Question

1. Verify Stock’s theorem for F~ = x^2i+xyj, where s is the area bonded by

a triangle whose vertices are at (0, 0), (1, −1) and (1, 1), respectively.

Answer 1

$\text{curl }\vec F=\begin{vmatrix} \vec i& \vec j & \vec k\\ \frac{\partial}{\partial x}& \frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\ x^2 & xy& 0 \end{vmatrix}=y\vec k,$

$\text{curl}\vec F\cdot \vec n=y,$

$\underset{S}{\iint }\text{curl}\vec F\cdot \vec ndS =\int_0^1\int_0^1ydydx+\int_0^1\int_{-1}^0ydydx=\int_0^1\frac{y^2}2|_0^1dx+\int_0^1\frac{y^2}2|_{-1}^0dx=\int_0^1\frac 12dx+\int_0^1(-\frac 12)dx=0.$