I would like to know if my following solution to a problem is valid or not.


Problem:


If S is a sphere and F satisfies the hypotheses of Stokes' theorem, show that \iint_{S}F\cdot ds =0.


**Solution:**


Stokes' theorem claims that if we "cap off" the curve C by any surface S (with appropriate orientation) then the line integral can be computed in terms of the curl. Solving mathematically, let S be the surface of the sphere and be bounded by a curve C and let \vec F be the vector field which satisfies Stokes' theorem. Then, for every closed path \int_{C}^{} \vec F \cdot \vec dr = 0 because \vec F is a conservative vector field in Stokes' theorem. Hence by Stokes' theorem we get \int_{C}^{} \vec F \cdot \vec dr = \int_{}^{}\int_{S}^{} \vec F \cdot \vec ds = 0 (from the definition of conservative field.)


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multivariable calculus - Validity of following solution - Mathematics Stack Exchange