Integral Theorems of Vector Calculus
Divergence Theorem, Stokes' Theorem, and Green's Theorem
Contents
1. Review of Line, Surface, and Volume Integrals
Line Integral
The line integral of a vector field $\mathbf{A}$ along a curve $C$ is defined by
$$\displaystyle\int_C \mathbf{A} \cdot d\mathbf{l}$$where $d\mathbf{l}$ is the infinitesimal tangent vector to the curve $C$. When $C$ is parametrized by $\mathbf{r}(t)$ $(a \le t \le b)$,
$$\displaystyle\int_C \mathbf{A} \cdot d\mathbf{l} = \displaystyle\int_a^b \mathbf{A}(\mathbf{r}(t)) \cdot \dfrac{d\mathbf{r}}{dt}\, dt$$Physically, this corresponds to the work done by a force field $\mathbf{A}$ in moving a particle along the curve $C$.
Surface Integral
The surface integral of a vector field $\mathbf{A}$ through a surface $S$ is defined by
$$\iint_S \mathbf{A} \cdot d\mathbf{S}$$where $d\mathbf{S} = \hat{\mathbf{n}}\, dS$ is the area element vector along the normal direction. When the surface is parametrized by $\mathbf{r}(u, v)$,
$$d\mathbf{S} = \left( \dfrac{\partial \mathbf{r}}{\partial u} \times \dfrac{\partial \mathbf{r}}{\partial v} \right) du\, dv$$The value of the surface integral depends on the orientation of the surface (the direction of the normal). For a closed surface, the outward normal is conventionally taken as positive.
Volume Integral
The volume integral of a scalar field $f$ over a region $V$ is defined by
$$\iiint_V f\, dV$$In Cartesian coordinates, $dV = dx\, dy\, dz$.
2. The Divergence Theorem (Gauss's Theorem)
Divergence Theorem (Gauss's Theorem) Proof
Let $V$ be a bounded region enclosed by a piecewise smooth closed surface $S = \partial V$, and let $\mathbf{A}$ be a $C^1$ vector field on $V$. Then
$$\iiint_V (\nabla \cdot \mathbf{A})\, dV = \iint_S \mathbf{A} \cdot d\mathbf{S} = \oint_S \mathbf{A} \cdot \hat{\mathbf{n}}\, dS$$where $\hat{\mathbf{n}}$ is the outward unit normal vector to $S$.
Physical Interpretation
The total flux of the vector field through the closed surface $S$ (right-hand side) equals the total amount of source (divergence) inside the region (left-hand side).
- Points where $\nabla \cdot \mathbf{A} > 0$ are sources.
- Points where $\nabla \cdot \mathbf{A} < 0$ are sinks.
In fluid dynamics, this means the net outflow of fluid through the closed surface equals the total amount produced by sources inside.
Corollaries
Gradient Form Proof
For a scalar field $f$,
$$\iiint_V \nabla f\, dV = \oint_S f\, d\mathbf{S}$$Curl Form Proof
For a vector field $\mathbf{A}$,
$$\iiint_V (\nabla \times \mathbf{A})\, dV = \oint_S (d\mathbf{S} \times \mathbf{A})$$Application: Gauss's Law
Application to Electromagnetism
Applying the divergence theorem to one of Maxwell's equations, $\nabla \cdot \mathbf{E} = \dfrac{\rho}{\varepsilon_0}$, gives
$$\oint_S \mathbf{E} \cdot d\mathbf{S} = \iiint_V \dfrac{\rho}{\varepsilon_0}\, dV = \dfrac{Q}{\varepsilon_0}$$This is the integral form of Gauss's law, which states that the flux of the electric field through a closed surface is proportional to the total enclosed charge $Q$.
Green's Identities
Green's First Identity Proof
Setting $\mathbf{A} = f \nabla g$ and applying the divergence theorem yields
$$\iiint_V \left( f \nabla^2 g + \nabla f \cdot \nabla g \right) dV = \oint_S f (\nabla g) \cdot d\mathbf{S}$$Green's Second Identity Proof
Subtracting the first identity with $f$ and $g$ swapped from itself gives
$$\iiint_V \left( f \nabla^2 g - g \nabla^2 f \right) dV = \oint_S \left( f \nabla g - g \nabla f \right) \cdot d\mathbf{S}$$This identity is used in proving uniqueness of solutions to partial differential equations (Laplace's and Poisson's equations) and in constructing Green's functions.
3. Stokes' Theorem
Stokes' Theorem Proof
Let $S$ be a piecewise smooth oriented surface with boundary curve $C = \partial S$, and let $\mathbf{A}$ be a $C^1$ vector field on $S$. Then
$$\iint_S (\nabla \times \mathbf{A}) \cdot d\mathbf{S} = \oint_C \mathbf{A} \cdot d\mathbf{l}$$Physical Interpretation
The flux of the curl through the surface $S$ (left-hand side) equals the circulation of the vector field along the boundary curve $C$ (right-hand side).
Intuitively, the total strength of the swirling inside the surface equals the flow that goes around its edge.
