The divergence theorem, also called Gauss’s theorem, says that the outward flux of a vector field F through a closed surface S equals the triple integral of div F over the solid E inside it: ∬ F · n dS over S = ∭ div F dV over E. It turns a hard surface integral into an easier volume integral.
Stokes’ theorem says the circulation of a vector field F around the boundary curve C of a surface S equals the flux of curl F through S: ∮ F · dr over C = ∬ (curl F) · n dS over S. C goes around S so that S stays on your left when your head points along n.
A surface integral adds up the values of a function over a surface S, as a line integral does along a curve. With a parametrization r(u, v), the piece of area is dS = |r_u × r_v| dA. For a vector field F, the surface integral of F · n is the flux: how much F flows through S.
A parametric surface is a surface traced out by a vector function of two parameters, r(u, v) = ⟨x(u, v), y(u, v), z(u, v)⟩, as (u, v) moves over a region D. Its tangent vectors r_u and r_v give a normal vector r_u × r_v, and its surface area is the double integral of |r_u × r_v| over D.
The divergence and curl are two derivatives of a vector field F = ⟨P, Q, R⟩. The divergence, P_x + Q_y + R_z, is a number that measures how much F spreads out from a point. The curl is a vector that measures how much F spins around a point, and about which axis.
Green’s theorem turns a line integral around a closed curve C into a double integral over the region D inside it. The curve must be traced counterclockwise, with D on the left. It often replaces several line integrals with one easy double integral, and it gives a formula for area.
A conservative vector field is the gradient of some function f, called a potential function: F = ∇f. For these fields, a line integral depends only on the endpoints, because the integral of ∇f · dr along C equals f(end) − f(start).
A line integral adds up a quantity along a curve C instead of along an interval. The scalar line integral of f ds gives things like the mass of a wire, and the vector line integral of F · dr gives the work done by a force along the curve.
A vector field assigns a vector to every point in a region. In the plane it has the form F(x, y) = ⟨P(x, y), Q(x, y)⟩, and in space F(x, y, z) = ⟨P, Q, R⟩. Vector fields model wind, water currents and forces.
A change of variables rewrites a multiple integral in new variables u and v to make the region or the integrand simpler. The Jacobian is the determinant that tells how much the substitution stretches area. You replace dA with |∂(x, y)/∂(u, v)| du dv and integrate over the new region.