Helmholtz-decomposition-code.md

@@ -2,23 +2,29 @@
 
 # Introduction
 
-Analysis of turbulence and vortices requires knowledge of the kinetic energy, is conservation and its modes. This tool performs the Helmholtz Decomposition, in order to extract the compressive and rotational components of kinetic energy of the flow as shown in **Tsubota, Fujimoto, Yui** (2017) [**Numerical Studies of Quantum Turbulence.**](https://arxiv.org/abs/1704.02566).
+Analysis of turbulence and vortices requires knowledge of the kinetic energy, is conservation and its modes. This tool performs the Helmholtz Decomposition, in order to extract the compressive and rotational components of kinetic energy of the flow as shown in **Tsubota, Fujimoto, Yui** (2017) [**Numerical Studies of Quantum Turbulence**](https://arxiv.org/abs/1704.02566).
 
-We define the effective velocity field as <a href="https://www.codecogs.com/eqnedit.php?latex=\omega(\textbf{r},t)=\sqrt{\rho(\textbf{r},t)}v(\textbf{r},t)" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\omega(\textbf{r},t)=\sqrt{\rho(\textbf{r},t)}v(\textbf{r},t)" title="\omega(\textbf{r},t)=\sqrt{\rho(\textbf{r},t)}v(\textbf{r},t)" /></a>. From this, we can calculate the kinetic energy as:
+We define the effective velocity field as $`\omega(\textbf{r},t)=\sqrt{\rho(\textbf{r},t)}v(\textbf{r},t)`$. From this, we can calculate the kinetic energy as: 
 
-<a href="https://www.codecogs.com/eqnedit.php?latex=E_k=\frac{m}{2V}\int&space;\omega(\textbf{r},t)^2d\textbf{r}=\frac{m}{2}\sum_{\textbf{k}}|\widetilde{\omega}(\textbf{k},t)|^2" target="_blank"><img src="https://latex.codecogs.com/gif.latex?E_k=\frac{m}{2V}\int&space;\omega(\textbf{r},t)^2d\textbf{r}=\frac{m}{2}\sum_{\textbf{k}}|\widetilde{\omega}(\textbf{k},t)|^2" title="E_k=\frac{m}{2V}\int \omega(\textbf{r},t)^2d\textbf{r}=\frac{m}{2}\sum_{\textbf{k}}|\widetilde{\omega}(\textbf{k},t)|^2" /></a>,
+```math
+E_k=\frac{m}{2V}\int\omega(\textbf{r},t)^2d\textbf{r}=\frac{m}{2}\sum_{\textbf{k}}|\widetilde{\omega}(\textbf{k},t)|^2,
+```
 
-where <a href="https://www.codecogs.com/eqnedit.php?latex=\widetilde{\omega}(\textbf{k},t)=\mathfrak{F}[\omega(\textbf{r},t)]=\int&space;\omega(\textbf{r},t)&space;\frac{e^{-i\textbf{k}\cdot\textbf{r}}}{V}d\textbf{r}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\widetilde{\omega}(\textbf{k},t)=\mathfrak{F}[\omega(\textbf{r},t)]=\int&space;\omega(\textbf{r},t)&space;\frac{e^{-i\textbf{k}\cdot\textbf{r}}}{V}d\textbf{r}" title="\widetilde{\omega}(\textbf{k},t)=\mathfrak{F}[\omega(\textbf{r},t)]=\int \omega(\textbf{r},t) \frac{e^{-i\textbf{k}\cdot\textbf{r}}}{V}d\textbf{r}" /></a> is the Fourier Transform of the effective velocity.
+where $`\widetilde{\omega}(\textbf{k},t)=\mathfrak{F}[\omega(\textbf{r},t)]=\int\omega(\textbf{r},t)\frac{e^{-i\textbf{k}\cdot\textbf{r}}}{V}d\textbf{r}`$ is the Fourier Transform of the effective velocity.
 
