# Math 294

# Lecture 8

### Given by Prof. Alan Zehnder on 9/18/95 at 10:10 in Kaufmann Auditorium

# Surface Integrals

Divergence Theorem

The Del Operator

NOTE: The first computing assignment is due on October 2, not any other
date listed on any of the sheets.

In Lecture 7 we introduced area and surface integrals
and gave an example of an area integral. In this lecture we do some examples of surface integrals and then learn a new
technique for evaluating such problems. The Divergence Theorem allows the
computation of surface integrals with considerably less effort than
before.

#### A Surface Integral Problem

Find the outward flux of **F** = xz**i** +
yz**j** + z^2**k** across the upper cap cut from the
hollow sphere `x^2 + y^2 + z^2 <=25` by the
plane `z=3`.

We can find **n** by inspection as `(x`**i** +
y**j** + z**k**) / 5, since we know that it points
radially outward and that all points on the sphere are a distance of 5 from
the origin.
We can find the radius of the region R in the xy plane, which we will be
projecting our surface onto, by a simple calculation:

`z = 3`

x^2 + y^2 + 3^2 = 25

x^2 + y^2 = 16

Therfore, the radius of R, our region in the xy plane, is
4.

Now, what if **F** = xz**i** + yz**j** +
**k**?

### The Del Operator

The Del operator is defined as:

If f is a scalar field then:

This is the Gradient vector field, which we've seen already.

If **F** is a vector field, **F** = M**i** + N**j** + P**k**

This is the divergence of **F**.

Finally, we can take the cross-product of del and **F**:

### Divergence Theorem

Recall Green's theorem, which stated that you can do the flux across a
closed path by taking the divergence over the area enclosed. The
Divergence Theorem genaralizes this concept into 3 dimensions by allowing
one to calculate the flux across a closed suface through taking the
divergence throughout the enclosed volume.

#### Some Examples

Calculate the flux across the surface of a cylinder with a paraboidal
hole, where **F** = y**i** + xy**j** - z**k**

where z=x^2+y^2 is the inner surface and x^2+y^2=4 is the outer surface.

Another example.

Third example - NOTE: ALL VOLUME INTEGRALS MUST HAVE A FACTOR "dV"

Lecture written by Lawrence C. Weintraub on Sunday, September 24, 1995

Edited by Aric Shafran on Sunday, September 24, 1995

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