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solution to this?

Let p: R^ 2 → R^ 3 be the smooth map defined as follows: p(u, v) = (u cos v, u sin v, hv), where h > 0 is a constant.

(a) Evaluate ∥pu × pv∥ where pu and pv are the partial derivatives. Hence show that p is regular at every point (u, v) ∈ R 2 .

(c) Let k > 0, and let Sk be the parametric surface obtained by restricting p to the rectangle: Uk = {(u, v) : 0 < u < k, 0 < v < 2π}. Calculate the surface integral: ∫∫ Sk √ x 2 + y 2 dA. Verify that when h = k = 1 the value of the integral is 2 3 π(2√ 2 − 1).

I need a solution to check mine against as I haven't been provided with one.

Reply 1

DFranklin

The way it works is to post your own solution and someone will (hopefully) check it.

Reply 2

simon0

For a, what methods can we use to distinguish a regular surface?
(Note, one requirement for a surface to be regular is it is smooth i.e. has no sharp "cusps").
For b) are you able to doulbe-check the integrand you provided in the question?

(edited 6 years ago)

Reply 3

atsruser

Original post by simon0

For a, what methods can we use to distinguish a regular surface?

What's the definition of a regular surface? A bit of googling suggests that it is one with a unique tangent plane/normal everywhere. Is there more to it than that?

Reply 4

DFranklin

Original post by atsruser

What's the definition of a regular surface? A bit of googling suggests that it is one with a unique tangent plane/normal everywhere. Is there more to it than that?

Call me cynical, but I think the OP has lost interest at the point where it appeared they'd actually have to provide some workings of their own...

Reply 5

atsruser

Original post by DFranklin

Call me cynical, but I think the OP has lost interest at the point where it appeared they'd actually have to provide some workings of their own...

Quite possibly. Still, I thought the final sentence showed a certain inventiveness as to justifying why we should solve the problem. I've seen weaker attempts.

So what is a regular surface. More than just smooth? In fact, I'm not even sure what area of maths this is - some kind of differential geometry, I guess, where you deal with osculating circles and Frenet-Serret formulae, and that kind of stuff?

Reply 6

simon0

Original post by atsruser

What's the definition of a regular surface? A bit of googling suggests that it is one with a unique tangent plane/normal everywhere. Is there more to it than that?

That is fine, there are a few tests to see if a surface is regular and I was trying to hint to one or to use the definition of a regular surface.

Also, you are correct, I did learn of surface parametrization in differential geometry.