# I cant´t find where I am wrong in my calculations

Consider the surfaces $$S_1=\{(x,y,z\in\mathbb{R^3}):x^2+y^2=9-z, \ z\geq0 \}$$ and $$S_2=\{(x,y,z)\in\mathbb{R^3}:x^2+y^2\leq9,\ z=0\}$$ and the vector field $$F=(y,2z,-3y^2)$$. I know that by stokes theorem I have that

$$\int\int_{s_1}\nabla \times F\cdot dS_1=\int\int_{s_2}\nabla\times F\cdot dS_2$$ My problem is the following: if I take the parameterizations $$\Phi_1=(\cos(\theta),\sin(\theta),z)\ \ 0\leq\theta\leq2\pi\ \ 0\leq z\leq9$$ for the surface $$S_1$$ and $$\Phi_2=(r\cos(\theta),r\sin(\theta),0)\ \ 0\leq\theta\leq2\pi\ \ 0\leq r\leq3$$ for $$S_2$$ I obtain that $$\int\int_{s_1}\nabla \times F\cdot dS_1=0$$ $$\int\int_{s_2}\nabla\times F\cdot dS_2=-9\pi$$ And this must be impossible. I don't know where is my mistake, can someone explain me?.

In my calculations a get that $$\nabla\times F=(-6y-2,0,-1),\ J(\Phi_1)=(\cos(\theta),\sin(\theta),0), \ J(\Phi_2)=(0,0,r)$$.

• I think your issue is with the parametrization of $\Phi_1$ – Andrei Oct 5 '18 at 17:20

Your $$\Phi_1$$ parametrization is going to produce a cylinder of radius 1 and height 9.
Instead, write the equation for $$S_1$$ as $$z = 9-x^2-y^2$$. Then you can parametrize it as $$\Phi_1(r,\theta) = (r \cos\theta, r \sin\theta,9-r^2)$$ for $$0 \leq \theta \leq 2\pi$$, $$0 \leq r \leq 3$$.