Question

Can someone please help me out on this one?

Answer 1

yeah sure! but, where is your question?

Answer 2

The catenoid C is a surface with parametrization x = cosh(v) cos(u), y = cosh(v) sin(u),
z = v (0 ≤ u ≤ π, −π ≤ v ≤ π). Recall sinh(x) = (e^x−e^−x)/2,
cosh(x) = (e^x−e^−x)/2.
The following identities may be useful
cosh2(x) − sinh2(x) = 1
cosh2(x) = (1 + cosh(2x))/2
(a) Find dS for this surface. [Hint: remember (ku) × v = u × (kv) = k(u × v),
ie constant factors can be taken out of cross products
(b) If the catenoid is a metal surface with density 1 +z^2
at point (x, y, z), write down a double integral for the mass of C.
(c) Evaluate the surface area of C. How does it compare with the area of the cylinder
with the same boundary?

Answer 3

take the cross product of the two.
Well, you're given the parametrization's already. Try setting up your bounds for D. Surgace area are double integrals. For surface area, I think it's
\[\int\limits \int\limits_{S}^{ } f(x,y,z)dS = \int\limits \int\limits_{D} f(r(u,v))||r_u \times r_v||dA\]

Answer 4

I think the hardest part is finding your range, D.

Answer 5

But, I think if I'm reading correctly, you're already given your D.

Answer 6

Isn't dS the magnitude of the x, y and z ?

Answer 7

that is where the given identities come to use.

Answer 8

I believe so, yes. So you find the criss prodct

Answer 9

You need:
\[||r_x \times r_y||\]

Answer 10

where D is the range of the parameters that trace out the surface S.

Answer 11

what is \[r_{v}\] and \[r_{v}\] ?

Answer 12

I think you want to use: \[
\int\limits \int\limits_{S}^{ } f(x,y,z)dS = \int\limits \int\limits_{D} f(x, y, g(x,y))\sqrt{\frac{\partial g}{\partial x} + \frac{\partial g}{\partial y} + 1}dA
\]

Answer 13

However, I'm thinking in this case that we don't want to parameterize z in terms of x and y...perhaps x in terms of y and z or y in terms of x and z

Answer 14

@Asad0000 Does it make sense?

Answer 15

\[ dS = \sqrt{\left[ \frac{\partial g}{\partial x } \right]^2+\left[ \frac{\partial g}{\partial y } \right]^2+1} \ dA\]

Answer 16

I'm thinking you want
either x to be in terms of z, y
or y in terms of z, x
Make sense?

Answer 17

so for the part b, the double integral should look like this: \[\int\limits_{}^{}\int\limits_{}^{} ((1 + z^{2}) . (\sqrt{r_{x}^{2} + r_{y}^{2} +1}\]

Answer 18

This is why they gave you those identities.

Answer 19

I've gotta go, but maybe I will be back. Sorry. Good luck.

Answer 20

\[||r_x \times r_y|| = \sqrt{(\frac{ ∂f }{ ∂x })^2+ (\frac{ ∂f }{ ∂y })^2 +1}\]