Orientation Convention: The Right-Hand Rule
The direction of the normal vector $\hat{\mathbf{n}}$ on the surface $S$ and the orientation of the boundary curve $C$ are related by the right-hand rule: when the thumb of the right hand points along $\hat{\mathbf{n}}$, the remaining fingers curl in the positive direction of traversal of $C$.
Corollary
Scalar Form
For a scalar field $f$,
$$\iint_S (d\mathbf{S} \times \nabla) f = \oint_C f\, d\mathbf{l}$$Application: Faraday's Law
The Law of Electromagnetic Induction
Applying Stokes' theorem to Maxwell's equation $\nabla \times \mathbf{E} = -\dfrac{\partial \mathbf{B}}{\partial t}$ gives
$$\oint_C \mathbf{E} \cdot d\mathbf{l} = -\dfrac{\partial}{\partial t} \iint_S \mathbf{B} \cdot d\mathbf{S} = -\dfrac{\partial \Phi_B}{\partial t}$$This is the integral form of Faraday's law of electromagnetic induction, stating that the electromotive force induced around a closed circuit equals the rate of change of the magnetic flux $\Phi_B$.
Connection to Conservative Fields
Conservative Field
If $\nabla \times \mathbf{A} = \mathbf{0}$, then by Stokes' theorem, for any closed curve $C$,
$$\oint_C \mathbf{A} \cdot d\mathbf{l} = 0$$In this case $\mathbf{A}$ is a conservative field (irrotational field), and there exists a scalar potential $\phi$ such that $\mathbf{A} = -\nabla \phi$. Equivalently, the line integral is independent of the path.
4. Green's Theorem
Green's Theorem (in the Plane) Proof
Let $D$ be a bounded closed region in the $xy$-plane, $C = \partial D$ its boundary (with positive orientation, i.e., counterclockwise), and $P(x,y)$, $Q(x,y)$ be $C^1$ functions on $D$. Then
$$\oint_C (P\, dx + Q\, dy) = \iint_D \left( \dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y} \right) dA$$Relation to Stokes' Theorem
Green's theorem is the two-dimensional special case of Stokes' theorem. Taking the vector field $\mathbf{A} = (P, Q, 0)$ and the surface $S$ to be a region $D$ in the $xy$-plane (with normal $\hat{\mathbf{z}}$), one has
$$(\nabla \times \mathbf{A}) \cdot \hat{\mathbf{z}} = \dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y}$$so Stokes' theorem reduces directly to Green's theorem.
Application to Area Computation
Area Formula
Setting $P = -y/2$ and $Q = x/2$ gives $\dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y} = 1$, so
$$A = \iint_D dA = \dfrac{1}{2} \oint_C (x\, dy - y\, dx)$$This formula computes the area enclosed by a closed curve directly from a boundary line integral. It is used in surveying and in computer graphics for polygon area calculation.
Example: Area of an Ellipse
Parametrizing the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ as $x = a\cos t$, $y = b\sin t$ $(0 \le t \le 2\pi)$ gives
$$A = \dfrac{1}{2} \oint_C (x\, dy - y\, dx) = \dfrac{1}{2} \displaystyle\int_0^{2\pi} (ab\cos^2 t + ab\sin^2 t)\, dt = \pi ab$$5. Relationships Among the Theorems
The Generalized Stokes Theorem
All three integral theorems are concrete instances of the generalized Stokes theorem (in the language of differential forms, $\displaystyle\int_M d\omega = \displaystyle\int_{\partial M} \omega$). Their common essence is the correspondence
"the integral of a derivative quantity over the interior = the integral over its boundary".
Hierarchy of the Theorems
- Green's theorem: 2D (planar region $D$ with boundary curve $\partial D$)
- Stokes' theorem: surface $S$ with boundary curve $\partial S$ (2D manifold with 1D boundary)
- Divergence theorem: volume $V$ with boundary surface $\partial V$ (3D manifold with 2D boundary)
Green's theorem is the special case of Stokes' theorem restricted to the $xy$-plane. Stokes' theorem and the divergence theorem correspond to the curl and the divergence respectively, and although they are independent statements at the surface, in the framework of the generalized Stokes theorem they are simply expressions of the same principle in different dimensions.
Helmholtz Decomposition
Helmholtz Decomposition Proof
A sufficiently smooth vector field $\mathbf{A}$ satisfying appropriate decay conditions admits a unique decomposition
$$\mathbf{A} = -\nabla \phi + \nabla \times \mathbf{B}$$using a scalar potential $\phi$ and a vector potential $\mathbf{B}$, where
- $-\nabla \phi$: irrotational component (curl-free, $\nabla \times (-\nabla \phi) = \mathbf{0}$)
- $\nabla \times \mathbf{B}$: solenoidal component (divergence-free, $\nabla \cdot (\nabla \times \mathbf{B}) = 0$)
This decomposition is a consequence of the divergence theorem and Stokes' theorem, and provides the mathematical foundation for separating electric and magnetic fields in electromagnetism.