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Homework Help: Multivariable Calculus: Applications of Grad (and the Chain Rule?)

  1. May 10, 2010 #1
    1. The problem statement, all variables and given/known data
    We say that a differentiable function [tex]f : \mathbb{R}^n \rightarrow \mathbb{R}[/tex] is homogenous of degree p if, for every [tex]\mathbf{x} \in \mathbb{R}^n [/tex] and every a>0,
    [tex]f(a\mathbf{x}) = a^pf(\mathbf{x}).[/tex]

    Show that, if f is homogenous, then [tex]\mathbf{x} \cdot \nabla f(\mathbf{x}) = p f(\mathbf{x})[/tex] .

    2. Relevant equations
    The chain rule (not sure if I need it): [tex]\displaystyle \frac{d}{dt} f(\mathbf{x}(t)) = \Sigma_{i = 1}^{n} f_{x_i}\dot{x_i} = \dot{\mathbf{x}} \cdot \nabla f[/tex]
    3. The attempt at a solution

    Well, I see the resemblance between the rightmost hand side of the chain rule I wrote down, but I don't really understand how the chain rule is applied in this situation, seeing as there isn't anything about x being a function of something else here.

    Any ideas?
    Cheers.
     
    Last edited: May 10, 2010
  2. jcsd
  3. May 10, 2010 #2

    vela

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    Try differentiating

    [tex]f(a\mathbf{x}) = a^pf(\mathbf{x})[/tex]

    with respect to a.
     
  4. May 11, 2010 #3
    [tex]\frac{\partial}{\partial a} f(a \mathbf{x}) = pa^{p-1}f(\mathbf{x})[/tex]

    Okay. I'm still clueless.

    EDIT:

    Hang on. [tex] \frac{\partial}{\partial a} f(a \mathbf{x}) = \frac{\partial}{\partial a} (a \mathbf{x}) \frac{\partial}{\partial \mathbf{x}} f(a\mathbf{x}) = \mathbf{x} \cdot \nabla f (a \mathbf{x}) [/tex] (if you'll forgive the abuse of notation), right?
     
    Last edited: May 11, 2010
  5. May 11, 2010 #4

    vela

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    Good. Now let a=1.
     
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