Possibly a variation of the increment theorem for functions of multiple variables Suppose that $F:\mathbb R^n\to\mathbb R$ is $C^\infty$. I'd like to prove that $\forall a=(a_1,\ldots,a_n)\in\mathbb R^n$, there exist $C^\infty$ functions $G_i,i=1\ldots,n,$ such that $\forall x=(x_1,\ldots,x_n)\in\mathbb R^n$, we have
$$F(x)=F(a)+\sum_{i=1}^n(x_i-a_i)G_i(x).$$
In fact, we have
$$G_i(a)=\frac{\partial F}{\partial x_i}(a)$$
for all $i$. This theorem is quoted in the appendix to my physics textbook, but I'm not sure about its legitimacy. According to my experience in the calculus course, the change in the value of $F$ from $a$ to $x$  should take the form
$$\Delta F=\sum_{i=1}^n[(x_i-a_i)\frac{\partial F}{\partial x_i}(a)+\epsilon_i(x_i-a_i)],$$
where each $\epsilon_i$ goes to zero as all $x_i-a_i$ tend to zero. Are these two statements consistent with each other? Thank you.
 A: Yes, the theorem is true as stated. It's all a matter of how much regularity you want to impose on your functions. Here's the general theorem:

Let $n,k\geq 1$ be integers (with $k=\infty$ also allowed), and let $F:\Bbb{R}^n \to \Bbb{R}$ be a $C^k$ function. Then, for each $a\in \Bbb{R}^n$, there are functions $G_1,\dots, G_n:\Bbb{R}^n\to \Bbb{R}$ of class $C^{k-1}$ such that for every $x\in \Bbb{R}^n$,
\begin{align}
F(x) -F(a) &= \sum_{i=1}^n (x_i-a_i)\cdot G_i(x),
\end{align}
and such that $G_i(a) = \dfrac{\partial F}{\partial x_i}(a)$.

The proof is actually pretty simple: fix a point $a \in \Bbb{R}^n$, and given any $x\in \Bbb{R}^n$, we consider the parametrized line $\gamma(t) = a + t(x-a)$, $t\in \Bbb{R}$ Clearly, $\gamma$ is $C^{\infty}$; now notice that
\begin{align}
F(x)-F(a) &= F(\gamma(1)) - F(\gamma(0)) \\
&= \int_0^1 (F\circ \gamma)'(t)\, dt \tag{by FTC} \\
&= \int_0^1 \sum_{i=1}^n (x_i-a_i) \dfrac{\partial F}{\partial x_i}(\gamma(t))\, dt \tag{chain rule} \\
&= \sum_{i=1}^n (x_i-a_i) \cdot \underbrace{\int_0^1 \dfrac{\partial F}{\partial x_i}(a + t(x-a))\, dt}_{G_i(x)}
\end{align}
Now, all you have to do is prove that $F$ being $C^k$ implies each $G_i$ is $C^{k-1}$; this is only slightly technical. Finally, it is obvious that $G_i(a) = \dfrac{\partial F}{\partial x_i}(a)$.

Note that the second equation you wrote down is nothing more than the definition of differentiability at a single point $a$. We're not assuming anything more than differentiability at a single point. In the theorem above we're making the much stronger assumption that of $F$ being $C^k$ on the whole of $\Bbb{R}^n$, which is why you can show the existence of functions $G_i$ with such and such property.
