Question

Complex Analysis

In all cases below marks will only be awarded for exact answers. No marks will be awarded for decimal approximations.

Consider the triangle S in the plane z = 1 that has vertices at the points(0,0,1), (2,0,1), (0,2, 1); and the vector field

\mathbf{v}=\left(x z, x^{3}, y^{2} z\right)

a. Compute directly the integral

\iint_{S}(\nabla \times \mathbf{v}) \cdot \mathrm{d} \mathbf{S}

where the positive z direction is taken as the positive normal direction. Enter the value of this integral below.

\iint_{S}(\nabla \times \mathbf{v}) \cdot \mathrm{d} \mathbf{S}=

Next calculate

\oint_{C_{i}} \mathbf{v} \cdot \mathrm{d} \mathbf{r}

for i = 1,2,3, where each C; is one of the edges surrounding S.Enter the values of this integral for each i = 1,2, 3: enter just three numbers, separated by commas. The order of your answers does not matter, but you should choose orientations compatible with the first part of this question.

C1, C2, C3 =%3D