Geometric interpretation of Euler's identity for homogeneous functions

A function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is called homogeneous of degree $d \geq 0$ if $$f(\lambda x_1, \ldots, \lambda x_n ) = \lambda^d f(x_1, \ldots, x_n)$$ Differentiating both sides with respect to $\lambda$ and then plugging in $\lambda=1$, we obtain the following equality: $$ \sum_{i=1}^n x_i \frac{ \partial f}{\partial x_i}(x_1, \ldots, x_n) = d \cdot f(x_1, \ldots, x_n) $$ This equation is usually called "Euler's identity." It feels it should have a clean geometrical interpretation, but I'm blanking out on what it might be.

Think at it in spherical polar coordinates. For any $0\ne x\in \mathbb{R}^n$, write $$ x=r\omega, $$ where $r=\lvert x\rvert$ and $\omega=x/\lvert x\rvert$. The function $f\colon \mathbb R^n\to \mathbb R$ is $d$-homogeneous if and only if $$\tag{1} f(r\omega)=r^d f(\omega),$$ and the operator $x\cdot \nabla$ reads $$ x\cdot \nabla = r\frac{\partial}{\partial r}.$$ So, if $f$ satisfies (1), then $$\tag{2} x\cdot \nabla f (x) = r\partial_r(r^d) f(\omega) = d\cdot f(x).$$

EDIT 2018 Actually, the point here is that $x\cdot\nabla$ is the generator of dilations, that is, $$ \left.\partial_\epsilon f(e^\epsilon x)\right|_{\epsilon=0} = x\cdot \nabla f(x).$$ With an old-fashioned language we can say that $x\cdot \nabla$ is an infinitesimal dilation. Equation (1) expresses a certain behavior with respect to finite dilations, while equation (2) expresses the analogous behavior with respect to infinitesimal dilations. Euler's theorem is the statement that the two points of view are equivalent.

Let me make an example, a kind of a rotation Euler's theorem. Consider in $\mathbb R^3$ the operator $R_z:= x\partial_y - y\partial_x$. This operator is an infinitesimal rotation around the $z$-axis, because $$ \partial_\epsilon f( \cos( \epsilon) x - \sin(\epsilon) y, \sin (\epsilon) x +\cos (\epsilon) y, z)|_{\epsilon=0}=R_z f(x, y, z).$$ Now it turns out that $$ R_z f(x, y,z ) =0, \quad \iff\quad f=f(x^2+y^2, z).$$ Here, again, $f=f(x^2+y^2, z)$ is a global behavior under finite rotations: it expresses the fact that $f$ is invariant under rotations around the $z$ axis. The infinitesimal version of this is $R_z f=0$.

It is in practice much easier to verify a local statement, such as the latter, instead of a global one.

My way of thinking about it is using exact differentials as an analogy

$$df=\frac{\partial f}{\partial x_1}dx_1+\frac{\partial f}{\partial x_1}dx_2+\cdots=\sum_{i=1}^n \frac{\partial f}{\partial x_i}dx_i----(1)$$

An exact differential is basically saying that the total change of a function is the sum of changes of each argument/components that form part of the function times the amount of infinitesimal changes of each arguments.

For Euler's Homogeneous Function Theorem, the LHS played an analogous role of Equation (1) $$ \sum_{i=1}^n x_i \frac{ \partial f}{\partial x_i}(x_1, \ldots, x_n) = d \cdot f(x_1, \ldots, x_n) $$

except that the degree d of the Homogeneous Function is not necessary 1 (i.e. not just 1st order homogenous functions like those in thermodynamics, i.e. not just additive) thus you can end up something like the sum of changes is d times the function itself

(think e.g. for the case for d=2, for every unit change in one of the arguments, your function got bigger by squared)

So geometrically, just as equation (1) is saying that the change is represented by a tangent hyperplane of the function at a point, Euler's Homogenous function theorem is saying something like each component of the function is like a x^d curve stacking head to tail, and the overall change represented by the stack is d times the original height (value of function f)

May not be a rigorous analogy, thus amendments are welcome