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8: Vector Calculus Theorems

The fundamental theorems connecting line integrals, surface integrals, and volume integrals.

Overview: The Big Three

TheoremRelatesEquation
Green’sLine integral ↔ Double integral∮CF⋅dr=∬DcurlzF dA\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_D \text{curl}_z \mathbf{F} \, dA
Stokes’Line integral ↔ Surface integral∮CF⋅dr=∬Scurl F⋅dS\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S \text{curl } \mathbf{F} \cdot d\mathbf{S}
DivergenceSurface integral ↔ Volume integral∬SF⋅dS=∭Ediv F dV\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_E \text{div } \mathbf{F} \, dV

These are all generalizations of the Fundamental Theorem of Calculus.

Curl and Divergence

Curl

For F=⟨P,Q,R⟩\mathbf{F} = \langle P, Q, R \rangle:

curl F=∇×F=∣ijk∂∂x∂∂y∂∂zPQR∣\text{curl } \mathbf{F} = \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix} =⟨∂R∂y−∂Q∂z,∂P∂z−∂R∂x,∂Q∂x−∂P∂y⟩= \left\langle \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right\rangle

Physical meaning: Measures rotation/circulation at a point. The curl vector points along the axis of rotation.

Divergence

div F=∇⋅F=∂P∂x+∂Q∂y+∂R∂z\text{div } \mathbf{F} = \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}

Physical meaning: Measures expansion/compression at a point.

  • div F>0\text{div } \mathbf{F} > 0: source (flow outward)
  • div F<0\text{div } \mathbf{F} < 0: sink (flow inward)
  • div F=0\text{div } \mathbf{F} = 0: incompressible

Key Identities

  • curl(∇f)=0\text{curl}(\nabla f) = \mathbf{0} (gradient fields are irrotational)
  • div(curl F)=0\text{div}(\text{curl } \mathbf{F}) = 0 (curl fields are incompressible)
  • div(∇f)=∇2f=fxx+fyy+fzz\text{div}(\nabla f) = \nabla^2 f = f_{xx} + f_{yy} + f_{zz} (Laplacian)

Green’s Theorem

Let CC be a positively oriented (counterclockwise), piecewise smooth, simple closed curve in the plane, and let DD be the region bounded by CC.

Circulation Form

∮CF⋅dr=∮CP dx+Q dy=∬D(∂Q∂x−∂P∂y)dA\oint_C \mathbf{F} \cdot d\mathbf{r} = \oint_C P \, dx + Q \, dy = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA

Interpretation: Circulation around CC equals the total curl inside.

Flux Form

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

Interpretation: Flux across CC equals the total divergence inside.

Applications

  1. Compute area: A=12∮Cx dy−y dxA = \frac{1}{2}\oint_C x \, dy - y \, dx

  2. Simplify line integrals by converting to double integrals (or vice versa)

  3. Prove path independence (if curl = 0 everywhere)

Stokes’ Theorem

Let SS be an oriented surface with boundary curve CC (oriented by right-hand rule).

∮CF⋅dr=∬S(curl F)⋅dS\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\text{curl } \mathbf{F}) \cdot d\mathbf{S}

Interpretation: Circulation around CC equals the total curl flowing through SS.

Special Cases

  • If SS is in the xyxy-plane: reduces to Green’s Theorem
  • If curl F=0\mathbf{F} = \mathbf{0}: line integral is path-independent

Applications

  1. Compute line integrals via surface integrals (or vice versa)

  2. Show independence of surface: If two surfaces share the same boundary, they give the same flux of curl F\mathbf{F}

  3. Physical: Relates circulation to vorticity

The Divergence Theorem (Gauss’s Theorem)

Let EE be a solid region with outward-oriented boundary surface SS.

∬SF⋅dS=∭E(div F) dV\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_E (\text{div } \mathbf{F}) \, dV

Interpretation: Total flux out of EE equals the total divergence inside.

Applications

  1. Compute flux through closed surfaces

  2. Derive physical laws: Conservation of mass, Gauss’s law in electromagnetism

  3. Compute volume: V=13∬Sr⋅dSV = \frac{1}{3}\iint_S \mathbf{r} \cdot d\mathbf{S} where r=⟨x,y,z⟩\mathbf{r} = \langle x, y, z \rangle

Example: Flux of Position Vector

For F=r=⟨x,y,z⟩\mathbf{F} = \mathbf{r} = \langle x, y, z \rangle:

  • div r=3\text{div } \mathbf{r} = 3
  • Flux through closed surface SS bounding volume VV: ∬Sr⋅dS=3V\iint_S \mathbf{r} \cdot d\mathbf{S} = 3V

Summary: When to Use Each Theorem

If you have…And want…Use…
Line integral in 2DDouble integralGreen’s
Line integral in 3DSurface integralStokes’
Surface integral (closed)Volume integralDivergence

The Generalized Stokes’ Theorem

All three theorems are special cases of:

∫∂Ωω=∫Ωdω\int_{\partial \Omega} \omega = \int_\Omega d\omega

“The integral of a form over the boundary equals the integral of its derivative over the interior.”

Physical Applications

Fluid Flow

  • Divergence theorem: Conservation of mass
  • Stokes’ theorem: Kelvin’s circulation theorem

Electromagnetism (Maxwell’s Equations)

  • ∬SE⋅dS=Qϵ0\iint_S \mathbf{E} \cdot d\mathbf{S} = \frac{Q}{\epsilon_0} (Gauss’s law)
  • ∮CB⋅dr=μ0I\oint_C \mathbf{B} \cdot d\mathbf{r} = \mu_0 I (Ampère’s law)

Heat Flow

  • Flux of heat = −k∇T-k\nabla T
  • Heat equation derived using divergence theorem

Quick Reference

ConceptFormula
Curl∇×F=⟨Ry−Qz,Pz−Rx,Qx−Py⟩\nabla \times \mathbf{F} = \langle R_y - Q_z, P_z - R_x, Q_x - P_y \rangle
Divergence∇⋅F=Px+Qy+Rz\nabla \cdot \mathbf{F} = P_x + Q_y + R_z
Green’s (circulation)∮CP dx+Q dy=∬D(Qx−Py) dA\oint_C P\,dx + Q\,dy = \iint_D (Q_x - P_y)\,dA
Stokes’∮CF⋅dr=∬S(∇×F)⋅dS\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}
Divergence∬SF⋅dS=∭E(∇⋅F) dV\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_E (\nabla \cdot \mathbf{F})\,dV