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Math Help - Multi-variable chain rule

  1. #1
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    Multi-variable chain rule

    Question

    Suppose z=f(x-ct) + g(x+ct), where c is a constant. Show that

    z(tt)=c^(2)z(xx)

    Where z(tt) means the second partial derivative of z with respect to t, as for z(xx) as well.

    I know this involves some application of the chainn rule but I can't seem to prove it at all... please help!
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  2. #2
    Super Member Random Variable's Avatar
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     \frac{\partial^{2} }{\partial x^{2}} z (x,t) = f''(x-ct) + g''(x+ct)

     \frac{\partial}{\partial t} z(x,t) = -cf'(x-ct) + cg'(x+ct) (chain rule)

     \frac{\partial^{2} }{\partial t^{2}} z(x,t)= c^{2}f''(x-ct) + c^{2}g''(x+ct) = c^{2} \frac{\partial^{2}}{\partial x^{2}} z(x,t)
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