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Math Help - chain rule question

  1. #1
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    chain rule question

    see if you can answer this one

    Let f = f(u,v) and u = x + y, v = x - y

    i) assume f to be twice differentiable and compute f_{xx} and f_{yy} in terms of f_{u}, f_{v}, f_{uu}, f_{uv} and f_{vv}.

    ii)express the wave equation:

    ((∂f)/(∂x))-((∂f)/(∂y))=0

    in terms of partial derivatives of f with respect to u and v.
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  2. #2
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    Quote Originally Posted by dexza666 View Post
    see if you can answer this one

    Let f = f(u,v) and u = x + y, v = x - y

    i) assume f to be twice differentiable and compute f_{xx} and f_{yy} in terms of f_{u}, f_{v}, f_{uu}, f_{uv} and f_{vv}.

    ii)express the wave equation:

    ((∂f)/(∂x))-((∂f)/(∂y))=0

    in terms of partial derivatives of f with respect to u and v.
    Well lets get started

    \frac{\partial f}{\partial x}=\frac{\partial f}{\partial u} \frac{\partial u}{\partial x}+ \frac{\partial f}{\partial v} \frac{\partial v}{\partial x}=f_u \cdot 1+f_v \cdot (1) = f_u + f_v

    Now we need the 2nd partial with respect to x

    f_{xx}=\frac{\partial}{\partial x}\frac{\partial f}{\partial x}=\frac{\partial }{\partial x} \left( f_u +f_v \right) = \frac{\partial f_{u}}{\partial x}+ \frac{\partial f_v}{\partial x} =

    \frac{\partial f_u}{\partial x}=\frac{\partial f_u}{\partial u} \frac{\partial u}{\partial x}+ \frac{\partial f_u}{\partial v} \frac{\partial v}{\partial x}=f_{uu} \cdot 1+f_{vu} \cdot (1) = f_{uu} + f_{vu}

    \frac{\partial f_v}{\partial x}=\frac{\partial f_v}{\partial u} \frac{\partial u}{\partial x}+ \frac{\partial f_v}{\partial v} \frac{\partial v}{\partial x}=f_{uv} \cdot 1+f_{vv} \cdot (1) = f_{uv} + f_{vv}

    so finally we have

    f_{xx}=(f_{uu}+f_{uv})+(f_{uv}+f_{vv})=f_{uu}+2f_{  uv}+f_{vv}

    The last step is by clairaut's theorem.

    You will need to do the same thing for partials with respect to y.

    Good luck.
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  3. #3
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    im a bit confused im assuming this is the answer to part i or is it a conjugation of part i and ii. if not what is part ii
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  4. #4
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    Quote Originally Posted by dexza666 View Post
    im a bit confused im assuming this is the answer to part i or is it a conjugation of part i and ii. if not what is part ii
    This half of number i.

    you need to find f_{yy}

    Then you plug both f_{xx},f_{yy}
    into the wave equaiton for part ii.
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