Generalised euler theorem A function f(x, y) is said to be homogeneous of degree m if $f(tx, ty) = t^mf(x, y)$
for any real number $t \neq 0$. Euler’s theorem states that if f is homogeneous of
degree m and has all partial derivatives of first order, then 
$x\frac{\partial{f}}{\partial{x}}
%x
+y\frac{\partial{f}}{\partial{y}}
%f
%y
= mf(x, y)$.
a. Verify Euler’s theorem for $f(x, y) = Ax^2 + Bxy + Cy^2$ and for
$g(x, y) = tan^{−1} (y/x) (x ,= 0)$.
Done
b. Prove Euler’s theorem.
Take the derivative of first equation w.r.t t and sub t = 1
c. Generalise the theorem and prove your generalisation
I've done part 1 and part 2. Not sure how to approach the third part, as in what to generalise. Thank you

Generalization:
If $\;f(\textbf x)\;,\;\;\textbf x=(x_1,...,x_n)\;$, is homogeneous of degree $\;m\;$ and has all its partial derivatives of first order, then
$$mf(\textbf  x)=\sum_{k=1}^n x_k\frac{\partial f}{\partial x_k}$$
Proof . We define new variables $x_k':=x_kt\;,\;\;k=1,2,...,n\;$ , and then:
$$t^mf (\textbf x)=f(t\textbf x)\stackrel{\text{diff. wrt}\; t}\implies mt^{m-1}f(\textbf x)=\sum_{k=1}^n\frac{\partial f}{\partial x'_k}\frac{\partial x_k'}{\partial t}=\sum_{k=1}^n x_k\frac{\partial f}{\partial x'_k}$$
The theorem follows by choosing $\;t=1\;$ and thus $\;x'_k=x_k\;$ .
