MULTIVARIABLE CALCULUS 11

Surface Integrals

A surface integral accumulates a scalar or vector quantity over a curved surface in space.

The main idea

Ordinary integrals accumulate along intervals.

Double integrals accumulate over flat regions.

Surface integrals extend this idea to curved surfaces.

\[ \boxed{ \iint_S f\,dS } \]

Here \(S\) is a surface and \(dS\) represents a small element of surface area.

1. From area to surface area

For a flat region,

\[ dA \]

represents a small area element.

For a curved surface,

\[ dS \]

represents a small surface-area element.

\[ \boxed{ \text{flat region} \rightarrow dA } \] \[ \boxed{ \text{curved surface} \rightarrow dS } \]

2. Scalar surface integral

Suppose

\[ f(x,y,z) \]

is a scalar field defined on a surface \(S\).

Then

\[ \boxed{ \iint_S f\,dS } \]

accumulates the scalar quantity over the whole surface.

3. Physical interpretation

Suppose a thin curved sheet has surface density

\[ \rho(x,y,z). \]

A small surface element has approximate mass

\[ dm=\rho\,dS. \]

Therefore, the total mass is

\[ \boxed{ M = \iint_S \rho\,dS } \]

4. Parameterising a surface

A surface can be described using two parameters:

\[ \mathbf{r}(u,v) = \begin{pmatrix} x(u,v)\\ y(u,v)\\ z(u,v) \end{pmatrix}. \]

The pair

\[ (u,v) \]

selects a point on the surface.

5. Two tangent directions

Differentiate the parameterisation:

\[ \mathbf{r}_u = \frac{\partial\mathbf{r}}{\partial u}, \]

and

\[ \mathbf{r}_v = \frac{\partial\mathbf{r}}{\partial v}. \]

These vectors lie tangent to the surface.

6. Surface-area element

The cross product

\[ \mathbf{r}_u\times\mathbf{r}_v \]

is perpendicular to the surface.

Its magnitude gives the local area-scaling factor.

Therefore,

\[ \boxed{ dS = \left\| \mathbf{r}_u\times\mathbf{r}_v \right\| \,du\,dv } \]

7. Scalar surface integral in parameter form

Therefore,

\[ \boxed{ \iint_S f\,dS = \iint_D f(\mathbf{r}(u,v)) \left\| \mathbf{r}_u\times\mathbf{r}_v \right\| \,du\,dv } \]

where \(D\) is the parameter region in the \(uv\)-plane.

8. Surface given as \(z=g(x,y)\)

A common surface has the form

\[ z=g(x,y). \]

We can parameterise it by

\[ \mathbf{r}(x,y) = \begin{pmatrix} x\\ y\\ g(x,y) \end{pmatrix}. \]

Then

\[ \mathbf{r}_x = \begin{pmatrix} 1\\ 0\\ g_x \end{pmatrix}, \]

and

\[ \mathbf{r}_y = \begin{pmatrix} 0\\ 1\\ g_y \end{pmatrix}. \]

9. Compute the cross product

We obtain

\[ \mathbf{r}_x\times\mathbf{r}_y = \begin{pmatrix} -g_x\\ -g_y\\ 1 \end{pmatrix}. \]

Therefore,

\[ \left\| \mathbf{r}_x\times\mathbf{r}_y \right\| = \sqrt{ 1+g_x^2+g_y^2 }. \]

\[ \boxed{ dS = \sqrt{ 1+ \left(\frac{\partial g}{\partial x}\right)^2 + \left(\frac{\partial g}{\partial y}\right)^2 } \,dx\,dy } \]

10. Surface-area formula

If we set

\[ f=1, \]

then the surface integral gives surface area:

\[ \boxed{ \text{Area}(S) = \iint_S dS } \]

For

\[ z=g(x,y), \]

this becomes

\[ \boxed{ \text{Area}(S) = \iint_R \sqrt{ 1+g_x^2+g_y^2 } \,dx\,dy } \]

11. Simple example: horizontal plane

Suppose

\[ z=3. \]

Then

\[ g_x=0, \qquad g_y=0. \]

Therefore,

\[ dS = \sqrt{1}\,dx\,dy = dx\,dy. \]

So for a horizontal plane, surface area reduces to ordinary planar area.

