TITLE: Fundamental Theorem of Calculus in Multivariate Case
QUESTION [7 upvotes]: From the FTC we have, for continuously differentiable $f: \mathbb{R} \to \mathbb{R}$, 
$$
f(a) - f(b) = \int_b^a \frac{d}{dx} f(x) dx
$$
I'm trying to write the difference between a vector function in similar terms, that is, given $g : \mathbb{R}^d \to \mathbb{R}^d$, having known Jacobian $J_f(x)$, what can we say about
$$
g(x_1) - g(x_2) = ? 
$$

REPLY [2 votes]: I would refer you to Spivak's Calculus on Manifolds.   The real generalization of the fundmental theorem of calculus is the general Stokes theorem.  Spivak does a nice job of assembling the machinery to understand this deep and interesting result.