Surface Integrals, Stokes & Divergence

Study Sheet

Surface Integrals, Stokes & Divergence

Flux through surfaces and the two capstone theorems

The Capstone Theorems

Tip
Flux: how much field crosses a surface

The surface integral ∬SF⋅dS=∬SF⋅n dS\iint_S \mathbf F\cdot d\mathbf S = \iint_S \mathbf F\cdot\mathbf n\,dS measures net flow through SS: only the component of F\mathbf F along the unit normal n\mathbf n counts.

For a FLAT surface with constant field this collapses to (normal component) ×\times (area) — the flux of (0,0,5)(0,0,5) up through a radius-33 disk is 5⋅9π=45π5\cdot9\pi = 45\pi, no machinery required. For curved surfaces, parametrize and integrate; for CLOSED surfaces there is usually a shortcut, and it is the next box.

Tip
The divergence theorem and Stokes’ theorem

DIVERGENCE THEOREM: outward flux through a closed surface equals ∭∇⋅F dV\iiint \nabla\cdot\mathbf F\,dV over the enclosed solid — total outflow equals the sum of all the little sources inside.

STOKES: circulation ∮F⋅dr\oint\mathbf F\cdot d\mathbf r around a closed curve equals the flux of ∇×F\nabla\times\mathbf F through ANY surface spanning it — total rotation around the rim equals the sum of the little whirlpools on the membrane. Green's theorem is Stokes flattened into the plane; the fundamental theorem of calculus is the 1-dimensional case of everything. One slogan covers the whole family: the integral of a derivative over a region equals the original quantity summed over the boundary.

Side note
Why physicists write Maxwell’s equations twice

Each of Maxwell's equations has a differential form (about ∇⋅\nabla\cdot and ∇×\nabla\times at a point) and an integral form (about flux and circulation over regions) — and the divergence and Stokes theorems are precisely the dictionary between them. Gauss's law "flux of E\mathbf E = enclosed charge" IS the divergence theorem applied to ∇⋅E=ρ/ε0\nabla\cdot\mathbf E = \rho/\varepsilon_0. Learning these two theorems is learning to read electromagnetism.

Try it
Try it: flux through a sphere without parametrizing anything

Find the outward flux of F=(x,y,z)\mathbf F = (x, y, z) through the sphere of radius 22. Work: ∇⋅F=3\nabla\cdot\mathbf F = 3, so flux =3×volume=3⋅43π⋅8=32π= 3\times\text{volume} = 3\cdot\tfrac43\pi\cdot8 = 32\pi.

Cross-check on the unit sphere: there F=n\mathbf F = \mathbf n on the surface, so flux == area =4π= 4\pi — and the theorem gives 3⋅43π=4π3\cdot\tfrac43\pi = 4\pi ✓. When the divergence is constant, flux problems are volume problems.

Proofs & Why It Matters

Tip
Proof sketch: the divergence theorem on a box

On a box, split the flux into three pairs of opposite faces. For the zz-pair: ∬[R(x,y,ztop)−R(x,y,zbot)]dA=∬ ⁣ ⁣∫∂zR dz dA\iint\left[R(x, y, z_{\text{top}}) - R(x, y, z_{\text{bot}})\right]dA = \iint\!\!\int \partial_z R\,dz\,dA by the one-variable fundamental theorem — the flux difference across the pair equals the integral of ∂R/∂z\partial R/\partial z through the inside. Adding the three pairs assembles ∭(∂xP+∂yQ+∂zR) dV\iiint(\partial_xP + \partial_yQ + \partial_zR)\,dV. A general solid is tiled by small boxes: fluxes across interior walls cancel in pairs (shared walls, opposite normals), leaving the outer boundary. ■\blacksquare SIGNIFICANCE: "interior contributions cancel, only the boundary survives" is the single deepest pattern in integration — it is why conservation laws exist in physics.

Side note
The grand unification

Line up the family: FTC (∫abf′=f(b)−f(a)\int_a^b f' = f(b) - f(a)), FTL for gradients, Green, Stokes, divergence. Each says: derivative integrated over the inside == function summed over the boundary. In advanced mathematics all five become ONE statement about differential forms (∫Ωdω=∫∂Ωω\int_\Omega d\omega = \int_{\partial\Omega}\omega, the generalized Stokes theorem) — this course has been climbing the rungs of a single ladder the whole time.

Reminder — The Fundamental Theorem of Calculus:  ∫abf(x) dx=F(b)−F(a)where F′=f\int_a^b f(x)\,dx = F(b)-F(a) \quad\text{where } F'=f

Going Deeper: Worked Problems

Example
Worked: flux through a closed box with the divergence theorem

Find the outward flux of F=(x2,y,z)\mathbf F = (x^2, y, z) through the surface of the box [0,2]×[0,1]×[0,3][0,2]\times[0,1]\times[0,3].

Step 1 — divergence: ∇⋅F=2x+1+1=2x+2\nabla\cdot\mathbf F = 2x + 1 + 1 = 2x + 2.

Step 2 — integrate over the box: ∫03 ⁣∫01 ⁣∫02(2x+2) dx dy dz\displaystyle\int_0^3\!\int_0^1\!\int_0^2 (2x + 2)\,dx\,dy\,dz. The inner integral is [x2+2x]02=8\left[x^2 + 2x\right]_0^2 = 8.

Step 3 — the outer two multiply by the remaining side lengths: 8⋅1⋅3=248\cdot1\cdot3 = 24.

Step 4 — direct check of one face pair: on x=2x = 2 the flux is ∬4 dA=4⋅3=12\iint 4\,dA = 4\cdot3 = 12 outward, on x=0x = 0 it is 00; the yy-faces contribute 1⋅(2⋅3)=61\cdot(2\cdot3) = 6 net and the zz-faces 3⋅2−0=63\cdot2 - 0 = 6: total 12+6+6=2412 + 6 + 6 = 24 ✓. Six face integrals or one volume integral — the theorem lets you choose.

Example
Worked: Stokes' theorem picking the easy surface

Compute ∮CF⋅dr\displaystyle\oint_C \mathbf F\cdot d\mathbf r for F=(−y,x,z2)\mathbf F = (-y, x, z^2) around the circle x2+y2=4x^2 + y^2 = 4 in the plane z=1z = 1, counterclockwise from above.

Step 1 — curl: ∇×F=(∂y(z2)−∂z(x),;∂z(−y)−∂x(z2),;∂x(x)−∂y(−y))=(0,0,2)\nabla\times\mathbf F = \left(\partial_y(z^2) - \partial_z(x),; \partial_z(-y) - \partial_x(z^2),; \partial_x(x) - \partial_y(-y)\right) = (0, 0, 2).

Step 2 — cap the circle with the flat disk z=1z = 1, upward normal (0,0,1)(0,0,1): the flux of the curl is ∬2 dA=2⋅π⋅4=8π\iint 2\,dA = 2\cdot\pi\cdot4 = 8\pi.

Step 3 — by Stokes, the circulation is 8π8\pi.

Step 4 — verify directly: parametrize r=(2cos⁡t,2sin⁡t,1)\mathbf r = (2\cos t, 2\sin t, 1); then F⋅r′=(−2sin⁡t)(−2sin⁡t)+(2cos⁡t)(2cos⁡t)+1⋅0=4\mathbf F\cdot\mathbf r' = (-2\sin t)(-2\sin t) + (2\cos t)(2\cos t) + 1\cdot0 = 4, and ∫02π4 dt=8π\int_0^{2\pi}4\,dt = 8\pi ✓. The z2z^2 component never mattered — its curl contribution was zero, and Stokes said so before any parametrizing.