In $\mathbb{R}^3$, $f=\left(\frac{x}{2}\right)^2+\left(\frac{y}{2}\right)^2+\left(\frac{z}{4}\right)^2$, Surface $S$ is defined by $S=\{(x,y,z)|f(x,y,z)=1, z>0\}$, and the vector field $A$ is defined by $A=(yz,x^3(z+1),x^2+y^2)^T$, where T is short for the transfer of a vector.
How to solve the integrate $$\int_S(\nabla\times A)\cdot\frac{\nabla f}{|\nabla f|}dS$$ where dS is the area element of surface S and $\nabla$ is the Hamilton Operator.
I have checked the book to find that $\nabla\times A$ denotes the curl of A, which formula is here: curl. But I don't know how to deal with dS in this surface integral.
