Explaining Line Integrals and Gradient in Conservative Fields

[Willa]

I have a question which asked me to evalute the line integral around the curve x^2+y^2=r^2 (z=z0 (a constant)) of the following vectors:

(0, z^2, 2yz)
and
(yz^2, yx^2, xyz)

the first one I get as 0, and the second one I get as: -pi(r*z0)^2

Those answers I'm pretty sure are right

The next part of the problem asks to find grad(yz^2) which I calculate to be: (0,z^2, 2yz).

The problem then asks to use this result to explain the answers to the two line integrals in the first part of the question. Now the first integral is easy to explain...since it is the integral of the gradient of a scalar...hence a conservative field, hence the integral around a closed loop i.e. a circle, is 0!

But I can't seem to explain the 2nd integral, any ideas anyone?

 

[HallsofIvy]

Have you had Green's theorem? since z remains constant (z0) thoughout the integration, you can treat this as a problem in the xy-plane. What would Green's theorem tell you about that integral?

(Oh, by the way, it's easy to integrate xy over a disk with center at (0,0) isn't it?_

 

[Willa]

well i thought about it being something to do with greens theorem...hence just do the curl of the vector over the area but the curl doesn't come out to -z0...i just can't link it to the grad of that scalar i gave. Do you care to elaborate on what you're thinking?