Consider a function $$ f:\mathbb{R}^n\rightarrow\mathbb{R}^m $$ given by $m$ functions $f_i:\mathbb{R}^n\rightarrow \mathbb{R}$ that we can assume to be polynomials in $x_1,\dots,x_n$.

Does there exist any formula expressing $f$ as a linear combination of $\frac{\partial f}{\partial x_1}(x_1,\dots,x_n),\dots,\frac{\partial f}{\partial x_n}(x_1,\dots,x_n)$, $f(x_1,0,\dots,0), \frac{\partial f}{\partial x_1}(x_1,0,\dots,0),\dots,\frac{\partial f}{\partial x_n}(x_1,0,\dots,0)$, where the coefficients of the linear combination are functions of $x_1,\dots,x_n$? Here $\frac{\partial f}{\partial x_i} = \left(\frac{\partial f_1}{\partial x_i},\dots,\frac{\partial f_m}{\partial x_i}\right)$.

Basically I am asking for an analogue of Euler's formula for homogeneous polynomials which says that if $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a homogeneous polynomial of degree $d$ then $d\cdot f = x_1\frac{\partial f}{\partial x_1}(x_1,\dots,x_n) + \dots + x_n\frac{\partial f}{\partial x_n}(x_1,\dots,x_n)$.

Thank you.

homogeneouspolynomials? $\endgroup$