Unit 8: Surface Integrals, Stokes' Theorem & Gauss' Divergence Theorem

Just as a space curve is traced out by a vector function of a single parameter $t$, a two-dimensional surface in $\mathbb{R}^3$ is described by a vector function of two parameters $u$ and $v$:

where $(u, v)$ varies over a planar parameter region $D \subset \mathbb{R}^2$.

1. Tangent Vectors and Normal Vectors

Holding $v = v_0$ constant, $\vec{r}(u, v_0)$ traces a grid curve on the surface whose tangent vector is:

Similarly, holding $u = u_0$ constant gives the grid curve tangent vector:

The normal vector to the tangent plane at $(u_0, v_0)$ is the cross product:

The surface is called smooth if $\vec{r}_u \times \vec{r}_v \neq \vec{0}$ everywhere.

2. The Surface Area Differential and Total Area

An infinitesimal parameter rectangle $\Delta u \times \Delta v$ maps to a curved patch on the surface approximated by the tangent parallelogram spanned by $\vec{r}_u \Delta u$ and $\vec{r}_v \Delta v$.

The area of this parallelogram is:

In differential form:

Total Surface Area:

Explicit Cartesian Surfaces $z = g(x, y)$:

Parameterize with $u = x, v = y$: $\vec{r}(x, y) = \langle x, y, g(x, y) \rangle$. $\vec{r}_x = \langle 1, 0, g_x \rangle$, $\vec{r}_y = \langle 0, 1, g_y \rangle$. $\vec{r}_x \times \vec{r}_y = \langle -g_x, -g_y, 1 \rangle$.

1. Surface Integrals of Scalar Fields

If $f(x, y, z)$ is a continuous scalar function defined on a smooth surface $S$:

If $f(x, y, z) = \rho(x, y, z)$ is surface mass density, the integral computes the total mass of the curved shell.

2. Oriented Surfaces and Unit Normals

A surface $S$ is orientable (two-sided) if it is possible to define a continuous unit normal vector field $\hat{n}$ across the entire surface (excluding non-orientable manifolds like the Möbius strip).

For a parameterized smooth surface, the unit normal is:

3. Surface Integrals of Vector Fields (Flux Integrals)

Let $\vec{F}$ be a continuous vector field on an oriented surface $S$ with unit normal $\hat{n}$.

Definition: The flux of $\vec{F}$ across $S$ is the surface integral of the normal component of $\vec{F}$:

Notice that the magnitude $|\vec{r}_u \times \vec{r}_v|$ in the unit normal cancels directly with the area element $dS = |\vec{r}_u \times \vec{r}_v| du dv$!

Physical Meaning:

If $\vec{F} = \rho \vec{v}$ is the mass flow rate per unit area of a moving fluid, $\iint_S \vec{F} \cdot d\vec{S}$ represents the total mass of fluid passing through surface $S$ per unit time.

1. Stokes' Theorem (Curl Circulation Theorem)

Sir George Gabriel Stokes (1854) established the higher-dimensional generalization of Green's theorem relating the circulation around a boundary curve in 3D space to the flux of curl through any spanning surface.

Theorem (Stokes' Theorem): Let $S$ be an oriented piecewise-smooth surface bounded by a simple, closed, piecewise-smooth space curve $C = \partial S$ whose orientation is positive relative to the normal vector $\hat{n}$ of $S$ (by the right-hand rule). If $\vec{F}$ is a vector field with continuous partial derivatives:

Fundamental Invariance (Surface Independence):

If $S_1$ and $S_2$ are any two distinct surfaces that share the exact same boundary curve $\partial S_1 = \partial S_2 = C$, then:

The flux of curl through a surface depends solely on its boundary rim, not on its bulging shape!

2. Gauss' Divergence Theorem

Carl Friedrich Gauss (1813) proved the fundamental theorem connecting the flux of a vector field across a closed bounding surface to the volume integral of its divergence throughout the enclosed solid.

Theorem (The Divergence Theorem): Let $E$ be a simple solid region in $\mathbb{R}^3$, and let $S = \partial E$ be its boundary surface, oriented with outward-pointing unit normal $\hat{n}$. If $\vec{F}$ is a vector field with continuous partial derivatives on an open region containing $E$:

Physical Interpretation:

The total net fluid volume exiting a closed container across its skin must equal the sum of all internal sources (divergence) within the container.

Major Applications in Physics:

1. Gauss' Law in Electromagnetism: $\iint_{\partial E} \vec{E} \cdot d\vec{S} = \frac{1}{\epsilon_0}\iiint_E \rho \, dV = \frac{Q_{enc}}{\epsilon_0} \implies \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$ (Maxwell's First Equation).

2. Fluid Continuity Equation: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0$.

3. The Grand Unification: Generalized Stokes' Theorem

All the fundamental integral theorems of calculus across all dimensions are manifestations of one single, universal topological equation for differential forms on an oriented manifold $\Omega$ with boundary $\partial \Omega$:

| Dimension | Manifold $\Omega$ | Boundary $\partial \Omega$ | Classical Theorem | | :---: | :---: | :---: | :---: | | $n = 1$ | Curve $[a, b]$ | Points $\{a, b\}$ | Fundamental Theorem of Calculus: $\int_a^b F'(x) dx = F(b) - F(a)$ | | $n = 2$ | Plane Region $D$ | Boundary Curve $C$ | Green's Theorem: $\oint_C P dx + Q dy = \iint_D (Q_x - P_y) dA$ | | $n = 2$ | Spatial Surface $S$ | Space Curve $C$ | Stokes' Theorem: $\oint_{\partial S} \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot d\vec{S}$ | | $n = 3$ | Solid Volume $E$ | Closed Surface $S$ | Divergence Theorem: $\iint_{\partial E} \vec{F} \cdot d\vec{S} = \iiint_E (\nabla \cdot \vec{F}) dV$ |