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Thread: A Partial Derivatives Proof Question.

  1. #1
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    Red face A Partial Derivatives Proof Question.

    Let:
    A Partial Derivatives Proof Question.-f.jpg
    be a differentiable function, where:
    A Partial Derivatives Proof Question.-.jpg
    then show that
    A Partial Derivatives Proof Question.-sdfdsf.jpg


    I don't know what the train of thought is to working through this proof is, could somebody please shed some light on how I should go about proving this.

    All those partial derivatives look daunting and strange, I can assure you my algebra and high school calculus skills are decent, but this question just doesn't make any sense to me.

    Thank you for helping me out I really do appreciate it.
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  2. #2
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    Re: A Partial Derivatives Proof Question.

    In case a picture helps...



    ... where (key in spoiler) ...

    Spoiler:


    is the chain rule for two inner functions, i.e...

    \displaystyle{\frac{d}{du}\ f(x(u), y(u)) = \frac{\partial f}{\partial x} \frac{dx}{du} + \frac{\partial f}{\partial y} \frac{dy}{du}}

    As with...



    ... the ordinary chain rule, straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case u and v in turn), and the straight dashed line similarly but with respect to the (corresponding) dashed balloon expression which is (one of) the inner function(s) of the composite expression (in this case x and y).

    But for the second derivative the chain rule is wrapped inside the left leg of each of two occurances of the legs-uncrossed version of...



    ... the product rule, where, again, straight continuous lines are differentiating downwards with respect to u or v.



    Add the bottom rows and simplify.


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    Last edited by tom@ballooncalculus; May 19th 2012 at 06:11 AM.
    Thanks from happyfingerrs and jsndacruz
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