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