Chapter 36 — Key Takeaways
A surface integral is the next rung on the ladder of integration: after intervals, flat regions, and curves, we now integrate over curved two-dimensional surfaces in space. The whole strategy is "double integral plus a stretching factor" — flatten the surface back to its parameter domain and pay for the distortion.
1. Parametrized Surfaces (Section 36.2)
A surface needs two parameters:
$$\mathbf{r}(u,v) = \langle x(u,v),\ y(u,v),\ z(u,v)\rangle, \qquad (u,v)\in D.$$
Holding one parameter fixed traces a curve on $S$; the two families form a curvilinear grid. Standard parametrizations to know cold:
| Surface | $\mathbf{r}(u,v)$ | Domain |
|---|---|---|
| Plane | $\mathbf{r}_0 + u\mathbf{a} + v\mathbf{b}$ | $\mathbb{R}^2$ |
| Graph $z = g(x,y)$ | $\langle x, y, g(x,y)\rangle$ | shadow $D$ |
| Sphere, radius $R$ | $\langle R\sin\phi\cos\theta, R\sin\phi\sin\theta, R\cos\phi\rangle$ | $0\le\phi\le\pi$, $0\le\theta<2\pi$ |
| Cylinder, radius $R$ | $\langle R\cos\theta, R\sin\theta, z\rangle$ | $\theta$ around, $z$ along |
| Torus $(R, r)$ | $\langle (R+r\cos\phi)\cos\theta, (R+r\cos\phi)\sin\theta, r\sin\phi\rangle$ | $\phi,\theta\in[0,2\pi)$ |
2. The Surface Area Element $dS$ (Section 36.3)
A small parameter rectangle maps to a parallelogram spanned by $\mathbf{r}_u\,du$ and $\mathbf{r}_v\,dv$, so its area is
$$\boxed{\,dS = \|\mathbf{r}_u\times\mathbf{r}_v\|\,du\,dv\,}.$$
The cross product $\mathbf{r}_u\times\mathbf{r}_v$ does double duty: its magnitude is the area-stretching factor; its direction is normal to $S$. Standard values:
| Surface | $dS$ |
|---|---|
| Graph $z = g(x,y)$ | $\sqrt{1 + g_x^2 + g_y^2}\,dA$ |
| Sphere, radius $R$ | $R^2\sin\phi\,d\phi\,d\theta$ |
| Cylinder side, radius $R$ | $R\,d\theta\,dz$ |
The graph factor $\sqrt{1+g_x^2+g_y^2}$ is the 2D analog of the arc-length factor $\sqrt{1+f'^2}$ from Chapter 18.
Standard closed-form areas:
| Shape | Area |
|---|---|
| Sphere, radius $R$ | $4\pi R^2$ |
| Cylinder side $(R, h)$ | $2\pi R h$ |
| Cone (base $R$, slant $\ell$) | $\pi R\ell$ |
| Torus $(R, r)$ | $4\pi^2 R r$ |
3. Scalar Surface Integrals (Section 36.4)
$$\iint_S f\,dS = \iint_D f(\mathbf{r}(u,v))\,\|\mathbf{r}_u\times\mathbf{r}_v\|\,du\,dv.$$
The right side is an ordinary double integral over the flat domain $D$. Uses: $f\equiv1$ gives area; $f = \sigma$ gives mass or total charge of a thin sheet; weighting by coordinates gives center of mass; dividing by area gives average value.
4. Orientation and the Unit Normal (Section 36.5)
To orient a surface is to pick a continuous unit normal $\hat{\mathbf{n}}$. From a parametrization,
$$\hat{\mathbf{n}} = \frac{\mathbf{r}_u\times\mathbf{r}_v}{\|\mathbf{r}_u\times\mathbf{r}_v\|};$$
for an implicit surface $G(x,y,z)=0$, use $\hat{\mathbf{n}} = \nabla G/\|\nabla G\|$. Conventions:
- Closed surface (sphere, box): outward normal.
- Graph $z = g(x,y)$: upward normal $\langle -g_x, -g_y, 1\rangle/\sqrt{1+g_x^2+g_y^2}$.
- Open surface with boundary: orientation compatible with the boundary (right-hand rule, needed for Stokes in Chapter 37).
A Möbius band is non-orientable — no consistent $\hat{\mathbf{n}}$ exists — so flux is undefined on it.
