Summary of Vector Calculus
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 6.9: “The Divergence Theorem” (summary remarks) |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/6-9-the-divergence-theorem |
| Textbook used in class | Stewart, Calculus, Section 16.10: “Summary” |
Quick Reference
All four theorems share the same structure: an integral of a derivative over a region equals an integral of the original function over the boundary of that region.
| Theorem | Interior | Boundary | Relation |
|---|---|---|---|
| Fundamental Theorem of Calculus | $[a,b]$ (1D) | $\{a,b\}$ (0D) | $\int_a^b f'(x)\,dx = f(b)-f(a)$ |
| Fundamental Theorem for Line Integrals | curve $C$ (1D) | endpoints (0D) | $\int_C\nabla f\cdot d\mathbf{r} = f(B)-f(A)$ |
| Green’s Theorem | region $D$ (2D) | closed curve $\partial D$ (1D) | $\oint_{\partial D}\mathbf{F}\cdot d\mathbf{r} = \iint_D(\operatorname{curl}\mathbf{F})\cdot\mathbf{k}\,dA$ |
| Stokes’ Theorem | surface $S$ (2D) | closed curve $\partial S$ (1D) | $\oint_{\partial S}\mathbf{F}\cdot d\mathbf{r} = \iint_S\operatorname{curl}\mathbf{F}\cdot d\mathbf{S}$ |
| Divergence Theorem | solid $E$ (3D) | closed surface $\partial E$ (2D) | $\oiint_{\partial E}\mathbf{F}\cdot d\mathbf{S} = \iiint_E\operatorname{div}\mathbf{F}\,dV$ |
The Unifying Idea
Every theorem in this chapter is a version of the same principle: to find the total effect of a derivative throughout a region, you only need to look at the original function on the boundary. The derivative integrates out; the boundary captures everything.
This principle goes by the name of Stokes’ Theorem in its most general form (using differential forms), which unifies all five rows of the table above into a single equation. In MATH241, you work with the three-dimensional incarnations: Green’s, Stokes’, and Divergence.
When to Use Each Theorem
Fundamental Theorem for Line Integrals: Use when $\mathbf{F} = \nabla f$. Find $f$, evaluate at the endpoints, subtract. Never integrate along the curve.
Green’s Theorem: Use when the path is a closed curve in the $xy$-plane. Replace the line integral with a double integral (or vice versa). The curl integrand is $Q_x - P_y$.
Stokes’ Theorem: Use when the path is a closed curve in 3D space, or when you have a surface integral of a curl. The two surfaces with the same boundary give the same answer, so pick the simplest surface.
Divergence Theorem: Use when you need the flux through a closed surface. Replace with a triple integral of the divergence, or add a cap to close an open surface and subtract.
The Operators in Context
Three differential operators connect the theorems:
- Gradient: $\nabla f$ turns a scalar into a vector field. $\int_C \nabla f\cdot d\mathbf{r}$ depends only on endpoints.
- Curl: $\operatorname{curl}\mathbf{F} = \nabla\times\mathbf{F}$ measures rotation. $\operatorname{curl}(\nabla f) = \mathbf{0}$ always.
- Divergence: $\operatorname{div}\mathbf{F} = \nabla\cdot\mathbf{F}$ measures spreading. $\operatorname{div}(\operatorname{curl}\mathbf{F}) = 0$ always.
These two identities -- curl of gradient is zero, divergence of curl is zero -- explain why conservative fields have zero circulation (Green’s/Stokes’), and why curl fields have zero net flux (Divergence Theorem applied to any closed surface around a simply connected region).
Decision Flowchart
Given an integral involving F -- what to do?
Is it a LINE INTEGRAL?
Is the path closed AND F is a curl field? --> Stokes
Is F conservative (F = grad f)? --> FThm for line integrals
Otherwise --> direct computation
Is it a SURFACE INTEGRAL of F (flux)?
Is S a closed surface? --> Divergence Theorem
Is S open with boundary curve C? --> Stokes (go to curl integral) or direct
Is F = curl G? --> Stokes gives zero if boundary is empty
Is it a SURFACE INTEGRAL of a scalar f? --> direct computation (no theorem shortcuts)Check for Understanding
Question: You want to compute $\oint_C (2x-y)\,dx + (x+3y)\,dy$ where $C$ is the ellipse $x^2/4+y^2=1$. Which theorem applies?
Show answer
This is a line integral around a closed plane curve, so Green’s Theorem applies. With $P = 2x-y$ and $Q = x+3y$: $Q_x - P_y = 1 - (-1) = 2$. The integral equals $\iint_D 2\,dA = 2\cdot\text{Area}(D) = 2\cdot\pi\cdot 2\cdot 1 = 4\pi$.
Common Misconceptions
Green’s theorem, Stokes’ theorem, and the Divergence theorem are three unrelated results that must each be memorized separately.
This is the concept-image-conflicts-definition error. All three theorems are instances of the same organizing principle: the integral of a derivative over a region equals the integral of the original quantity over the boundary of that region. Green’s theorem applies in 2D (curl integrated over a region equals the line integral around its boundary); Stokes’ theorem lifts this to surfaces in 3D; the Divergence theorem relates divergence integrated over a solid to the flux through its closed surface. Recognizing the common structure reduces memorization to one idea with three specializations.