If , show that . is a vector field and is a scalar field.
I know it's an obvious question but I can't remember how to formally prove it.
I know it's true if and in Cartesian coordinates, I've showed it for but not for a general form of . I think I'm over-complicating this.
Could you give me a tip?
I do not know how to relates Stokes' theorem with this problem.
I know that .
In the example given, I do not know where and are defined.
By intuition I could take as a circle with radius R tending to infinite and S being the upper half sphere of radius R but I'm not sure it makes sense nor how do I use . Am I, at least, on the right direction? If so I'll think more about it.
Edit: Now I'm thinking about using Clairaut's theorem. But I'd have to show that is continuous and it is not given...