3.II.9A

Let $C$ be the closed curve that is the boundary of the triangle $T$ with vertices at the points $(1,0,0),(0,1,0)$ and $(0,0,1)$.

Specify a direction along $C$ and consider the integral

$\oint_{C} \mathbf{A} \cdot d \mathbf{x}$

where $\mathbf{A}=\left(z^{2}-y^{2}, x^{2}-z^{2}, y^{2}-x^{2}\right)$. Explain why the contribution to the integral is the same from each edge of $C$, and evaluate the integral.

State Stokes's theorem and use it to evaluate the surface integral

$\int_{T}(\boldsymbol{\nabla} \times \mathbf{A}) \cdot d \mathbf{S},$

the components of the normal to $T$ being positive.

Show that $d \mathbf{S}$ in the above surface integral can be written in the form $(1,1,1) d y d z$.

Use this to verify your result by a direct calculation of the surface integral.

*Typos? Please submit corrections to this page on GitHub.*