Post Made Community Wiki by S. Carnahan
1

I really like the multi-index notation for derivatives which is quite often found in the theory of PDEs.

$D^\alpha f = \frac{\partial^{\vert\alpha\vert} f}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}}$

Where $\alpha$ is defined as:

$\alpha = (\alpha_1,\cdots,\alpha_n),$ $\vert \alpha\vert = \alpha_1 + \cdots +\alpha_n$

It makes definining Sobolev Spaces (and their corresponding Norms) so much cleaner and straight forward:

$W^{k,p} = \lbrace f\in L^p : D^\alpha f\in L^p \text{ for all } \vert\alpha\vert \le k \rbrace$