Kelvin–Stokes theorem!

The Kelvin–Stokes theorem states that the total curl around a closed surface is equal to the line integral of the vector field around the boundary of the surface. In other words, it relates the circulation of a vector field around a closed curve to the flux of its curl through the surface bounded by that curve.

observe below:

$\displaystyle \iint _{\Sigma }(\nabla \times \mathbf {F} )\cdot \mathrm {d} \mathbf {\Sigma } =\oint _{\partial \Sigma }\mathbf {F} \cdot \mathrm {d} \mathbf {\Gamma }$

$\displaystyle \iint _{\Sigma }(\nabla \times \mathbf {F} )\cdot \mathrm {d} \mathbf {\Sigma } =0$

Note that there may be small differences in LHS and RHS due to numerical inaccuracies.
We approximate the Stokes-theorem computationally using a framework: $$ \iint_S (\nabla \times \vec{F}) \cdot d\vec{S} \approx \sum_{u,v} \left[ (\nabla \times \vec{F})(\vec{r}(u,v)) \cdot \left( \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v} \right) \right] \Delta u\, \Delta v $$ and: $$ \oint_{\partial S} \vec{F} \cdot d\vec{\Gamma} \approx \sum_{\partial S} \left[ \vec{F}(\vec{r}(u,v)) \cdot \left( \frac{\partial \vec{r}}{\partial u} \Delta v + \frac{\partial \vec{r}}{\partial v} \Delta u \right) \right] $$

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Curve parameters:

$r_x(u, v)$ =

$r_y(u, v)$ =

$r_z(u, v)$ =

Vector field parameters:

$F_x(x, y, z)$ =

$F_y(x, y, z)$ =

$F_z(x, y, z)$ =