| | 2025.01.28

7: Surface Integrals

Integrating functions over surfaces in 3D.

Parametric Surfaces

A surface $S$ can be parametrized by:

\mathbf{r}(u, v) = \langle x(u, v), y(u, v), z(u, v) \rangle

where $(u, v)$ varies over some region $D$ in the $uv$ -plane.

Examples

Sphere of radius $a$ :

\mathbf{r}(\phi, \theta) = \langle a\sin\phi\cos\theta, a\sin\phi\sin\theta, a\cos\phi \rangle

Cylinder of radius $r$ :

\mathbf{r}(\theta, z) = \langle r\cos\theta, r\sin\theta, z \rangle

Graph $z = f(x, y)$ :

\mathbf{r}(x, y) = \langle x, y, f(x, y) \rangle

Tangent Planes and Normal Vectors

The tangent vectors to the surface:

\mathbf{r}_u = \frac{\partial \mathbf{r}}{\partial u}, \quad \mathbf{r}_v = \frac{\partial \mathbf{r}}{\partial v}

The normal vector:

\mathbf{n} = \mathbf{r}_u \times \mathbf{r}_v

This is perpendicular to the surface at each point.

For a Graph $z = f(x, y)$

\mathbf{n} = \langle -f_x, -f_y, 1 \rangle

Surface Area

A = \iint_D |\mathbf{r}_u \times \mathbf{r}_v| \, dA

For a Graph $z = f(x, y)$

A = \iint_D \sqrt{1 + f_x^2 + f_y^2} \, dA

Surface Integrals of Scalar Functions

\iint_S f \, dS = \iint_D f(\mathbf{r}(u, v)) |\mathbf{r}_u \times \mathbf{r}_v| \, dA

For a Graph

\iint_S f \, dS = \iint_D f(x, y, f(x,y)) \sqrt{1 + f_x^2 + f_y^2} \, dA

Applications

Surface area: $\iint_S 1 \, dS$
Mass of a shell: $\iint_S \rho \, dS$
Center of mass: $\bar{x} = \frac{1}{m}\iint_S x\rho \, dS$

Oriented Surfaces

An oriented surface has a chosen “positive” side (direction of normal).

Closed surface: Convention is outward normal
Surface with boundary: Use right-hand rule with boundary curve

Surface Integrals of Vector Fields (Flux)

The flux of $\mathbf{F}$ through surface $S$ :

\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_S \mathbf{F} \cdot \mathbf{n} \, dS

Computation

\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D \mathbf{F}(\mathbf{r}(u,v)) \cdot (\mathbf{r}_u \times \mathbf{r}_v) \, dA

For a Graph $z = f(x, y)$ with Upward Normal

\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D \mathbf{F} \cdot \langle -f_x, -f_y, 1 \rangle \, dA

Or if $\mathbf{F} = \langle P, Q, R \rangle$ :

\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D (-Pf_x - Qf_y + R) \, dA

Physical Interpretation of Flux

If $\mathbf{F}$ is a velocity field of a fluid:

\text{Flux} = \iint_S \mathbf{F} \cdot d\mathbf{S} = \text{volume of fluid passing through } S \text{ per unit time}

Positive flux: flow in direction of normal
Negative flux: flow opposite to normal
Zero flux: flow parallel to surface

Computing Flux Through Common Surfaces

Sphere of Radius $a$ (Outward Normal)

\mathbf{n} = \frac{\mathbf{r}}{|\mathbf{r}|} = \frac{\langle x, y, z \rangle}{a}

dS = a^2 \sin\phi \, d\phi \, d\theta

Cylinder (Outward Normal)

\mathbf{n} = \frac{\langle x, y, 0 \rangle}{r}

Plane

Normal is constant, $dS = dA$ .

Summary

Concept	Formula
Normal vector	$\mathbf{n} = \mathbf{r}_u \times \mathbf{r}_v$
Surface area	$\iint_D
Scalar surface integral	$\iint_S f , dS = \iint_D f
Flux	$\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D \mathbf{F} \cdot (\mathbf{r}_u \times \mathbf{r}_v) \, dA$
Graph $z=f(x,y)$	$dS = \sqrt{1 + f_x^2 + f_y^2} \, dA$

7: Surface Integrals

Parametric Surfaces

Examples

Tangent Planes and Normal Vectors

For a Graph z=f(x,y)z = f(x, y)z=f(x,y)

Surface Area

For a Graph z=f(x,y)z = f(x, y)z=f(x,y)

Surface Integrals of Scalar Functions

For a Graph

Applications

Oriented Surfaces

Surface Integrals of Vector Fields (Flux)

Computation

For a Graph z=f(x,y)z = f(x, y)z=f(x,y) with Upward Normal

Physical Interpretation of Flux

Computing Flux Through Common Surfaces

Sphere of Radius aaa (Outward Normal)

Cylinder (Outward Normal)

Plane

Summary

For a Graph $z = f(x, y)$

For a Graph $z = f(x, y)$

For a Graph $z = f(x, y)$ with Upward Normal

Sphere of Radius $a$ (Outward Normal)