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Math Help - A Multivariable chain rule question which is stomping me :(

  1. #1
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    A Multivariable chain rule question which is stomping me :(

    The question goes like this:

    Use the chain rule to verify the identity y * (∂g/∂x) + x * (∂g/∂y) = 0 if g(x,y) = f(x^2 - y^2, y^2 - x^2) and f is differentiable.

    No answer key to help, so any support would be appreciated, thanks!
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  2. #2
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    Re: A Multivariable chain rule question which is stomping me :(

    Hey echodot.

    What is the function g in this case?
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  3. #3
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    Re: A Multivariable chain rule question which is stomping me :(

    Quote Originally Posted by echodot View Post
    Use the chain rule to verify the identity y * (∂g/∂x) + x * (∂g/∂y) = 0 if g(x,y) = f(x^2 - y^2, y^2 - x^2) and f is differentiable.
    Denote u=x^2-y^2,\;v=y^2-x^2. Then,

    \begin{Bmatrix} \dfrac{{\partial g}}{{\partial x}}=\dfrac{{\partial f}}{{\partial u}}\dfrac{{\partial u}}{{\partial x}}+\dfrac{{\partial f}}{{\partial v}}\dfrac{{\partial v}}{{\partial x}}=\dfrac{{\partial f}}{{\partial u}}(2x)+\dfrac{{\partial f}}{{\partial v}}(-2x)\\ \\\dfrac{{\partial g}}{{\partial y}}=\dfrac{{\partial f}}{{\partial u}}\dfrac{{\partial u}}{{\partial y}}+\dfrac{{\partial f}}{{\partial v}}\dfrac{{\partial v}}{{\partial y}}=\dfrac{{\partial f}}{{\partial u}}(-2y)+\dfrac{{\partial f}}{{\partial v}}(2y) \end{matrix}
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    Re: A Multivariable chain rule question which is stomping me :(

    @chiro, g is just some function.

    @Fernando, thanks for the quick help, that makes complete sense, not sure why I hadn't done that earlier.
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