Last hope = this forum (Math Physics Calc C)

[Fox]

Alright, it's 1:42 am here on the East Coast... I've been working on a set of problems for my Math Physics class for the whole weekend. Four have got me stuck and well, I really need to get them done in order to pass this class (I'm not doing so hot) Thing is, all this is Calc C, and I haven't taken Calc C. I'm just praying you guys can help.

1A. Find the volume of the region bounded by the parabolic cylinder z=4-x^2 and the planes x=0, y=6, y=0, and z=0. Use rectangular coordinates. Make an accurate sketch of the geometry involved.
B. Find the coordinates of the centroid of the above volume.

Where I got stuck here: Well, I know I have to intergrate something, but what? And how many times? I'm not used to 3d geometry.

2. Calculate the moment of inertia, I(z), of that portion of the ellipsoid with surface x^2 + y^2 + z = 4, lying above the plane z=0, with mass density ρ = (1-r^(-2)). Use cylindrical coordinates.

Where I got stuck here: I know was moment of inertia is, that isn't that bad. Just what's a ellipsoid and how the heck do I use cylindrical coordinates?

3. Give that x = r cos φ sin θ , y = r sin θ sin φ , and z = r cos θ , derive the expression for the Jacobian of the transformation from the varibles x, y, & z to r, θ, and φ by Jacobi's method.

What went wrong here: I don't know Jacobi's method and looking it up on the internet didn't help any... and yes I tried MathWorld. :)

4. Calculate the electric potential at the point x = 4 meters, y = 8 meters due to a linear charge density along a rod located between z' = 0 and z' = b = 10 meters. The charge density is 2+(z')^2 C/m. Work out the solution in terms of the variables. Substitute the values of x, y, and b only at the end.

What went wrong here: I know electric potential, that isn't hard. but the 3d geometry is screwing me over again :(

Honestly, thank you thank you thank you for ANY and all help :)

 

[Crosson]

Hmm, I was myself in a similar position; doing multi-variable calc in physics before math (I think it is better that way anyway).

1A. Draw three coordinate axis and think of z = 4-x^2. This does not depend on y, so at every y = constant plane there will be a parabola centered around z. Hence the name cylinder. Surely you can imagine the x=0, y=6, y=0, and z=0 planes if you put in the effort. Here is the integral you need:


$$V = \int_V dv = \int_V dx dy dz = \int_0 ^6 \int_0 ^p \int_0 ^{4 -x^2} dy dx dz$$

Where p is some kind of x limit of integration (bounded by the plane x = 0 and what?).

These are some fairly weighty exercises, and there is only one way that you can get through them: genuine mental effort. It does no good to go through the motions, playing with calculus notation; you are going to have to think very hard about spatial shapes and Reimann sums.

 

[Fox]

I have solved 1A, 2, and 3.

1A. No problem once I got my geometery striaght and with a special thanks to Crosson :)

2. I pulled myself together and just dove head in again after taking a break... (on two I got a negative inertia... umm... I don't remember if that's possible?)

3. Easy once I actually found out what the hell a Jacobian was.

 

[Fox]

solved 1b now too...

still uneasy about negative interia and I check my work as best as I could. I'm stuck on #4 I know I have to use V = (k dq)/r Now I replace dq with lambda d(?)

can anyone fill in the question mark? And what do I expand "r" out to be? I could do this problem if it was two dimensional... :(