Homogeneous functions and inner products

Question

I am having trouble with understanding how homogeneous functions are related to the inner product. I'm trying to prove the following: For a functions $f:\mathbb{R}^n\rightarrow\mathbb{R}$, $f$ is homogeneous of degree $d$ if and only if $\langle x ,\nabla f(x)\rangle=df(x)$, where $x$ is an $n$-dimensional vector. How do homogeneous functions express themselves in derivatives or in the inner product?

Answer 1

Defn. $f(\mathbf x)$ is homogeneous of degree $k$ if $f(a\mathbf x) = a^kf(\mathbf x)$

Euler's homogeneous function theorem.

$f(\mathbf x)$ is homogeneous of degree $k \iff$ $\mathbf x \cdot \nabla f(\mathbf x) = kf(\mathbf x)$

To make things a little easier, suppose $\mathbf x = (x,y)$ and $a\mathbf x = (ax,ay) = (x',y')$

$\frac {d}{da} f(a\mathbf x) = \frac {df}{dx'}\frac {dx'}{da} + \frac {df}{dy'}\frac {dy'}{da} = \frac {df}{dx'}x + \frac {df}{dy'}y = ka^{k-1} f(\mathbf x)$

and let $a$ approach $1.$
$\frac {df}{dx}x + \frac {df}{dy}y = k f(\mathbf x)$

and $\frac {df}{dx}x + \frac {df}{dy}y = \mathbf x \cdot \nabla f(\mathbf x)$