Let be the surface of the cap (see figure). Let be the differentiable curve of the cap in the plane. Let be the portion of the plane xy that C contains. And let be the value of the area of . Consider oriented outward. Determine with arrows the sense of the curve and the value of , where .
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My attempt :
The arrow is easy to draw : counter clockwise.
I've calculated the curl of , but now I realize it is likely useless. Anyway here it comes : .
I'm sure I have to calculate (which is worth by Stokes' theorem) but how to proceed? I have almost no information about , I mean I don't know its parametrization. The only way I think I might solve the problem is to realize something obvious that imply that , but I don't see anything.
Thanks for the answer NCA, I wouldn't have find it by myself within the next 6 remaining days my exam. The problem is indeed nice because it is different from most.
Just a little clarification of my first post : I mean C has the clockwise sense and not anticlockwise as I said. I sketched it clockwise but I don't know why I said anticlockwise.