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Does this notation for partial derivatives work?

  1. Oct 28, 2009 #1
    I noticed that for partial calculus equations I can use Jacobians
    [tex]
    \left(\frac{\partial A}{\partial B}\right)_C=\frac{\partial(A,C)}{\partial(A,B)}
    [/tex]
    You immediately get the triple product rule
    (http://en.wikipedia.org/wiki/Triple_product_rule)
    [tex]
    \frac{\partial(A,C)}{\partial(A,B)}=\frac{\partial(A,C)}{\partial(A,B)}\frac{\partial(B,C)}{\partial(B,C)}=-\frac{\partial(A,C)}{\partial(B,C)}\frac{\partial(B,C)}{\partial(B,A)}
    [/tex]
    and of course the other rule
    [tex]
    \frac{\partial(A,C)}{\partial(A,B)}=\frac{\partial(A,C)}{\partial(A,B)}\frac{\partial(A,D)}{\partial(A,D)}=\frac{\frac{\partial(A,C)}{\partial(A,D)}}{\frac{\partial(A,B)}{\partial(A,D)}}
    [/tex]

    Now I was surprised to see that even the pseudo-equation for the chain rule
    [tex]
    \partial(A,B)=\frac{\partial(A,D)\partial(E,B)-\partial(A,E)\partial(D,B)}{\partial(E,D)}
    [/tex]
    works well even if I treat all terms as a real variable.

    My question is: Does this algebra for partial derivatives always work despite crazy manipulations and cancellations?
     
  2. jcsd
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