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Little confused with tensor index manipulation

  1. Feb 16, 2015 #1

    kau

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    I am making mess of the following expression..
    i have following expression ## \frac{\partial{g}}{\partial{g_{\mu j}}} *g_{\nu j}=g \delta^{\mu} _{\nu} ##
    then I have sum over j only in the above expression.
    But above expression is nonzero only when ##{\mu}## is equal to ##\nu##.
    So we have ## \frac{\partial{g}}{\partial{g_{\mu j}}} *g_{\mu j}=g ## ....(a)
    I can understand that there is no ##\mu## dependence in right hand side so we have to sum over ##{\mu}## also. But doing the mathematical step just summing over ##{\nu}## gives equation in ##{\mu}## but in addition how the summation in that implied????
    Also now if I take partial derivative of the above expression (a) how it gives
    ## \frac{\partial{g}}{\partial{g_{\mu j}}} *\frac{\partial{g_{\mu j}}}{\partial{x^{i}}}=\frac {\partial{g}}{\partial{x^{i}}} ##
    ?? why the term like ## \frac{\partial^{2}{g}}{{\partial{g_{\mu j}}}{\partial{x^{i}}}} ## vanishes???
    please tell me what I am missing??
     
  2. jcsd
  3. Feb 16, 2015 #2

    mathman

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    Science Advisor
    Gold Member

    I believe you could clear things up for yourself if you explicitly used the sum sign rather than relying on the convention.
     
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