Proof of Taylor expansion in multiple variables Given a smooth function $f:\mathbb R^n\rightarrow \mathbb R$, can I expand it at $a\in \mathbb R^n$so that 

$f(x)=f(a)+\sum(x_i-a)g_i(x)$ where $g_i$ are smooth functions with $g_i(a)=\frac {\partial f}{\partial x_i}(a)$?

If so, I would like to see some reference for(or direct here) the proof. 
 A: The result you wrote is true : it is called Hadamard's Lemma, but it is not truly a Taylor expansion of $f$ in a neighbourhood of $0$. Actually, the proof of your result relies heavily on the Taylor formula with integral rest. See the Wikipedia page of the Hadamard's lemma for more details on the proof.
If you are looking for "true" Taylor formulas in case of functions in multiple variables, you should check this Wikipedia page about them which will contain the formulas. As for how you can prove them, I can't think of an english reference as of now, but the usual way of proving them relies on the following principle :
Take a function $f : \mathbb{R}^n \to \mathbb{R}$, and also $a,x$ $\in \mathbb{R}^n$ (with $a$ the point where you're looking for a Taylor expansion). Then define $g: \mathbb{R} \to \mathbb{R}$ with $g(t) = f(a + t(x-a))$, and write down Taylor's formula for $g$ (where any version would work - just pay attention to the Taylor Lagrange one, you can find more information about why by searching on Google about the mean value theorem in multiples variables).
I guess I went a bit overboard with this answer, and not that precise. I hope it helped though, and feel free to ask any question.
A: Yes, you can expand $f:\mathbb{R}^n \rightarrow \mathbb{R}$ in the way posted because $\mathbb{R}^n$ is a neighborhood of the point $a$. 

If so, I would like to see some reference for(or direct here) the proof.

A proof can be found in Loring Tu's An Introduction to Manifolds (Second Edition) on page no.6.
