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Math Help - Grad of a scalar field problem

  1. #1
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    Grad of a scalar field problem

    Hi,
    I am having trouble understanding the following question:
    There is a scalar field f(r) depending on r = (x,y,z) through r = \sqrt[2]{x^2+y^2+z^2}.
    Show that \nabla f = \frac{f'}{r}r

    I know that grad f is normally each partial derivative i.e. (df/dx, df/dy, df/dz) but how do I calculate it when it only has r as the parameter?
    I'm also unsure as to how you would calculate f' in this situation, as I can't see an equation to differentiate.

    Anything that helps me understand this better is greatly appreciated.
    Thanks.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    This is a confusing problem because of bad notation.

    It is given that f is a function whose domain is \mathbb{R}^3, but whose value only depends on r. So what is meant, really, is that there is a function g:\mathbb{R}\to \mathbb{R} such that f(x,y,z)=g(r(x,y,z))=g(\sqrt{x^2+y^2+z^2}).

    Now by f'(x,y,z), what is meant is g'(r(x,y,z)), where g' is the usual derivative

    Essentially, you just have to take the gradient on both sides of the equation relating f to g, using the chain rule.
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  3. #3
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    That makes it so much clearer - thanks a lot for your help
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