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


Problem:


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


**Solution:**


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


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multivariable calculus - Validity of following solution - Mathematics Stack Exchange