Question

Let S be the part of the cone z^2 = x^2 + y^2 where z is between 1 and 2. (a) Write down the Monge parameterization of S and compute the surface element.

Answer 1

not sure how to do the monge parameterization

Answer 2

The parametrization is
\[x(u, v) = <u, v, \pm \sqrt{u ^{2}+v ^{2} }>.\]
For y > 0, we take the positive xquare root, and for y < 0, we take the negative square root.

Answer 3

okay so u=x v=y and z=sqrt(u^2+v^2)

Answer 4

I an uncertain about how to deal with the case y = 0. The parametrization as it stands now coners the lateral surface of the frustum except where the cone intersects y = 0., so it fails to completely cover that lateral surface. At least it is an open set like it's supposed to be.

Answer 5

yeah well its close enough for me. thanks for the help!

Answer 6

Cool beans:^)

Answer 7

do you know how to Parameterize S using polar angle theta and z would it be 1<z<2 and 0<theta<2pi?

Answer 8

In cylindrical coordinater, the cone has the equation, z = r. We disregard z = -r xince 1 < z <2.
\[x = r \cos \theta ,y = r \sin \theta.\]

Answer 9

The parametrization is now
\[x(r,\theta) = <r \cos \theta, r \sin \theta, r)>.\]

Answer 10

okay good thats what i ended up with. Im trying to find the bounds for the flux integral would theta be from 0 to 2pi and r and z are both from 1 to 2?

Answer 11

They sure are.

Answer 12

Great! thanks so much for the help :)

Answer 13

:^)