Added first two sections of optics.

config/en/mkdocs.yaml

@@ -117,7 +117,17 @@ nav:
     - physics/index.md
     - 'Mathematical physics': 
       - 'Signal analysis': 
-        - 'Signals': mathematics/mathematical-physics/signal-analysis/signals.md
+        - 'Signals': physics/mathematical-physics/signal-analysis/signals.md
+        - 'Fourier series': 
+        - 'Fourier transform': 
+        - 'Discrete Fourier transform': 
+    - 'Electromagnetism': 
+      - 'Electrostatics':
+      - 'Magnetostatics': 
+      - 'Maxwell-equations': physics/electromagnetism/maxwell-equations.md
+      - 'Optics':
+        - 'Waves': physics/electromagnetism/optics/waves.md
+        - 'Electromagnetic waves': physics/electromagnetism/optics/electromagnetic-waves.md
     
   - 'Chemistry':
     - chemistry/index.md

docs/en/physics/electromagnetism/maxwell-equations.md

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+# Maxwell equations

docs/en/physics/electromagnetism/optics/electromagnetic-waves.md

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+# Electromagnetic waves
+
+This section is a direct follow up on the section [Maxwell equations](../maxwell-equations.md). Where the Laplacian of the electric field $\mathbf{E}: U \to \mathbb{R}^3$ and magnetic field $\mathbf{B}: U \to \mathbb{R}^3$ in vacuum ($\varepsilon = \varepsilon_0, \mu = \mu_0$) have been determined, given by
+
+$$
+\begin{align*}
+    &\nabla^2 \mathbf{E}(\mathbf{v}, t) = \varepsilon_0 \mu_0 \partial_t^2 \mathbf{E}(\mathbf{v}, t) \\\\
+    &\nabla^2 \mathbf{B}(\mathbf{v}, t) = \varepsilon_0 \mu_0 \partial_t^2 \mathbf{B}(\mathbf{v}, t)
+\end{align*}
+$$
+
+for all $(\mathbf{v}, t) \in U$. 
+
+It may be observed that the eletric and magnetic field comply with the $3 + 1$ dimensional wave equation posed in the section [waves](waves.md). Obtaining the speed $v \in \mathbb{R}$ given by
+
+$$
+    v = \frac{1}{\sqrt{\varepsilon_0 \mu_0}} = c,
+$$
+
+defined by $c$ the speed of light, or more generally the speed of information in the universe. Outside vacuum we have 
+
+$$
+    v = \frac{1}{\sqrt{\varepsilon \mu}} = \frac{c}{n},
+$$
+
+with $n = \sqrt{K_E K_B}$ the index of refraction. 
+
+> *Proposition*: let $\mathbf{E},\mathbf{B}: U \to \mathbb{R}^3$, a solution for the wave equations of the electric and magnetic field may be harmonic linearly polarized plane waves satisfying Maxwell's equations given by
+>
+> $$
+> \begin{align*}
+> \mathbf{E}(\mathbf{v}, t) &= \text{Im}\Big(\mathbf{E}_0 \exp i \big(\langle \mathbf{k}, \mathbf{v} \rangle - \omega t+ \varphi\big) \Big) \\ \\ \mathbf{B}(\mathbf{v}, t) &= \text{Im} \Big(\mathbf{B}_0 \exp i \big(\langle \mathbf{k}, \mathbf{v} \rangle - \omega t+ \varphi\big) \Big)
+> \end{align*}
+> $$
+>
+> for all $(\mathbf{v}, t) \in U$ with $\mathbf{E}_0, \mathbf{B}_0 \in \mathbb{R}^3$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+The above proposition gives an example of a light wave, but note that there are much more solutions that comply to Maxwell's equations.
+
+> *Law*: the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ for all solutions of the posed wave equations are orthogonal to the direction of propagation $\mathbf{k}$. Therefore electromagnetic waves are transverse.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+> *Law*: the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ in a electromagnetic wave are orthogonal to each other; $\langle \mathbf{E}, \mathbf{B} \rangle = 0$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+> *Corollary*: it follows from the above law that the magnitude of the electric and magnetic fields $E, B: U \to \mathbb{R}$ in a electromagnetic wave are related by 
+>
+> $$
+>   E(\mathbf{v}, t) = v B(\mathbf{v}, t)
+> $$
+>
+> for all $(\mathbf{v}, t) \in U$ with $v = \frac{c}{n}$ the wave speed. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+## Energy flow
+
+> *Law*: the energy flux density $\mathbf{S}: U \to \mathbb{R}^3$ of an electromagnetic wave is given by
+>
+> $$
+>   \mathbf{S}(\mathbf{v}, t) = \frac{1}{\mu_0} \mathbf{E}(\mathbf{v}, t) \times \mathbf{B}(\mathbf{v}, t),
+> $$
+>
+> for all $(\mathbf{v}, t) \in U$. $\mathbf{S}$ is also called the Poynting vector.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+> *Definition*: the time average of the magnitude of $\mathbf{S}$ is called the irradiance. 
+
+<br>
+
+> *Proposition*: the irradiance $I \in \mathbb{R}$ for harmonic linearly polarized plane electromagnetic waves is given by
+>
+> $$
+>   I = \frac{\varepsilon_0 c}{2} E_0^2,
+> $$
+>
+> with $E_0$ the magnitude of $\mathbf{E}_0$. 
+
+??? note "*Proof*:"
+
+    Will be added later.