12. Why curved surfaces have a larger area element

If the surface slopes,

\[ g_x \]

\[ g_y \]

is nonzero.

Then

\[ \sqrt{1+g_x^2+g_y^2}>1. \]

So the true surface patch is larger than its flat projection onto the \(xy\)-plane.

13. Diagram

14. Vector surface integrals

Surface integrals can also involve vector fields.

Suppose

\[ \mathbf{F}(x,y,z) \]

is a vector field and

\[ \mathbf{n} \]

is a unit normal vector to the surface.

Then

\[ \boxed{ \iint_S \mathbf{F}\cdot\mathbf{n}\,dS } \]

measures the flux of the field through the surface.

15. What is flux?

Flux measures how much of a vector field passes through a surface.

If the field points directly through the surface, the contribution is large.

If the field is tangent to the surface, the contribution is zero.

This is why the dot product appears:

\[ \mathbf{F}\cdot\mathbf{n}. \]

16. Orientation matters

A surface has two possible normal directions.

For a closed surface, we usually choose the outward normal.

Reversing the orientation changes the sign of the flux integral:

\[ \boxed{ \mathbf{n}\rightarrow-\mathbf{n} \quad\Rightarrow\quad \text{flux changes sign} } \]

17. Vector area element

Instead of writing

\[ \mathbf{n}\,dS, \]

we can write the vector surface element

\[ d\mathbf{S}. \]

Then the flux integral becomes

\[ \boxed{ \iint_S \mathbf{F}\cdot d\mathbf{S} } \]

18. Parameter form of flux

Using a parameterisation

\[ \mathbf{r}(u,v), \]

we may write

\[ \boxed{ \iint_S \mathbf{F}\cdot d\mathbf{S} = \iint_D \mathbf{F}(\mathbf{r}(u,v)) \cdot \left( \mathbf{r}_u\times\mathbf{r}_v \right) \,du\,dv } \]

The cross product provides both the local area scale and a normal direction.

19. Closed surfaces

For a closed surface, flux is often written

\[ \oiint_S \mathbf{F}\cdot\mathbf{n}\,dS. \]

It measures the net outward flow through the boundary.

20. Connection with divergence

Divergence measures local expansion inside a region.

Flux measures the total flow through its boundary.

These ideas are connected by the divergence theorem:

\[ \boxed{ \oiint_{\partial V} \mathbf{F}\cdot\mathbf{n}\,dS = \iiint_V \nabla\cdot\mathbf{F}\,dV } \]

The total outward flux through the boundary equals the total divergence inside the volume.

21. Why this is a natural conservation statement

If a vector field represents the flow of material, then positive divergence inside a region means material is being produced or spreading locally.

The divergence theorem says that this net internal production is exactly reflected by the net flow through the boundary.

22. Scalar versus vector surface integrals

\[ \boxed{ \iint_S f\,dS = \text{accumulation of a scalar over a surface} } \] \[ \boxed{ \iint_S \mathbf{F}\cdot\mathbf{n}\,dS = \text{flux of a vector field through a surface} } \]

23. The pattern of integration

\[ \boxed{ \int_C f\,ds = \text{accumulation along a curve} } \] \[ \boxed{ \iint_S f\,dS = \text{accumulation over a surface} } \] \[ \boxed{ \iiint_V f\,dV = \text{accumulation through a volume} } \]

The underlying idea is always the same: divide into small pieces, calculate a local contribution, and accumulate.

24. The central idea

\[ \boxed{ \text{surface} + \text{small area element }dS } \] \[ \downarrow \] \[ \boxed{ \text{accumulate a scalar or flux} } \] \[ \downarrow \] \[ \boxed{ \text{surface integral} } \]

Surface integrals extend the idea of integration to curved two-dimensional objects embedded in three-dimensional space.