Home / Courses / Calculus III (Multivariable) / Unit 8

Calculus III (Multivariable) Unit 8: Surface integrals, Stokes & divergence

Surface integrals measure flux through a surface. Stokes' and the divergence theorem connect them to line and volume integrals.

Covers: Parametric surfaces, flux, Stokes' theorem, divergence theorem

Study guide

Surfaces can be parametrized as r(u, v) = ⟨x(u, v), y(u, v), z(u, v)⟩, such as a sphere in spherical angles or a graph z = f(x, y) using x and y as parameters. The vectors r_u and r_v are tangent to the surface, and their cross product r_u × r_v is a normal vector. Surface area is ∬ |r_u × r_v| dA.

A surface integral of a scalar function, ∬_S f dS, uses dS = |r_u × r_v| du dv. The flux of a vector field through an oriented surface, ∬_S F·dS = ∬ F·(r_u × r_v) du dv, measures how much of the field flows through it. Orientation matters: choose the normal (outward for closed surfaces) consistently.

The divergence of F, ∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z, measures how much the field spreads out from a point. The curl, ∇ × F, measures rotation. A conservative field has zero curl.

Stokes' theorem relates the circulation around a closed curve to the curl through any surface it bounds: ∮_C F·dr = ∬_S (curl F)·dS, with the orientation of C and S linked by the right-hand rule. The divergence theorem relates the outward flux through a closed surface to the divergence inside: ∯_S F·dS = ∭_E div F dV. Both generalize Green's theorem and the Fundamental Theorem of Calculus: the integral of a derivative over a region equals a sum over its boundary.

For a surface z = f(x, y) with an upward normal, dS = ⟨−f_x, −f_y, 1⟩ dA, so the flux is ∬ (−P f_x − Q f_y + R) dA over the surface's shadow in the xy-plane. curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩, the determinant with rows (i, j, k), (∂/∂x, ∂/∂y, ∂/∂z) and (P, Q, R).

In Stokes' theorem you may use any surface with the given boundary curve, so pick the simplest one, often a flat disc. Orient it so the curve runs counterclockwise when viewed from the tip of the normal. To find the flux through an open surface, you can add a simple lid to make a closed surface, apply the divergence theorem, and subtract the lid's flux.

Key ideas

• Parametrize a surface with r(u, v); the normal is r_u × r_v.
• Flux ∬_S F·dS measures flow through a surface.
• curl F measures rotation; div F measures outward spreading.
• Stokes' theorem: circulation around a boundary equals flux of the curl through the surface.
• Divergence theorem: outward flux through a closed surface equals the integral of div F over the solid.

Formulas and rules

Common mistakes

• Using the divergence theorem on a surface that isn't closed.
• Mismatching surface orientation and boundary direction in Stokes' theorem.

Words to know

Divergence of F

∇·F

Curl of F

∇ × F

Stokes' theorem

∮_C F·dr = ∬_S (curl F)·dS

Divergence theorem

Outward flux through a closed surface = ∭ div F dV

Normal vector for a parametrized surface

r_u × r_v

Parametric surface

r(u, v) = ⟨x(u, v), y(u, v), z(u, v)⟩

Surface area (parametric)

∬ |r_u × r_v| dA

Surface integral

∬_S f dS, with dS = |r_u × r_v| du dv.

Flux

∬_S F·dS: how much of F passes through S.

Orientation

The chosen normal direction of a surface (outward for closed surfaces).

Curl of a conservative field

0

Sample practice questions

17 more questions like these, from level 1 to 5, are in the practice set.