If $F(x,y,z) \in \mathbb{C}[x,y,z]$ is a homogeneous polynomial of degree $n$, prove the Euler's formula.
$$XF_X+YF_Y+ZF_Z=nF$$
Could you give me a hint how we could show this?
EDIT: Could we show it like that?
Let $F(x,y,z)=\sum_{i+j+k=n} a_{ijk} \ x^i \ y^j \ z^k$ $$F_x(x,y,z)=\sum_{i+j+k=n} a_{ijk} i x^{i-1} y^j z^k$$
$$xF_x(x,y,z)=x\sum_{i+j+k=n} a_{ijk} i x^{i-1} y^j z^k=\sum_{i+j+k=n} a_{ijk} i x^{i} y^j z^k$$
$$yF_x(x,y,z)=y\sum_{i+j+k=n} a_{ijk} x^{i}j y^{j-1} z^k=\sum_{i+j+k=n} a_{ijk} x^{i} j y^j z^k$$
$$zF_x(x,y,z)=z\sum_{i+j+k=n} a_{ijk} x^{i} y^{j} k z^{k-1}=\sum_{i+j+k=n} a_{ijk} x^{i} y^j k z^k$$
$$xF_x+yF_y+zF_z=(i+j+k) \sum_{i+j+k=n} a_{ij} x^i y^j z^k=nF$$
Hint: prove that it is true for a monomial $X^iY^jZ^k$ of degree $n=i+j+k$, and that both sides are linear in $F$.