Surface Integrals — Theory
1. What is a surface integral?
A surface integral adds a quantity over a curved surface \(S\).
There are two main types:
- Scalar surface integrals: add a scalar function over a surface.
- Flux surface integrals: measure how much a vector field passes through a surface.
\[
\iint_S g(x,y,z)\,dS
\]
\[
\iint_S \mathbf F\cdot\mathbf n\,dS
\]
2. Scalar surface integrals
A scalar surface integral has the form:
\[
\iint_S g(x,y,z)\,dS.
\]
Here \(g(x,y,z)\) is a scalar function defined on the surface.
If \(g=1\), the integral gives the surface area:
\[
\operatorname{Area}(S)=\iint_S 1\,dS.
\]
If \(g\) is a surface density, then the scalar surface integral gives total mass on the surface.
3. Flux surface integrals
A flux integral has the form:
\[
\iint_S \mathbf F\cdot\mathbf n\,dS.
\]
The vector field \(\mathbf F\) represents a flow, force field, electric field, or another vector quantity.
The vector \(\mathbf n\) is a unit normal vector to the surface.
Flux measures how strongly the vector field passes through the surface.
If the field points in the same direction as the chosen normal, the flux is positive.
If it points opposite the normal, the flux is negative.
4. Parametric surfaces
A parametric surface is written as
\[
\mathbf r(u,v)
=
\langle X(u,v),Y(u,v),Z(u,v)\rangle.
\]
The tangent vectors are:
\[
\mathbf r_u=\frac{\partial \mathbf r}{\partial u},
\qquad
\mathbf r_v=\frac{\partial \mathbf r}{\partial v}.
\]
Their cross product gives an oriented area vector:
\[
\mathbf r_u\times\mathbf r_v\,du\,dv.
\]
Reversing the order reverses the orientation:
\[
\mathbf r_v\times\mathbf r_u
=
-(\mathbf r_u\times\mathbf r_v).
\]
5. Surface area element for parametric surfaces
The surface area element is the magnitude of the cross product:
\[
dS
=
\left\|\mathbf r_u\times\mathbf r_v\right\|\,du\,dv.
\]
Therefore, a scalar surface integral over a parametric surface becomes:
\[
\iint_S g\,dS
=
\iint_D
g(\mathbf r(u,v))
\left\|\mathbf r_u\times\mathbf r_v\right\|
\,du\,dv.
\]
A flux integral becomes:
\[
\iint_S \mathbf F\cdot\mathbf n\,dS
=
\iint_D
\mathbf F(\mathbf r(u,v))
\cdot
\left(\mathbf r_u\times\mathbf r_v\right)
\,du\,dv.
\]
6. Explicit surfaces
An explicit surface is written as:
\[
z=h(x,y).
\]
The upward oriented area vector is:
\[
\langle -h_x,-h_y,1\rangle\,dx\,dy.
\]
The downward oriented area vector is:
\[
\langle h_x,h_y,-1\rangle\,dx\,dy.
\]
The scalar surface area element is:
\[
dS=\sqrt{1+h_x^2+h_y^2}\,dx\,dy.
\]
7. Flux through an explicit surface
Suppose
\[
\mathbf F=\langle P,Q,R\rangle,
\qquad
z=h(x,y).
\]
For upward orientation:
\[
\iint_S \mathbf F\cdot\mathbf n\,dS
=
\iint_D
\langle P,Q,R\rangle
\cdot
\langle -h_x,-h_y,1\rangle
\,dx\,dy.
\]
The vector field must be evaluated on the surface, meaning \(z=h(x,y)\).
8. Orientation
Orientation is essential for flux integrals. A surface has two possible normal directions.
Choosing the opposite direction changes the sign of the flux:
\[
\iint_S \mathbf F\cdot(-\mathbf n)\,dS
=
-
\iint_S \mathbf F\cdot\mathbf n\,dS.
\]
However, orientation does not affect scalar surface integrals because they use \(dS\), not
the oriented vector \(\mathbf n\,dS\).
9. Example: flux through a unit sphere
A common outward parametrization of the unit sphere can be written as:
\[
\mathbf r(u,v)
=
\langle
\sin v\cos u,\,
\sin v\sin u,\,
\cos v
\rangle,
\]
where
\[
0\le u\le 2\pi,
\qquad
0\le v\le \pi.
\]
For this parametrization, \(\mathbf r_v\times\mathbf r_u\) gives the outward orientation.
If \(\mathbf F=\langle x,y,z\rangle\), then on the unit sphere the field points outward.
The flux is:
\[
\iint_S \mathbf F\cdot\mathbf n\,dS
=
4\pi.
\]
This also agrees with the Divergence Theorem, because
\(\nabla\cdot\mathbf F=3\) and the unit ball has volume \(4\pi/3\).
10. Example: surface area of a sphere
Surface area is the scalar surface integral with \(g=1\):
\[
\operatorname{Area}(S)
=
\iint_S 1\,dS.
\]
For a sphere of radius \(a\), the surface area is:
\[
4\pi a^2.
\]
For the unit sphere, this becomes:
\[
4\pi.
\]
12. Common mistakes
- Using \(dA\) instead of \(dS\): curved surfaces need the surface area factor.
- Forgetting to evaluate the vector field on the surface: substitute \(x,y,z\) from the parametrization or from \(z=h(x,y)\).
- Mixing scalar and flux integrals: scalar integrals use \(g\,dS\), while flux integrals use \(\mathbf F\cdot\mathbf n\,dS\).
- Ignoring orientation: reversing orientation changes the sign of flux.
- Using the wrong cross-product order: \(\mathbf r_u\times\mathbf r_v\) and \(\mathbf r_v\times\mathbf r_u\) point in opposite directions.
- Assuming scalar surface integrals change sign: they do not, because \(dS\) is nonnegative.
- Choosing too few panels: highly curved surfaces or oscillating fields need more numerical panels.
