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Step 18: Multivariable calculus

Green's, Stokes', and the divergence theorems

Step 18 · Multivariable calculus topics

Use the three big theorems of vector calculus to trade an integral around a boundary for an integral over the region inside, and compute both sides for simple shapes.

Green's theorem

For a region D in the plane whose boundary C runs counterclockwise, the circulation of F = ⟨P, Q⟩ around C equals the integral of Qx−PyQ_x - P_y over D. Tiny swirls inside cancel each other, leaving only the swirl along the edge.

∮CP dx+Q dy=∬D(∂Q∂x−∂P∂y)dA\oint_C P\,dx + Q\,dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dA

Example: Checking the swirl

Use Green's theorem for F = ⟨−y, x⟩ around the circle of radius 2.

  1. Qx−Py=1−(−1)=2Q_x - P_y = 1 - (-1) = 2.
  2. Multiply by the disc's area, 4π.

Answer8π, matching the line integral

Example: Area from the boundary

Show that ∮Cx dy\oint_C x\,dy is the area inside C, and use it for the unit square.

  1. With P = 0 and Q = x, Qx−Py=1Q_x - P_y = 1, so the double integral is the area.
  2. For the unit square that is 1.

Answer1

Stokes' theorem

Stokes' theorem lifts Green's theorem into space. For a surface S with boundary curve C, the circulation of F around C equals the flux of curl F through S.

∮CF⋅dr=∬S(∇×F)⋅n dS\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n}\,dS

Example: Both sides

F = ⟨−y, x, 0⟩, and S is the upper half of the sphere of radius 1, whose edge is the unit circle in the xy-plane. Check Stokes' theorem.

  1. Around the circle, F · r′ = sin²t + cos²t, which is 1, so the circulation is 2π.
  2. curl F = ⟨0, 0, 2⟩. Its flux through the dome equals its flux through the flat unit disc with the same edge: 2 × π.

AnswerBoth sides give 2π

The divergence theorem

For a solid E with closed boundary surface S, the outward flux of F through S equals the integral of the divergence over E. The divergence measures how much F spreads out from each point.

∬SF⋅n dS=∭E∇⋅F dV∇⋅F=Px+Qy+Rz\iint_S \mathbf{F} \cdot \mathbf{n}\,dS = \iiint_E \nabla \cdot \mathbf{F}\,dV \qquad \nabla \cdot \mathbf{F} = P_x + Q_y + R_z

Example: A sphere again

Find the outward flux of F = ⟨x, y, z⟩ through the sphere of radius 2.

  1. div F = 1 + 1 + 1, which is 3.
  2. Multiply by the ball's volume, 32π3\frac{32\pi}{3}.

Answer32π, matching the surface integral

Example: A cube

Find the outward flux of F = ⟨x², 0, 0⟩ through the cube 0 ≤ x, y, z ≤ 1.

  1. div F = 2x.
  2. The integral of 2x over the cube is 1.

Answer1

TheoremBoundary sideInside side
Fundamental theoremf(b) − f(a)Integral of f′
Green'sCirculation around CCurl over D
Stokes'Circulation around CFlux of curl through S
DivergenceFlux through SDivergence over E

Try it yourself

Work each one on paper first, then open the answer to check.

  1. F = ⟨−y, x⟩. What is Qx−PyQ_x - P_y?

    Show answer

    2

  2. Use Green's theorem to find the circulation of ⟨−y, x⟩ around the unit square.

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    2

  3. What is the divergence of ⟨x, y, z⟩?

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    3

  4. Find the outward flux of ⟨x, y, z⟩ through the unit cube 0 ≤ x, y, z ≤ 1.

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    3

  5. What is the divergence of ⟨y, z, x⟩?

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    0

  6. curl F = ⟨0, 0, 4⟩. What is the circulation of F counterclockwise around the unit circle in the xy-plane?

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    4π

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