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

[gadje]

Homework Statement

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] .

Homework 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]

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.

 

[vela]

Try differentiating

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

with respect to a.

 

[gadje]

[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?

 

[vela]

Good. Now let a=1.