8: Vector Calculus Theorems

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

Overview: The Big Three

These are all generalizations of the Fundamental Theorem of Calculus.

Curl and Divergence

Curl

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

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

Divergence

Physical meaning: Measures expansion/compression at a point.

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

Key Identities

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

Green’s Theorem

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

Circulation Form

Interpretation: Circulation around $C$ equals the total curl inside.

Flux Form

Interpretation: Flux across $C$ equals the total divergence inside.

Applications

1. Compute area: $A = \frac{1}{2}\oint_C x \, dy - y \, dx$

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

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

Stokes’ Theorem

Let $S$ be an oriented surface with boundary curve $C$ (oriented by right-hand rule).

Interpretation: Circulation around $C$ equals the total curl flowing through $S$.

Special Cases

• If $S$ is in the $xy$-plane: reduces to Green’s Theorem
• If curl $\mathbf{F} = \mathbf{0}$: 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 $\mathbf{F}$

3. Physical: Relates circulation to vorticity

The Divergence Theorem (Gauss’s Theorem)

Let $E$ be a solid region with outward-oriented boundary surface $S$.

Interpretation: Total flux out of $E$ 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: $V = \frac{1}{3}\iint_S \mathbf{r} \cdot d\mathbf{S}$ where $\mathbf{r} = \langle x, y, z \rangle$

Example: Flux of Position Vector

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

• $\text{div } \mathbf{r} = 3$
• Flux through closed surface $S$ bounding volume $V$: $\iint_S \mathbf{r} \cdot d\mathbf{S} = 3V$

Summary: When to Use Each Theorem

The Generalized Stokes’ Theorem

All three theorems are special cases of:

“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)

• $\iint_S \mathbf{E} \cdot d\mathbf{S} = \frac{Q}{\epsilon_0}$ (Gauss’s law)
• $\oint_C \mathbf{B} \cdot d\mathbf{r} = \mu_0 I$ (Ampère’s law)

Heat Flow

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

Quick Reference