Intuition about Euler's Theorem on homogeneous equations

Question

I wonder, what would be the intuition or motivation to studying Euler Formula for homogeneous function

$f:\mathbb{R}^k \to \mathbb{R}$ such that $f(tx) = t^n f$, for all $t>0$ .

$\sum x_i \frac{\partial f}{\partial x_i} = n f$

I understand its proof and can do some problem but it feels really artificial or rather just manipulation process in doing such problems.

Kindly share the intuition or importance of Euler theorem, or share the sources where I can read about it.

It would be helpful for me. Thanks in advanced.

Answer 1

"Motivation" for Euler formula can be found in the framework of Linear Algebra with the matrix form of the equation of a conic curve (as mentionned by @peek-a-boo):

$$Ax^2+By^2+2Cxy+2Dx+2Ey+F=0$$

Let us homogenize it $(x=\frac{X}{T},y=\frac{Y}{T})$ under the following form:

$$\varphi(X,Y,T)=AX^2+BY^2+2CXY+2DXT+2EYT+FT^2=0\tag{1}$$

which is homogeneous of degree $n=2$.

Partial derivatives of $\varphi$ with respect to $X,Y,T$ are:

$$\begin{cases}\partial \varphi/\partial X =2(AX+CY+DT)\\\partial \varphi/\partial Y =2(BX+CY+ET)\\\partial \varphi/\partial T =2(DX+EY+FT) \end{cases}\tag{2}$$

i.e., under matrix form

$$\varphi(X,Y,T)=\begin{pmatrix}X&Y&T\end{pmatrix}\begin{pmatrix}A&B&D\\B&C&E\\D&E&F\end{pmatrix}\begin{pmatrix}X\\Y\\T\end{pmatrix}=0$$

or

$$\varphi(X,Y,T)=\begin{pmatrix}X&Y&T\end{pmatrix}\begin{pmatrix}AX+CY+DT\\BX+CY+ET\\DX+EY+FT\end{pmatrix}=0$$

$$\varphi(X,Y,T)=\begin{pmatrix}X&Y&T\end{pmatrix}\begin{pmatrix}\tfrac12\partial \varphi/\partial X\\\tfrac12\partial \varphi/\partial Y \\\tfrac12\partial \varphi/\partial T\end{pmatrix}=0$$

$$2\varphi(X,Y,T)=\begin{pmatrix}X&Y&T\end{pmatrix}\begin{pmatrix}\partial \varphi/\partial X\\\partial \varphi/\partial Y \\\partial \varphi/\partial T\end{pmatrix}=0$$

(using (2)) which is Euler formula.

Remark : taking "half the coefficients of the partial derivatives" was rather usual "in yester time" for the presentation of the matrix associated with a conic curve (see for example here).

Answer 2

In short, the Euler theorem says that the "radial" derivative of a homogeneous function behaves like the "usual" derivative of the scalar function $f(t)= c t^n$, which satisfies $t\frac{d}{d t} f=nf$.

In more details:

The vector field $F(x)=(x_1, \ldots, x_k)$ is "radial" -- you can see this by observing that it is normal to the spheres through the origin (of any radius), since it is parallel to the gradient of the function $h(x_1, \ldots, x_k)=x_1^2+\ldots+x_k^2$, for which the spheres are the level sets.

The differential operator $\sum_j x_j \frac{\partial }{\partial x_j}$ is associated to this $F$. When applied to $f$ at some point, it takes directional derivative of this $f$ in the "radially out" direction and multiplies it by the $|F(x)|=|x|$. On the other hand, let's restrict $f$ to an "outward" ray through the origin. The ray is given by direction vector $\hat{u}$ and parameterized by distance to the origin $t$. By homogeneity, when restricted to this ray, the function $f(x)$ is simply $f(t\hat{u})=t^n f(\hat{u})$. Since $f$'s directional derivative at $x=t\hat{u}$ in the radial direction is just the derivative of this restriction to the ray, it is equal to $n t^{n-1} f(\hat{u})$. Multiplied by $|x|=|t \hat{u}|=t$ it gives $n t^n f(\hat{u})=n f(x)$. So $\sum_j x_j \frac{\partial }{\partial x_j} f=n f$.