Integration (Intermediate)

Integral theorems of vector calculus (upper-undergraduate level)

Overview of the Intermediate Level

Figure 1. The overall picture of integration at the intermediate level. Built on line integrals along curves (work, circulation) and surface integrals over surfaces (flux), the three great integral theorems — Green's (2D), Stokes' (surface), and Gauss's divergence theorem (3D) — are all unified as special cases of the generalized Stokes theorem $\int_{\partial\Omega}\omega=\int_\Omega d\omega$.

At the intermediate level, we study the integral theorems of vector calculus: line integrals along curves, surface integrals over surfaces, and the three great integral theorems of Green, Stokes, and Gauss.

Learning Goals

Understand the definition and computation of line integrals
Understand surface integrals and flux
Understand and apply Green's theorem
Understand Stokes' theorem
Understand the Gauss divergence theorem
Understand the three integral theorems as unified through the generalized Stokes theorem

Chapter 1 Line Integrals of Scalar Fields
Integration along a curve, arc-length parameter
Chapter 2 Line Integrals of Vector Fields
Work, circulation, conservative fields
Chapter 3 Green's Theorem
Planar regions and boundary curves, applications
Chapter 4 Surface Integrals
Parametrization of surfaces, flux
Chapter 5 Stokes' Theorem
Surfaces and boundary curves, curl
Chapter 6 The Gauss Divergence Theorem
Closed surfaces, divergence, physical applications

Differentiation under the integral sign (Feynman's trick) — Explains differentiation under the integral sign (the Leibniz integral rule, Feynman's trick).
Evaluating the Gaussian integral — Evaluates the Gaussian integral $\int e^{-x^2}\,dx=\sqrt{\pi}$ by several methods: polar-coordinate change, differentiation under a parameter, and complex integration.
Real integrals via residues — Explains how to compute real definite integrals using the residue theorem.

Prerequisites

The content of Integration (Basic)
Vector differentiation (gradient, divergence, curl)
Parametrization of curves and surfaces

The Three Great Integral Theorems

Green's theorem

When $D$ is a region in the plane and $\partial D$ is its boundary:

$$\oint_{\partial D} (P\,dx + Q\,dy) = \iint_D \left(\dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y}\right) dA$$

Stokes' theorem

When $S$ is a surface and $\partial S$ is its boundary curve:

$$\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$$

Gauss's divergence theorem

When $V$ is a three-dimensional region and $\partial V$ is its boundary surface:

$$\iint_{\partial V} \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F}) \, dV$$

Reading

The boundary tells of the interior — three theorems were one [Reading] — How Green, Stokes, and Gauss are really a single statement: “the integral on the boundary = the integral over the interior.” Tells of the beauty of unification revealed by the generalized Stokes theorem.
The freedom to change coordinates — multiple integrals, polar coordinates, and Gauss's trick [Reading] — The same integral can come apart at once simply by changing coordinates. Why a double integral can change its clothes, and why the Gaussian integral $\int e^{-x^2}\,dx=\sqrt{\pi}$ resolves so brilliantly in polar coordinates — told all the way to the true nature of the Jacobian.
The integral that forgets its path — conservative fields and a single hole in space [Reading] — There are line integrals whose value does not change no matter which path joins the same two points. Why an integral can “forget the path and remember only the endpoints” — the conservative fields and potentials behind it, and how a single hole opened in space breaks that magic.

Frequently Asked Questions

What do you learn in intermediate integration?

You study the integral theorems of vector calculus: line integrals along curves (work and circulation), surface integrals over surfaces (flux), and the three great integral theorems of Green, Stokes, and Gauss. The focus is on how to capture a vector field through integration.

How do line integrals and surface integrals differ?

A line integral is an integral along a curve $C$, representing the work $\int_C \mathbf{F}\cdot d\mathbf{r}$ done by a vector field or its circulation. A surface integral is an integral over a surface $S$, representing the flux $\iint_S \mathbf{F}\cdot d\mathbf{S}$ the field carries through the surface. The essential difference is whether the object of integration is a curve or a surface.

Why can Green's, Stokes', and Gauss's theorems be unified into one?

Each expresses the same statement — that the integral over the boundary equals the integral over the interior — and they are all special cases of the generalized Stokes theorem $\int_{\partial\Omega}\omega=\int_\Omega d\omega$. The two-dimensional case is Green's theorem, the surface case is Stokes' theorem, and the three-dimensional case is Gauss's divergence theorem.