Math Basics

Physics can be seen as the try to express nature in term of mathematics.

This category intends to provide some basic mathematical tools to cope with the physics problems on the site. Remember, it's a physics site, so it might be much more leaned towards actual calculations than mathematicians might outline the topics.

Subcategories

When dealing with Maxwell's equations, we often need to switch between $$(\mathbf{r},t)$$- and $$(\mathbf{k},\omega)$$-space, i.e. between space-time and Fourier domain. This is advantageous, since derivatives in space and time are simple products in Fourier domain.

The Fourier transformation, or often short Fourier transform, of the electric field is given by:

\begin{eqnarray*}\mathbf{E}\left(\mathbf{r},t\right)&=&\intop_{-\infty}^{\infty}\mathbf{\overline{E}}\left(\mathbf{r},\omega\right)e^{-i\omega t}d\omega\\\mathbf{\overline{E}}\left(\mathbf{r},\omega\right)&=&\frac{1}{2\pi}\intop_{-\infty}^{\infty}\mathbf{E}\left(\mathbf{r},t\right)e^{i\omega t}dt \end{eqnarray*} and \begin{eqnarray*} \mathbf{E}\left(\mathbf{r},t\right)&=&\intop_{-\infty}^{\infty}\mathbf{E}\left(\mathbf{k},t\right)e^{i\mathbf{kr}}d\mathbf{r}\\\mathbf{E}\left(\mathbf{k},t\right)&=&\frac{1}{2\pi}\intop_{-\infty}^{\infty}\mathbf{E}\left(\mathbf{r},t\right)e^{-i\mathbf{kr}}d\mathbf{k}\end{eqnarray*}