How do I use Stokes' theorem to find curl How do I solve questions that ask me to use Stokes' Theorem to find curl F
For example: 
Use Stoke's Theorem to find curl F: 
$$F(x, y, z) = \langle e^x+y^2, y^2+z^2, \sin(z)+x^2\rangle$$
$\iint_S \operatorname{curl}(F) * n\, dS$  where s is the portion of the paraboloid $z = 7 - 3x^2-2y^2$ above the plane z = 1, oriented with normal vector pointing upward. 
 A: An approach to the definition of the basic 3D vector differential operators by using the Gauss-Green and Stokes theorems has been sometimes used by mathematicians and mathematical physicists: to my knowledge, the most complete work on such topic was done by Claus Müller in the monograph [1]. The approach is of some importance since it does not require any differentiability condition on the fields involved.
Precisely, without the use of distribution theory, and instead by using the integral definition of standard vector operators, he defines a sort of weak derivatives in the following way:
$$
\nabla\times \boldsymbol{F} =\lim_{S_i\to x}\frac{1}{\Vert S_i\Vert}\int\limits_{\partial{S_i}} \boldsymbol{n}\times \boldsymbol{F}\,\mathrm{d}\sigma\triangleq\mathrm{curl}\,\boldsymbol{F}\tag{1}\label{1}
$$
where 


*

*$\boldsymbol{F}$ is a (non-differentiable) vector field in $\mathbb{R}^3$,

*$\{S_n\}$ is an contractible indexed family of smooth sets in $\mathbb{R}^3$ converging to the point $x\in\mathbb{R}^3$, whose volume is $\Vert S_n\Vert$ and whose boundary surface is $\partial{S}_n$,

*$\boldsymbol{n}$ is the inward normal vector to the surface $\partial{S}_n$.


Considering your case, you should chose a contractible surface $S$ in order to contract to the point $(x,y,z)$ where you want to find the value of $\mathrm{curl}\,\boldsymbol{F}$, and then compute the limit in \eqref{1}
[1] Claus Müller (1969)[1957], Foundations of the Mathematical Theory of Electromagnetic Waves, Grundlehren der mathematischen Wissenschaften 155, Springer-Verlag, pp. VIII+353, MR0253638, Zbl 0181.57203.
