MHB Is Stokes' Theorem Applicable to a Sphere in This Solution?

[kalish1]

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.)

I have cross-posted this question here and here:
http://math.stackexchange.com/questions/740899/validity-of-following-solution
Validity of following solution - Math Help Forum

 

[I like Serena]

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.)

I have cross-posted this question here and here:
http://math.stackexchange.com/questions/740899/validity-of-following-solution
Validity of following solution - Math Help Forum

Hi kalish!

Your problem statement does not seem to be quite correct.

If $S$ is a sphere it would not be a capped off closed curve.
In other words, Stokes would not apply.

Perhaps Gauss's theorem is intended?
$$\iint_{\partial V} F \cdot dS = \iiint_V \nabla\cdot F dV$$Furthermore, can you clarify what you mean by the hypotheses of Stokes' theorem?
The only thing I can think of, is the condition of Stokes' theorem that says that the curl of the vector field must be properly defined on the surface.

Either way, something more must have been intended, otherwise the integral would not be guaranteed to be zero.