# Expressing a vector valued function in terms of its derivatives

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.

• Consider the constant functions. May 15 at 16:57
• are the $f_i$'s homogeneous polynomials? May 15 at 16:59
• I think the constant function is ok since we can use also $f(x_1,0,\dots,0)$. No, in general the polynomials are not homogeneous.
– R_O
May 15 at 17:19
• Missed that inclusion given the title. Sorry. May 15 at 18:12
