal since
+
+$$
+    \nabla \cdot \big(\nabla \times \mathbf{u}(\mathbf{x}) \big) = 0,
+$$
+
+for all $\mathbf{x} \in \mathbb{R}^3$.
+
+> *Proposition*: a solenoidal vector field $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ has a vector potential $\mathbf{u}: \mathbb{R}^3 \to \mathbb{R}^3$ such that
+>
+> $$
+>   \mathbf{v}(\mathbf{x}) = \nabla \times \mathbf{u}(\mathbf{x}),
+> $$
+>
+> for all $\mathbf{x} \in \mathbb{R}^3$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+The theorem below is the Helmholtz decomposition theorem and states that every vector field can be written in terms of two potentials. 
+
+> *Theorem*: every vector field $\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3$ can be written in terms of a scalar $\Phi: \mathbb{R}^3 \to \mathbb{R}$ and a vector $\mathbf{u}: \mathbb{R}^3 \to \mathbb{R}^3$ potential as
+>
+> $$
+>   \mathbf{v}(\mathbf{x}) = \nabla \Phi(\mathbf{x}) + \nabla \times \mathbf{u}(\mathbf{x}),
+> $$
+>
+> for all $\mathbf{x} \in \mathbb{R}^3$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+It then follows that the scalar and vector potentials can be determined for a volume $V \subset \mathbb{R}^3$ with a boundary surface $A \subset \mathbb{R}^3$ that encloses the domain $V$. 
+
+> *Corollary*: the scalar $\Phi: \mathbb{R}^3 \to \mathbb{R}$ and vector $\mathbf{u}: \mathbb{R}^3 \to \mathbb{R}^3$ potentials for a volume $V \subset \mathbb{R}^3$ with a boundary surface $A \subset \mathbb{R}^3$ that encloses the domain $V$ are given by
+>
+> $$
+> \begin{align*}
+>   \Phi(\mathbf{x}) &= \frac{1}{4\pi} \int_V \frac{\nabla \cdot \mathbf{v}(\mathbf{r})}{\|\mathbf{x} - \mathbf{r}\|}dV - \frac{1}{4\pi} \oint_A \bigg\langle \frac{1}{\|\mathbf{x} - \mathbf{r}\|} \mathbf{v}(\mathbf{r}), d\mathbf{A} \bigg\rangle, \\
+> \\
+> \mathbf{u}(\mathbf{x}) &= \frac{1}{4\pi} \int_V \frac{\nabla \times \mathbf{v}(\mathbf{r})}{\|\mathbf{x} - \mathbf{r}\|}dV - \frac{1}{4\pi} \oint_A  \frac{1}{\|\mathbf{x} - \mathbf{r}\|} \mathbf{v}(\mathbf{r}) \times d\mathbf{A},
+> \end{align*}
+> $$
+>
+> for all $\mathbf{x} \in \mathbb{R}^3$.
+
+??? note "*Proof*:"
+
+    Will be added later.