5. Flux: Vector Surface Integrals (Sections 36.6–36.7)
The flux of $\mathbf{F}$ through an oriented surface is
$$\iint_S\mathbf{F}\cdot d\mathbf{S} = \iint_S(\mathbf{F}\cdot\hat{\mathbf{n}})\,dS = \iint_D \mathbf{F}(\mathbf{r}(u,v))\cdot(\mathbf{r}_u\times\mathbf{r}_v)\,du\,dv.$$
The integrand $\mathbf{F}\cdot\hat{\mathbf{n}}$ is the piercing component of $\mathbf{F}$. The working formula uses the un-normalized normal $\mathbf{r}_u\times\mathbf{r}_v$ — no square roots. Three standard cases (with $\mathbf{F} = \langle P,Q,R\rangle$):
| Surface | Flux formula |
|---|---|
| Graph $z = g(x,y)$, upward | $\iint_D(-P g_x - Q g_y + R)\,dA$ |
| Sphere, radius $R$, outward | $\iint(\mathbf{F}\cdot\mathbf{r}/R)\,R^2\sin\phi\,d\phi\,d\theta$ |
| Cylinder, radius $R$, outward | $\iint(P\cos\theta + Q\sin\theta)\,R\,d\theta\,dz$ |
Physical readings of flux: fluid velocity → volume flow rate; heat-flux field $\mathbf{q}=-k\nabla T$ → heat energy per time (Section 36.8); electric field → electric flux (Section 36.9).
6. Gauss's Law (Section 36.9)
For the inverse-square field $\mathbf{F} = \mathbf{r}/\|\mathbf{r}\|^3$, the outward flux through any centered sphere is $4\pi$ — independent of radius. Scaled for electricity, this is Gauss's law:
$$\oiint_S\mathbf{E}\cdot d\mathbf{S} = \frac{Q_{\text{enc}}}{\varepsilon_0}.$$
Only enclosed charge counts; charges outside contribute zero net flux. The magnetic twin is $\oiint_S\mathbf{B}\cdot d\mathbf{S} = 0$ (no monopoles).
7. Common Pitfalls
- Dropping the stretching factor. Writing $dS = dA$ for a tilted or curved surface — the $\sqrt{1+g_x^2+g_y^2}$ (or $\|\mathbf{r}_u\times\mathbf{r}_v\|$) is the whole point.
- Wrong orientation. On a closed surface, every face's normal must point outward; the bottom cap of a cylinder points down. Mixing orientations silently flips signs.
- Confusing flux with surface area. The integrand is $\mathbf{F}\cdot\hat{\mathbf{n}}$, not $\|\mathbf{F}\|$ — only the normal component crosses.
- Forgetting orientability. Flux is undefined on non-orientable surfaces (Möbius band).
- Normalizing unnecessarily. For flux, use $\mathbf{r}_u\times\mathbf{r}_v$ directly; the magnitude cancels.
8. Connections
- Chapter 32 — double integrals: the flat-region analog; every surface integral reduces to one over $D$.
- Chapter 33 — Jacobians: $\|\mathbf{r}_u\times\mathbf{r}_v\|$ is the change-of-variables Jacobian living on a curved surface.
- Chapter 34 — vector fields: the fields $\mathbf{F}$ whose flux we compute.
- Chapter 35 — line integrals and Green's Theorem: the one-dimension-down analog; work dots $\mathbf{F}$ with the tangent, flux dots it with the normal.
- Chapter 37 — Stokes' Theorem and the Divergence Theorem: both take the flux integrals built here as one of their two sides. Stokes relates $\iint_S(\nabla\times\mathbf{F})\cdot d\mathbf{S}$ to a boundary circulation; the Divergence Theorem relates $\oiint_S\mathbf{F}\cdot d\mathbf{S}$ to $\iiint_E\nabla\cdot\mathbf{F}\,dV$.
What's Next
Chapter 37 is the climax of vector calculus. Stokes' Theorem and the Divergence Theorem are higher-dimensional cousins of the Fundamental Theorem of Calculus (Chapter 14): the integral over a region equals something on its boundary. Chapter 38 then unifies the FTC, Green's, Stokes', and the Divergence Theorem into one statement, $\int_{\partial M}\omega = \int_M d\omega$.
Reflection
Strip away the parametrizations and cross products, and the flux integral asks one plain question: how much of $\mathbf{F}$ passes through $S$? Whether $\mathbf{F}$ is flowing water, escaping heat, an electric field, or light in a rendered scene, the answer is always a surface integral. That single skill — flatten the surface, weigh each patch by the piercing component, sum — is exactly what the great theorems of Chapter 37 are built to exploit.