Use Stokes’ Theorem to evaluate int int curl F * dS. F(x,y,z)=2ycos zi+e^(x)sin zj+xe^(y) k, S is the hemisphere
x^(2)+y^(2)+z^(2)=9, z>= 0,
oriented upward

1 Answers

Best Answer

When applying the Stokes's theorem the first thing you need to do is to identify the correct boundary of the surface. In our case, this boundary is the circle of radius 3 at the xy plane. We immediately parametrize it as:
gamma(t)=(3cos t, 3sin t, 0) -> gamma'(t)=(-3sin t, 3cos t, 0)
Next, recall that the Stokes's theorem states that:
int int_(S)curl F*dS=int_(∂S)F*ds=int_(0)^(2pi)F(gamma (t))*gamma'(t)
The integral becomes:
int_(0)^(2pi)-18sin^(2) t dt=-18pi
Result:
-18pi