Parametric Surfaces, Part 4

The change-of-coordinates discussion in Part 2 can be considered a special case of parameterizing surfaces, where all the surfaces are actually in the xy-plane. For example, polar coordinates transform a rectangle in the r-plane into a circular region in the xy-plane. Explain in your own words how this fits into the more general context of Part 3 and how the planar nature of the parameterized surface is reflected in the formulas for fundamental vector product and Jacobian.

Let S be a circular cylinder of height h and radius a. Write a parameterization of S using as parameters the cylindrical coordinates z and , where 0 < z < h and 0 < < 2. You probably don't need calculation to figure out the local change-in-area factor -- what do you think it is and why? (If you're not sure, do the calculation, and then explain why you didn't really need the calculation.)

Explain why the fundamental vector product of a parameterization is always perpendicular to the parameterized surface.

There is a famous geometry theorem called the Second Theorem of Pappus, dating from the early 4th century CE, that can be stated as follows:

Suppose a curve C in the plane is rotated around an axis that does not intersect C to form a surface of revolution. Then the area of the surface generated is sd, where s is the length of C, and d is the circular distance traveled by the centroid of C.

Now "centroid" of a curve is not a trivial matter in general, but there are some easy cases: The centroid of a circle is its center, and the centroid of a line segment is its midpoint. In this module you encountered at least one case of Pappus's Theorem, and possibly others. Identify the case(s), and explain the connection(s).