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Math Help - change of variable in a PDE

  1. #1
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    change of variable in a PDE

    I'm having some troubles figuring this out:

    Given g( x(u,v) , y(u,v) ) = h( u,v),
    and that
    u = {\sigma^2 \over 2} (T-x)
    v = ln {y \over k} - (0.5 \sigma^2 -r)(T-x)

    how do we find {\partial g \over \partial x}
    and {\partial g \over \partial y}?
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  2. #2
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    can anyone explain how is it that:
    {\partial g \over \partial x} = - {\sigma^2 \over 2} {\partial h \over \partial u} - (r - 0.5 \sigma^2) { \partial h \over \partial v}

    my guess is:
    {\partial g \over \partial x} = ({\partial  \over \partial u}h){\partial u  \over \partial x} + ({\partial  \over \partial v}h){\partial v  \over \partial x}

    normally, we have {\partial  \over \partial u}h as the solution, but this time round, we have {\partial u  \over \partial x} instead.
    Can someone please verify if my reasoning is correct?
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  3. #3
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    If g=h and h=f_1(u,v) with u=f_2(x,y) and v=f_3(x,y), then using the general chain rule:

    \frac{\partial}{\partial x} g=\frac{\partial}{\partial x} h

    and:

    \frac{\partial}{\partial x} h=\frac{\partial h}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial h}{\partial v}\frac{\partial v}{\partial x}
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