The *divergence theorem* only applies for *closed*
surfaces *S*. However, we can sometimes work out a flux integral
on a surface that is not closed by being a little sneaky. *NOTE
that this is NOT always an efficient way of proceeding.* However,
it sometimes is, and this is a nice example of both the divergence
theorem and a flux integral, so we'll go through it as is.

**Example**

Find the flux of the vector field
**F** = *x y* **i** +
*y z* **j** + *x z* **k**
through the surface
*z* = 4 -
*x*^{2} -
*y*^{2}, for
*z* >= 3.

**Solution**

The surface is shown in the figure to the right. Because this is not
a closed surface, we can't use the divergence theorem to evaluate the
flux integral. However, if we had a closed surface, for example the
second figure to the right (which includes a bottom surface, the
yellow section of a plane) we could. We'll consider this in the following.

The divergence theorem says

where the surface *S* is the surface we want plus the bottom
(yellow) surface. So we can find the flux integral we want by finding
the right-hand side of the divergence theorem and then subtracting off
the flux integral over the bottom surface. This gives us nice
practice both applying the divergence theorem and finding a surface
integral, so we'll do it.

**The divergence theorem** part of the integral:
Here div **F** = *y* + *z* +
*x*. Note that here we're evaluating the divergence over the
entire enclosed volume, so we *can't* evaluate it on the
surface. Doing the integral in cylindrical coordinates, we get

**The flux** through the bottom boundary: Note that here
we have a very easy parameterization of the surface,
**r** = <*x*, *y*, 3>. The normal vector
**N** = <0, 0, -1> (because we want an *outward*
normal), and *dS* = *dx* *dy*. Thus on the
surface
**F** = **F** = *x y* **i** +
*y z* **j** + 3 *x* **k**, and
the surface integral becomes

**Putting it together**: here, things dropped out
nicely. Using the divergence theorem, we get the value of the flux
through the top and bottom surface together to be 5 *pi* / 3,
and the flux calculation for the bottom surface gives zero, so that
the flux just through the top surface is also 5 *pi* / 3.

*(How were the figures here generated? In Maple, with this
maple worksheet.)*

using the divergence theorem

Last modified: Mon Apr 19 12:06:56 EDT 2004

Comments to:glarose@umich.edu

©2004 Gavin LaRose, UM Math Dept.