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 = xzi + yzj + z^2k 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 (xi + yj + zk) / 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 = xzi + yzj + 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 = Mi + Nj + Pk

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 = yi + xyj - zk

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

Another example.


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