Im struggling to solve this problem :
Consider F$(x,y,z) = (y,z,x)$ and the surface $\phi(u,v) = (u,v,sin(u^2 + v^2))$ where $(u,v)$ belongs to some disc centred at $(0,0)$ and $r=3$. Find the surface integral using Stoke's Theorem.
So far I have parametrised the curve so that $$ \gamma(t) = (3cos(t),3sin(t), 0)$$
I'm slightly unsure of the method used to solve this integral using Stoke's Theorem. What should i be taking the curl of?
The correct answer should be zero which is not what im getting.
Hint: You can write $$F=\nabla\times G$$ where $$G(x,y,z)=\left (\frac{z^2}{2},\frac{x^2}{2},\frac{y^2}{2}\right )$$
in order to use Stokes' theorem to express $$\int_S F\cdot \mathrm d\Sigma=\int_S \left (\nabla \times G\right ) \cdot \mathrm d \Sigma=\oint_{\partial S}G\cdot \mathrm d s$$