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Notation for partial derivatives using indexes

  1. Jul 13, 2013 #1

    Stephen Tashi

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    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?
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  3. Jul 13, 2013 #2

    Simon Bridge

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    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.

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


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    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

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