# Notation for partial derivatives using indexes

1. Jul 13, 2013

### Stephen Tashi

Is there a standard notation for partial derivatives that uses indexes instead of letters to denote ideas such as the 3 rd partial derivative with respect to the the 2nd argument of a function?

As soon as a symbol gets superscripts and subscripts like $\partial_{2,1}^{3,1} \ f$ the spectre of tensors appears. Is a particular choice of what the super and subscripts mean consistent with the idea of tensors? Or are tensors irrelevant?

2. Jul 13, 2013

### Simon Bridge

Taking the 3rd partial wrt the second argument would be:

$$\frac{\partial^3}{\partial x_2^3}f(\vec{x})\; : \vec{x}=(x_1,x_2,\cdots)$$
Which, indeed, simplifies to:
$$\partial_2^3 f$$
You do have to be careful ... what would $\partial_\mu f^\mu$ mean?
When you get mor subscripts and superscripts you may need to use some sort of delimiter to keep the roles separate.

http://en.wikipedia.org/wiki/Multi-index_notation

Last edited: Jul 13, 2013
3. Jul 13, 2013

### lurflurf

Yes, one of which uses superscript list to denote differentiations f^(i,j,k) is the ith derivative w/respect 1st variable jth w/respect second and so on. Naturally this is problematic for functions with unequal mixed partials.

$$f^{(0,3)}=\dfrac{\partial ^3}{\partial x_2^3}f$$

$$f^{(11,17)}=\dfrac{\partial ^{17}}{\partial x_2^{17}} \dfrac{\partial ^{11}}{\partial x_1^{11}} f$$

it is also a problem for functions with many variables so we can use a multilist

$$f^{((70352,3),(1924518,2))}=\dfrac{\partial ^{2}}{\partial x_{1924518}^{2}} \dfrac{\partial ^3}{\partial x_{70352}^3} f$$