Where C is the loop with z=2
with a clockwise orientation when viewed from above.
As usual, I have trouble with the third dimension. I have a problem like this without the third dimension, and I have problems with the third dimension but other shapes (parabola, etc). I'm confused with this one.
Edited: can you show me both clockwise and counterclockwise orientations? I'm not really following how to differentiate that part.
Once you have determined that the field is conservative (that the differential is exact), for example, by observing that , , and that , you don't really have to find an "anti-derivative"- the integral around any closed path is 0.
ban26ana, if the differential were not exact, any one dimensional path can be written in terms of a single parameter. Here, since the path is the circle with center at (0, 0), in the z= 1 plane, with radius 1, so we can use the "standard" parameterization for the unit circle, , with . Then , and . . Integrating around the path in the "positive orientation" would be integrating from 0 to and in the "negative orientation", from to 0 (or, equivalently, from 0 to ).
Here, so .
Since it is true, for any function G(x,y,z) with continuous second derivatives, that , Chris L T521's observation that this for a specific G, is the same as saying and so, by Stoke's theorem, .