# Math Abuse

When I shows a mathematician some of the homework problems I have done, he was a little shocked. So I went on and looked for other notations that physicists use differently or even naively.

## Multiple Integrals

This is a multiple integral in the regular notation:

$$\int_0^R \int_0^\pi \int_0^{2\pi} f(r, \theta, \phi) \, \mathrm d\phi \, \mathrm d\theta \, \mathrm dr$$

Theoretical physicists often use the following notation:

$$\int_0^R \mathrm d r \int_0^\pi \mathrm d \theta \int_0^{2\pi} \mathrm d \phi \, f(r, \theta, \phi)$$

The integrals is not $\int 1 \, \mathrm dx$, but the integrate everything
*after* the $\mathrm dx$! The advantage is that, just like with a summation
sign, you can see the bounds right away, and swap integrals easier.

## Summation Convention

Let me start with the regular inner product. A mathematician would write it either $\langle v, w \rangle$ or $(v, w)$.

Physicists might use the mathematician's notation, or write $\vec v \cdot \vec w$, since we write vector arrows (or bold vectors, then $\boldsymbol v \cdot \boldsymbol w$). I prefer bold vectors by now, so I will continue to use them.

If you assume a particular basis for your vector space, then you can access the components of your vectors with upper index, if they are regular (contravariant) vectors: $v^i$. If you transpose the vector, it will become a covector (covariant) and has a lower index: $v_i$. If your "transpose" is a "complex conjugate transpose", the inequality $v^i \neq v_i$ holds in the general case.

With that in mind, you can write the scalar product like so:

$$\langle \boldsymbol v, \boldsymbol w \rangle = \boldsymbol v^{\mathrm T} \boldsymbol w = \sum_i v_i w^i$$

And physicists like to omit the summation sign and just write $v_i w^i$ for the scalar product.

The transpose is induced by a metric tensor $\eta$, so the transpose works like this: $x_i = \eta_{ij} x^j$.

As a corollary, the physicist distinguishes between Latin and Greek indexes in the summation. A Latin index $i$ means $\sum_{i=1}^3$, where a Greek index $\mu$ means $\sum_{\mu=0}^3$. This is important for the theory of special relativity, where the zeroth coordinate is the time. That way, one can construct the Laplace ($\triangle$) and d'Alambert ($\Box$) Operators:

$$\begin{aligned} \triangle &= \partial_i \partial^i = \sum_{i=1}^3 \frac{\partial^2}{\partial x_i^2} \ \Box &= \partial_\mu \partial^\mu = \sum_{\mu=0}^3 \frac{\partial^2}{\partial x_i^2} = \frac{\partial^2}{\partial t^2} - \triangle \end{aligned}$$

Where I have used the metric tensor of special relativity:

$$\begin{aligned} \eta = \begin{pmatrix} 1 & 0 & 0 & 0 \ 0 & -1 & 0 & 0 \ 0 & 0 & -1 & 0 \ 0 & 0 & 0 & -1 \ \end{pmatrix} \end{aligned}$$

## Separation of Variables

Say a physicist is given the following ordinary differential equation: $f'(x) = f(x)$. He might do the following:

$$\begin{aligned} \frac{\mathrm df}{\mathrm dx} &= f \ \mathrm d f &= \mathrm d x f \ \frac{\mathrm d f}{f} &= \mathrm d x \ \int \frac{\mathrm d f}{f} &= \int \mathrm d x \ \ln(f) &= x + C \ f(x) &= C \exp(x) \end{aligned}$$

## Symbol overloading

In some programming languages, it is possible to overload a function with different arguments:

int f(); int f(int x); int f(int x, int y);

Those all have the same name, but the compiler will be able to distinguish that
from your arguments. `f(1)`

and `f(1, 1)`

will call two distinct functions.

The same applies for different types. So this is also possible:

int h(int x); int h(std::string x);

When you call `h(1)`

and `h("")`

, it will pick out the right function to call.
Unless the functions do a roughly the same thing, this will lead to confusion
pretty easily.

Now physicists are doing it even worse! Say you have a function $f \colon X \subseteq \mathbb R \mapsto Y \subseteq \mathbb R$. Then you use $x \in X$ and write stuff like $f(x)$ which is fine. Now you create the Fourier transform that I will denote with a $\mathcal F$. With $\omega \in \Omega \subseteq \mathbb R$ you could write:

$$g(\omega) = \mathcal F f = \frac1{\sqrt{2\pi}} \int \mathrm dx \, f(x) \exp(- \mathrm i x \omega)$$

This seems correct and unambiguous to me. Physicists like decorators more than different letters, so they would write $\hat f$ instead of $g$, which should be fine as well.

But even that seems to much writing. So they write $f(\omega)$ for the Fourier transformed function and $f(x)$ for the original function. That means that even if $\omega = x$, it does not need to follow that $f(\omega) \neq f(x)$, since they are different functions! The function is overloaded, and the correct one is chosen by the symbol you write the argument with. I think this is a gross violation of scoping.

I have seen a case where somebody wrote $f(\omega = 0)$ to make sure the reader understands that the transformed function is meant. Just write $\hat f(0)$!