Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Notation for partial derivatives using indexes

  1. Jul 13, 2013 #1

    Stephen Tashi

    User Avatar
    Science Advisor

    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 [itex] \partial_{2,1}^{3,1} \ f [/itex] 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. jcsd
  3. Jul 13, 2013 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    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
  4. Jul 13, 2013 #3

    lurflurf

    User Avatar
    Homework Helper

    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

    $$f^{((70352,3),(1924518,2))}=\dfrac{\partial ^{2}}{\partial x_{1924518}^{2}} \dfrac{\partial ^3}{\partial x_{70352}^3} f$$
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Notation for partial derivatives using indexes
Loading...