23.1 Surface integrals of scalar fields

Surface Integrals and Scalar Fields

Surface integrals of scalar fields extend the idea of single-variable integration to curved surfaces in 3D. Just as you can integrate a function along a curve (a line integral), you can integrate a scalar field over a surface to accumulate quantities like mass, charge, or total heat across that surface.

Definition and Components

A scalar field assigns a single number to each point in space. Think of temperature $T(x,y,z)$ at every point in a room, or mass density $\rho(x,y,z)$ across a thin shell.

A surface integral of a scalar field adds up the values of that field over every point on a surface $S$:

$\iint_S f(x,y,z)\, dS$

The key pieces you need:

• A surface $S$ in $\mathbb{R}^3$, described by a parametrization $\vec{r}(u,v)$
• A scalar field $f(x,y,z)$ defined on that surface
• The differential area element $dS$, which accounts for how the surface stretches and curves

How to Compute a Surface Integral

Here's the process, step by step:

1. Parametrize the surface. Write $\vec{r}(u,v) = \big(x(u,v),\, y(u,v),\, z(u,v)\big)$ and identify the parameter domain $D$ in the $uv$-plane.

2. Compute the partial derivatives $\vec{r}_u = \frac{\partial \vec{r}}{\partial u}$ and $\vec{r}_v = \frac{\partial \vec{r}}{\partial v}$.

3. Take their cross product $\vec{r}_u \times \vec{r}_v$ and find its magnitude $|\vec{r}_u \times \vec{r}_v|$. This magnitude is the area distortion factor that converts $du\,dv$ in parameter space into actual surface area $dS$.

4. Substitute the parametrization into $f$. Replace $x, y, z$ with their expressions in $u$ and $v$ so that $f(x,y,z)$ becomes a function of $u$ and $v$.

5. Evaluate the double integral over the parameter domain:

$\iint_S f\, dS = \iint_D f\big(x(u,v),\, y(u,v),\, z(u,v)\big)\, |\vec{r}_u \times \vec{r}_v|\, du\, dv$

Applications

• Mass of a thin shell: If $\rho(x,y,z)$ is the surface mass density (mass per unit area), then $\iint_S \rho\, dS$ gives the total mass.
• Average value: The average of $f$ over $S$ is $\frac{1}{\text{Area}(S)} \iint_S f\, dS$.
• Total heat or charge: Any quantity distributed over a surface can be totaled this way.

Parametrization and Surface Area

Parametrizing Surfaces

The parametrization $\vec{r}(u,v)$ maps a flat region $D$ in the $uv$-plane onto the curved surface in 3D. Choosing a good parametrization makes the integral much easier.

Common examples:

• Sphere of radius $a$: Use $\vec{r}(\theta, \phi) = (a\sin\phi\cos\theta,\, a\sin\phi\sin\theta,\, a\cos\phi)$ with $\theta \in [0, 2\pi)$ and $\phi \in [0, \pi]$.
• Cylinder of radius $R$: Use $\vec{r}(\theta, z) = (R\cos\theta,\, R\sin\theta,\, z)$.
• Graph surface $z = g(x,y)$: Use $\vec{r}(x,y) = (x,\, y,\, g(x,y))$. This is often the simplest option when the surface is given explicitly.

Surface Area Calculation

The total surface area of $S$ is the special case where $f = 1$:

$\text{Area}(S) = \iint_D |\vec{r}_u \times \vec{r}_v|\, du\, dv$

For the common case of a graph surface $z = g(x,y)$, this simplifies nicely. The partial derivatives are $\vec{r}_x = (1, 0, g_x)$ and $\vec{r}_y = (0, 1, g_y)$, so:

$|\vec{r}_x \times \vec{r}_y| = \sqrt{1 + g_x^2 + g_y^2}$

This means $dS = \sqrt{1 + g_x^2 + g_y^2}\, dx\, dy$, a formula worth memorizing.

Flux (Preview)

This topic (23.1) focuses on scalar field surface integrals, but it's worth seeing how they connect to flux integrals, which you'll study in depth soon.

Flux measures the flow of a vector field $\vec{F}$ through a surface. The integral looks different:

$\iint_S \vec{F} \cdot \vec{n}\, dS = \iint_D \vec{F}\big(\vec{r}(u,v)\big) \cdot (\vec{r}_u \times \vec{r}_v)\, du\, dv$

Here $\vec{n}$ is the unit outward normal to the surface. Notice that the scalar surface integral uses $|\vec{r}_u \times \vec{r}_v|$ (the magnitude), while the flux integral uses the cross product vector $\vec{r}_u \times \vec{r}_v$ itself (via the dot product with $\vec{F}$). That distinction matters.

Key differences at a glance:

• Scalar integral $\iint_S f\, dS$: integrates a scalar field, result is a scalar, surface orientation doesn't matter.
• Flux integral $\iint_S \vec{F} \cdot \vec{n}\, dS$: integrates a vector field, result is a scalar, surface orientation does matter (flipping the normal flips the sign).

Flux shows up in Gauss's Law (electric flux through a closed surface), fluid dynamics (volume flow rate through a pipe cross-section), and Faraday's Law (magnetic flux through a loop). You'll need the scalar surface integral machinery from this section to handle all of those.