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Homework Help: Gradient Problem

  1. Jan 17, 2009 #1
    Define [tex] f: R^{2} \rightarrow R , by f(x,y) = \int^{sin(x sin(y sin z))}_{a} g(s) ds [/tex]


    where g:R -> R is continuous. Find the gradient of f.


    I tried using the FTC, and differentiating under the integral, but did not get anywhere,

    thanks for any suggestions.
     
  2. jcsd
  3. Jan 17, 2009 #2

    HallsofIvy

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    Science Advisor

    Yes, the FTC, together with the chain rule should work. Basically, you are saying that
    [tex]f(x,y)= \int_0^u(x,y) g(s)ds[/tex]
    where u(x,y)= x sin(x sin(y sin(x))).
    [tex]\frac{df}{du}= g(u)[/tex]
    and
    [tex]\frac{\partial f}{\partial x}= g(u)\frac{\partial u}{\partial x}[/tex]
    [tex]\frac{\partial f}{\partial y}= g(u)\frac{\partial u}{\partial y}[/tex]

    So the question is really just: What are [itex]\partial u/\partial x[/itex] and [itex]\partial u/\partial y[/itex]f?
     
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