Calculus in several variables culminates in three theorems — Green's, Stokes', and the Divergence theorem — that are all one theorem: the integral of a derivative over a region equals the integral of the function over the boundary. This page runs the standard Calculus III arc: partial derivatives and the gradient, multiple integrals, then the vector-field machinery those unifying theorems live in.
Partial derivatives measure change along coordinate axes; the gradient \nabla f = (\partial f/\partial x_1, \ldots, \partial f/\partial x_n) packages them, points in the direction of steepest ascent, and is normal to level sets. The directional derivative along unit vector u is \nabla f \cdot u. The multivariable chain rule is matrix multiplication of Jacobians — the cleanest way to see it (details and the Hessian in Differential Calculus). Optimization: interior critical points have \nabla f = 0, classified by the Hessian's definiteness; constrained optima come from Lagrange multipliers — solve \nabla f = \lambda \nabla g on the constraint g = 0, which encodes that no ascent direction remains inside the constraint surface.
Double and triple integrals are Riemann sums over regions; Fubini's theorem reduces them to iterated one-variable integrals, and the practical skill is choosing integration order and describing the region's bounds. Changing variables costs a volume-distortion factor:
The polar (dA = r\,dr\,d\theta), cylindrical, and spherical (dV = \rho^2 \sin\varphi\,d\rho\,d\varphi\,d\theta) Jacobians are the ones to know cold. The classic payoff: computing \int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi} by squaring and switching to polar coordinates.
A vector field \mathbf{F} assigns a vector to each point; the line integral \int_C \mathbf{F} \cdot d\mathbf{r} is the work done along a curve. \mathbf{F} is conservative if it is a gradient field \mathbf{F} = \nabla \varphi; then work is path-independent and equals \varphi(\text{end}) - \varphi(\text{start}) — the multivariable FTC. On simply connected domains, conservativity is testable componentwise (\text{curl}\,\mathbf{F} = \mathbf{0}); the standard counterexample (-y, x)/(x^2+y^2) on the punctured plane shows the domain's topology matters — the observation that grows into de Rham cohomology in differential geometry.
Two identities structure everything: \nabla \times (\nabla \varphi) = \mathbf{0} (gradients don't rotate) and \nabla \cdot (\nabla \times \mathbf{F}) = 0 (curls have no sources).
Green's theorem (plane): for a positively oriented closed curve C bounding region R,
Stokes' theorem (surfaces in space): \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} — circulation around the boundary equals curl flux through the surface.
Divergence theorem (Gauss): over the closed boundary surface \partial V of a solid region V,
— flux out of a closed surface equals total source strength inside.
All three are instances of the generalized Stokes theorem \int_{\partial M} \omega = \int_M d\omega for differential forms — "boundary of the region" pairs with "derivative of the field." This unification is the gateway to differential forms and manifolds.
These theorems are the vocabulary of the field equations of physics (Maxwell's equations exist in both integral and differential form precisely via Stokes/Gauss), the derivation route for the heat and wave equations (conservation laws + divergence theorem), and the geometric substrate of fluid dynamics and electromagnetism. Computationally, gradients and Jacobians are the daily bread of optimization and machine learning.