 From this, we can apply the Helmholtz Decomposition theorem to obtain the compressive and rotational components of the effective velocity field:
 
-<a href="https://www.codecogs.com/eqnedit.php?latex=\omega(\textbf{r},t)=&space;\omega_0(t)&plus;\omega_c(\textbf{r},t)&plus;\omega_i(\textbf{r},t)" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\omega(\textbf{r},t)=&space;\omega_0(t)&plus;\omega_c(\textbf{r},t)&plus;\omega_i(\textbf{r},t)" title="\omega(\textbf{r},t)= \omega_0(t)+\omega_c(\textbf{r},t)+\omega_i(\textbf{r},t)" /></a>
-
-<a href="https://www.codecogs.com/eqnedit.php?latex=\omega_c(\textbf{r},t)=\sum_{\textbf{k}\neq0}^{}&space;\frac{\textbf{k}\cdot&space;\widetilde{\omega}(\textbf{k},t)}{k^2}\;&space;\textbf{k}e^{i\textbf{k}\cdot\textbf{r}}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\omega_c(\textbf{r},t)=\sum_{\textbf{k}\neq0}^{}&space;\frac{\textbf{k}\cdot&space;\widetilde{\omega}(\textbf{k},t)}{k^2}\;&space;\textbf{k}e^{i\textbf{k}\cdot\textbf{r}}" title="\omega_c(\textbf{r},t)=\sum_{\textbf{k}\neq0}^{} \frac{\textbf{k}\cdot \widetilde{\omega}(\textbf{k},t)}{k^2}\; \textbf{k}e^{i\textbf{k}\cdot\textbf{r}}" /></a>
-
-<a href="https://www.codecogs.com/eqnedit.php?latex=\omega_i(\textbf{r},t)=\sum_{\textbf{k}\neq0}^{}&space;\left&space;\{&space;\widetilde{\omega}(\textbf{k},t)&space;-&space;\frac{\textbf{k}\cdot&space;\widetilde{\omega}(\textbf{k},t)}{k^2}\;&space;\textbf{k}\right&space;\}e^{i\textbf{k}\cdot\textbf{r}}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\omega_i(\textbf{r},t)=\sum_{\textbf{k}\neq0}^{}&space;\left&space;\{&space;\widetilde{\omega}(\textbf{k},t)&space;-&space;\frac{\textbf{k}\cdot&space;\widetilde{\omega}(\textbf{k},t)}{k^2}\;&space;\textbf{k}\right&space;\}e^{i\textbf{k}\cdot\textbf{r}}" title="\omega_i(\textbf{r},t)=\sum_{\textbf{k}\neq0}^{} \left \{ \widetilde{\omega}(\textbf{k},t) - \frac{\textbf{k}\cdot \widetilde{\omega}(\textbf{k},t)}{k^2}\; \textbf{k}\right \}e^{i\textbf{k}\cdot\textbf{r}}" /></a>
+```math
+\omega(\textbf{r},t)=\omega_0(t)+\omega_c(\textbf{r},t)+\omega_i(\textbf{r},t)
+```
+```math
+\omega_c(\textbf{r},t)=\sum_{\textbf{k}\neq0}^{}\frac{\textbf{k}\cdot\widetilde{\omega}(\textbf{k},t)}{k^2}\textbf{k}e^{i\textbf{k}\cdot\textbf{r}}
+```
+```math
+\omega_i(\textbf{r},t)=\sum_{\textbf{k}\neq0}^{}\{\widetilde{\omega}(\textbf{k},t)-\frac{\textbf{k}\cdot\widetilde{\omega}(\textbf{k},t)}{k^2}\textbf{k}\}e^{i\textbf{k}\cdot\textbf{r}}
+```
 
-Note that in our specific case, we assume the component <a href="https://www.codecogs.com/eqnedit.php?latex=\omega_0(t)" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\omega_0(t)" title="\omega_0(t)" /></a> to be zero across the whole lattice.
+Note that in our specific case, we assume the component $`\omega_0(t)`$ to be zero across the whole lattice.
 
 # Usage
 TODO: explain how to use...