I would like to know if my following solution to a problem is valid or not.
Problem:
If is a sphere and satisfies the hypotheses of Stokes' theorem, show that .
**Solution:**
Stokes' theorem claims that if we "cap off" the curve by any surface (with appropriate orientation) then the line integral can be computed in terms of the curl. Solving mathematically, let be the surface of the sphere and be bounded by a curve and let be the vector field which satisfies Stokes' theorem. Then, for every closed path because is a conservative vector field in Stokes' theorem. Hence by Stokes' theorem we get (from the definition of conservative field.)
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multivariable calculus - Validity of following solution - Mathematics Stack Exchange