Green's, Gauss's, and Stokes' Theorems

date2026-07-15document_iddoc_20772b8be8e12158fcca76f73a6d7b4fdescriptionGreen・Gauss・Stokes の定理を、局所的な微分量と境界上の積分量を結ぶ同一原理として整理し、適用条件と計算例も確認する。prerequisites線積分と保存場 / 面積分と流束type講義content_typelecturestatusactive

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1Introduction

This lecture presents Green's theorem, Gauss's divergence theorem, and Stokes' theorem as manifestations of one principle relating differential quantities in a region to integral quantities on its boundary.

2Overview

Green's theorem relates planar curl to circulation along the boundary of a planar region. The divergence theorem relates divergence in a volume to flux through its boundary surface. Stokes' theorem relates curl on a surface to circulation along the boundary curve.

3Preconditions for Applying the Theorems

Before applying a theorem, identify the integration domain, determine the induced orientation, and verify the smoothness of the field throughout the relevant region. Green's theorem requires a planar region with its positively oriented boundary. The divergence theorem requires a closed surface with outward unit normal. Stokes' theorem requires an oriented surface whose boundary orientation is induced by the right-hand convention.

4Standard Formulas

One form of Green's theorem is

\oint_{\partial D} P d x + Q d y = \iint_{D} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) d A .

Here $D$ is a bounded planar region whose boundary consists of finitely many piecewise $C^{1}$ Jordan curves, and $P, Q$ are $C^{1}$ on an open neighborhood of $\overline{D}$ . Outer boundary components are oriented counterclockwise, whereas boundaries of holes are oriented clockwise. The two sides are respectively the total boundary circulation and the integral of interior rotation density.

Gauss's divergence theorem is

\iint_{\partial V} F \cdot n d S = ∭_{V} \nabla \cdot F d V .

Here $V$ is a bounded region with piecewise $C^{1}$ boundary, and $F$ is $C^{1}$ on an open neighborhood of $\overline{V}$ . The unit normal points outward from $V$ ; on the boundary of a cavity, it points into the cavity. The two sides are respectively the total flux through all boundary components and the volume integral of divergence.

Stokes' theorem states

\oint_{\partial S} F \cdot d r = \iint_{S} (\nabla \times F) \cdot n d S .

Assume that $S$ is a compact piecewise $C^{1}$ regular surface with piecewise $C^{1}$ boundary, that an orientation $n$ has been selected, and that $F$ is $C^{1}$ on an open set containing $S$ . For several boundary components, the left side is their sum. The induced boundary direction follows the right-hand rule: with the right thumb pointing along $n$ , the curled fingers indicate the positive traversal direction. Equivalently, when an observer traverses the boundary with their head pointing in the direction of $n$ , the surface lies on the observer's left. Reversing $n$ reverses the boundary orientation.

5Roles of the Three Theorems

Theorem	Boundary quantity	Interior quantity
Green	circulation along a planar boundary	rotation density over the region
Gauss	flux through a closed surface	divergence density over the volume
Stokes	circulation along a boundary curve	normal component of curl over the surface

6The Common Principle

Partition a region into small pieces. Each artificial interface shared by adjacent pieces occurs with opposite induced orientations, so its two contributions cancel. Only the boundary of the entire region remains, including components surrounding holes or cavities.

7Example 1: Green's Theorem

Let $F = (- y / 2, x / 2)$ and let $D$ be the unit disk. Green's theorem gives

\oint_{\partial D} F \cdot d r = \iint_{D} 1 d A = π .

For a direct computation, set $r (t) = (\cos t, \sin t)$ , $0 [PARSE ERROR: Undefined("Command(\"le\")")] t [PARSE ERROR: Undefined("Command(\"le\")")] 2 π$ . Then

F (r (t)) = (- \frac{\sin t}{2}, \frac{\cos t}{2}), r^{'} (t) = (- \sin t, \cos t),

F (r (t)) \cdot r^{'} (t) = \frac{1}{2}, \oint_{\partial D} F \cdot d r = \int_{0}^{2 π} \frac{1}{2} d t = π .

Green's theorem replaces the direct boundary computation with an interior integral and thereby clarifies the structural relation between the two representations.

8Example 2: The Divergence Theorem

Let $F = (x, y, z)$ and let $V$ be the ball of radius $R > 0$ centered at the origin. Since $\nabla \cdot F = 3$ ,

∭_{V} \nabla \cdot F d V = 3 \cdot \frac{4 π R^{3}}{3} = 4 π R^{3.}

On the boundary sphere, $F \cdot n = R$ and its area is $4 π R^{2}$ , so the outward flux is also $4 π R^{3}$ . This example shows that when divergence has the constant value $3$ , total flux grows in proportion to enclosed volume. The theorem is especially useful when the divergence integral is simpler than direct surface integration.

9Example 3: Stokes' Theorem

Let $F = (- y / 2, x / 2, 0)$ , let $S$ be the unit disk in the $x y$ -plane, and select the upward normal. Since $\nabla \times F = (0, 0, 1)$ ,

\iint_{S} (\nabla \times F) \cdot n d S = π .

The induced boundary orientation is counterclockwise, and the boundary line integral is also $π$ . Green's theorem is the planar instance of this form of Stokes' theorem.

10Choosing a Computation Method

Direct computation is sufficient when the boundary and field have simple parameterizations. When the boundary is complicated but the interior differential quantity is simple, these theorems can replace the original integral with a more convenient one.

11Connection to Differential Forms

The dimensions and domains differ, but all three theorems relate an interior derivative to a boundary integral. Differential forms express and generalize this common structure.

data/lecture/math/exterior-algebra/general-stokes-theorem-and-vector-calculus-dictionary.lecture.n.md

12Connection to the Laplacian

Let $u$ be $C^{2}$ on an open neighborhood of $\overline{V}$ and substitute $F = \nabla u$ into the divergence theorem. Then

\iint_{\partial V} \nabla u \cdot n d S = ∭_{V} Δ u d V .

The left side integrates the outward normal derivative $\partial u / \partial n = \nabla u \cdot n$ . The next lecture develops the meaning and physical applications of $Δ u = \nabla \cdot \nabla u$ .

data/lecture/math/vector-calculus/laplacian-and-physical-applications.lecture.n.md

13Fields with Singularities

If a field has an interior singularity, remove a neighborhood of it and include the newly created inner boundary. For example,

F (x, y) = (- \frac{y}{x^{2} + y^{2}}, \frac{x}{x^{2} + y^{2}})

is undefined at the origin, so Green's theorem cannot be applied directly to a disk containing the origin. Removing a small disk produces an annulus with a counterclockwise outer boundary and clockwise inner boundary. The curl vanishes on the annulus, while the circulations $2 π$ and $- 2 π$ cancel. The newly created inner-boundary contribution must not be omitted.

14Related Lectures

data/lecture/math/vector-calculus/vector-calculus-portal.lecture.n.md data/lecture/math/vector-calculus/line-integrals-and-conservative-fields.lecture.n.md data/lecture/math/vector-calculus/surface-integrals-and-flux.lecture.n.md