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.
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:
Now, what if F = xzi + yzj +
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:
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.
Third example - NOTE: ALL VOLUME INTEGRALS MUST HAVE A FACTOR "dV"