Evaluating ∫cF⋅dr Using Stokes' Theorem

[happykamper21]

Homework Statement

Use Stokes' Theorem to evaluate ∫_cF ⋅ dr, where F(x, y, z) = x²zi + xy²j + z²k and C is the curve of the intersection of the plane x + y + z = 1 and the cylinder x² + y² = 9 oriented counterclockwise as viewed from above.

Homework Equations

Stoke's Theorem:
∫_cF ⋅ dr = ∫_s curlF ⋅ ds

The Attempt at a Solution

For this problem I am extremely confused of which variant of Stoke's theorem to use and when it is appropriate to use a certain variant. For this problem my teacher found the curlF and then dotted it with the ds. However there are problems in the same section where he uses the left side of Stoke's Theorem. Is it possible to use both? If so, would it be possible to say which would be more advantageous over the other?

 

[FactChecker]

Since both sides give identical answers, it is concievable to use either one or both. But often one is much easier than the other. You can probably get a feel for which is easier by looking at the examples and the one that your teacher did not use to see what the problems would be. In example problems, you are often given the information needed for one side and not for the other, so it is simple.

 

[Orodruin]

Since you are being told to use the theorem to compute the circulation integral, you should use it to go from a circulation integral to a surface integral. If you just computed the circulation integral you would not be using the theorem...