surface integral

A line integral spreads accumulation along a one-dimensional curve; a surface integral spreads it across a two-dimensional surface. Imagine a curved tarp on which the paint thickness varies from spot to spot, and you want the total amount of paint: you tile the surface into tiny patches, multiply the local thickness by each patch's area, and add them all. The surface integral is the limit of that sum as the patches shrink.

For a scalar field f over a surface S, the surface integral is written integral over S of f dS, where dS is an element of surface area. To compute it you parametrize the surface as r(u, v); the area element becomes the magnitude of the cross product of the two tangent vectors, dS = |r_u cross r_v| du dv, and the surface integral collapses to an ordinary double integral over the (u, v) parameter region. Setting f = 1 just recovers the surface area; using a mass density f gives the mass of a curved shell, and so on. The surface is one dimension up from the curve, so you now integrate two parameters instead of one.

Surface integrals are the second-to-last rung before the great theorems. Their most important special case is the flux integral, which uses a vector field and the surface's orientation to measure how much of the field passes through the surface; flux is the quantity that the divergence theorem links to divergence and that Stokes' theorem links to curl. Beyond physics they compute heat flow through a wall, electric flux through a Gaussian surface, and the mass throughput of a fluid across a membrane.