## Wednesday, 19 February 2020

### Microsoft Mathematics

I was recently doing a problem which involved calculating the electromagnetic energy momentum tensor$$T^{\mu\nu}=F^{\mu\lambda}F_{\ \ \ \lambda}^\nu-\frac{1}{4}\eta^{\mu\nu}F^{\lambda\sigma}F_{\lambda\sigma}$$In a previous problem I had already calculate$$F_{\mu\nu}=\left[\begin{matrix}0&-E_1&-E_2&-E_3\\E_1&0&B_3&-B_2\\E_2&-B_3&0&B_1\\E_3&B_2&-B_1&0\\\end{matrix}\right],\ F_{\ \ \ \lambda}^\nu=\eta^{\nu\rho}F_{\rho\lambda}=\left[\begin{matrix}0&E_1&E_2&E_3\\E_1&0&B_3&-B_2\\E_2&-B_3&0&B_1\\E_3&B_2&-B_1&0\\\end{matrix}\right]$$
and$$F^{\mu\nu}=F_{\ \ \ \lambda}^\mu\eta^{\lambda\nu}=\left[\begin{matrix}0&E_1&E_2&E_3\\-E_1&0&B_3&-B_2\\-E_2&-B_3&0&B_1\\{-E}_3&B_2&-B_1&0\\\end{matrix}\right]$$I was using $c=1$ and metric signature $-++++$
So I had to do a bit of unpleasant matrix multiplication. I found Microsoft Mathematics useful. It successfully tells me that $$F^{\mu\lambda}F_{\ \ \ \lambda}^\nu=\left[\begin{matrix}{E_1}^2+{E_2}^2+{E_3}^2&B_3E_2-B_2E_3&B_1E_3-B_3E_1&B_2E_1-B_1E_2\\B_3E_2-B_2E_3&{B_2}^2+{B_3}^2-{E_1}^2&-B_1B_2-E_1E_2&-B_1B_3-E_1E_3\\B_1E_3-B_3E_1&-B_1B_2-E_1E_2&{B_1}^2+{B_3}^2-{E_2}^2&-B_2B_3-E_2E_3\\B_2E_1-B_1E_2&-B_1B_3-E_1E_3&-B_2B_3-E_2E_3&{B_1}^2+{B_2}^2-{E_3}^2\\\end{matrix}\right]$$
(without working out that ${E_1}^2+{E_2}^2+{E_3}^2=E$) and was good at the other bits. The above matrix will easily paste back into a MS-Word equation. Sadly one cannot paste matrices from Word into it.

Subscripted variables can easily be entered as E_x becomes $E_x$. Greek letters are allowed too. Superscripts are always exponents. It also does differentiation, integration and quite a bit more which I have not explored.

It has some drawbacks on my computer at least.

As mentioned I was unable to copy/paste matrices from MS-Equations into MS-Mathematics, so all the data entry must be done in the latter. There is an addin which might do the trick. But it did not like my version of Word (Office 365). The main install program, crashed when it came to the addin part.

The 'keyboard' is tiny and the seven choices of colours are demented. The image on the right is about the right size but it is less legible than the real thing.