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

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+# Waves
+
+> *Definition*: a wave is a propagating disturbance transporting energy and momentum. A $1 + 1$ dimensional wave $\Psi: \mathbb{R}^2 \to \mathbb{R}$ travelling can be defined by a linear combination of a right and left travelling function $f,g: \mathbb{R} \to \mathbb{R}$ obtaining
+>
+> $$
+>   \Psi(x,t) = f(x - vt) + g(x + vt),
+> $$ 
+>
+> for all $(x,t) \in \mathbb{R}^2$ and $v \in \mathbb{R}$ the speed of the wave. Satisfies the $1 + 1$ dimensional wave equation
+>
+> $$
+>   \partial_x^2 \Psi(x,t) = \frac{1}{v^2} \partial_t^2 \Psi(x,t).
+> $$
+
+The derivation of the wave equation can be obtained in section...
+
+> *Theorem*: a right travelling harmonic wave $\Psi: \mathbb{R}^2 \to \mathbb{R}$ with $\lambda, T, A, \varphi \in \mathbb{R}$ the wavelength, period, amplitude and phase of the wave is given by
+>
+> $$
+> \begin{align*}
+>   \Psi(x,t) &= A \sin \big(k(x-vt) + \varphi\big), \\
+>             &= A \sin(kx-\omega t + \varphi), \\
+>             &= A \sin \Big(2\pi \Big(\frac{x}{\lambda} - \frac{t}{T} \Big) + \varphi \Big),
+> \end{align*}
+> $$
+>
+> for all $(x,t) \in \mathbb{R}^2$. With $k = \frac{2\pi}{\lambda}$ the wavenumber, $\omega = \frac{2\pi}{T}$ the angular frequency and $v = \frac{\lambda}{T}$ the wave speed. 
+
+A right travelling harmonic wave $\Psi: \mathbb{R}^2 \to \mathbb{R}$ can also be represented in the complex plane given by
+
+$$
+    \Psi(x,t) = \text{Im} \big(A \exp i(kx - \omega t + \varphi )\big),
+$$
+
+for all $(x,t) \in \mathbb{R}^2$.
+
+> *Theorem*: let $\Psi: \mathbb{R}^4 \to \mathbb{R}$ be a $3 + 1$ dimensional wave then it satisfies the $3 + 1$ dimensional wave equation given by
+>
+> $$
+>   \nabla^2 \Psi(\mathbf{x},t) = \frac{1}{v^2} \partial_t^2 \Psi(\mathbf{x},t),
+> $$
+>
+> for all $(\mathbf{x},t) \in \mathbb{R}^4$. 
+
+We may formulate various solutions $\Psi: \mathbb{R}^4 \to \mathbb{R}$ for this wave equation. 
+
+The first solution may be the plane wave that follows cartesian symmetry and can therefore best be described in a cartesian coordinate system $\mathbf{v}(x,y,z)$. The solution is given by
+
+$$
+    \Psi(\mathbf{v}, t) = \text{Im}\big(A \exp i(\langle \mathbf{k}, \mathbf{v} \rangle - \omega t + \varphi) \big),
+$$
+
+for all $(\mathbf{v}, t) \in \mathbb{R}^4$ with $\mathbf{k} \in \mathbb{R}^3$ the wavevector.
+
+The second solution may be the cylindrical wave that follows cylindrical symmetry and can therefore best be described in a cylindrical coordinate system $\mathbf{v}(r,\theta,z)$. The solution is given by
+
+$$
+    \Psi(\mathbf{v}, t) = \text{Im}\Bigg(\frac{A}{\sqrt{\|\mathbf{v}\|}} \exp i(k \|\mathbf{v} \| - \omega t + \varphi) \Bigg),
+$$
+
+for all $(\mathbf{v}, t) \in \mathbb{R}^4$. 
+
+The third solution may be the spherical wave that follows spherical symmetry and can therefore best be described in a spherical coordinate system $\mathbf{v}(r, \theta, \varphi)$. The solution is given by
+
+$$
+    \Psi(\mathbf{v}, t) = \text{Im}\Bigg(\frac{A}{\|\mathbf{v}\|} \exp i(k\|\mathbf{v}\| - \omega t + \varphi) \Bigg)
+$$
+
+for all $(\mathbf{v}, t) \in \mathbb{R}^4$. 
+
+> *Principle*: the principle of superposition is valid for waves, since the solution space of the wave equation is linear. 
+
+From this principle we obtain the property of constructive and destructive interference of waves.

docs/en/physics/index.md

@@ -1,6 +1,6 @@
 # Physics
 
-Welcome to the physics page. Some special physical environments that will be used in this seection are listed and explained below.
+Welcome to the physics page. Some special physical environments that will be used in this section are listed and explained below.
 
 * *Principles*: a fundamental rule or concept in physics serving as a basis for reasoning. 
 * *Assumptions*: a less fundamental rule or concept in physics that is taken to be true such that certain phenoma can be simplified. 

docs/en/physics/mathematical-physics/signal-analysis/fourier-series.md

@@ -0,0 +1 @@
+# Fourier series