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Confusing chain differentiation rule

  1. Aug 9, 2008 #1


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    If I have a function f from RxR to R, and a function g from RxR to RxR. What are the partial derivatives of the composition f(g)? I end up multiplying the derivative of f with g, but g is a vector? The partial derivative should have its image in R.
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  3. Aug 9, 2008 #2


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    Well, the total derivative of g is a matrix, whereas the total derivative of f is a vector. Together, they yield a vector.

    Let's take a concrete example:
    [tex]h(x,y)=f(u(x,y),v(x,y)), g(x,y)=(u(x,y),v(x,y))[/tex]
    We have therefore, for example:
    This is then one of the two partial derivatives of h, the other being differentiation with respect to y.
  4. Aug 9, 2008 #3


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    what does
    mean? Is the following formula correct?
    Last edited by a moderator: Aug 9, 2008
  5. Aug 9, 2008 #4


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    It means the partial derivative of f with respect to u, of course. What else could it mean?

    It doesn't make sense. If f(u(a,b),v(a,b)) makes any sense then f is a function of u and v, not x and y. You must mean [tex]\frac{\partial f}{\partial u}[/tex] not [tex]\frac{\partial f}{\partial x}[/tex].
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