2 Harder problems
(I feel like none of the problems involving Stokes' theorem can be called \"exercises\", given how much
there is going on here, so they're all in the \"harder problems\" section.)
2. Let $C$ be the closed curve made up of four straight line segments: from (1, 1, 1) to (-1, 1, -1),
from (-1,1,-1) to (-1,-1,1), from (-1,-1, 1) to (1,-1,-1), and from (1,-1,-1) back to
(1, 1, 1). This is shown in the diagram on the left.
Fun fact: this weird piecewise linear curve $C$ actually lies entirely on the graph of $z = xy$,
shown in the diagram on the right.
(a) Use this fun fact, and Stokes' theorem, to find the circulation integral
$\int_C (x^2 \mathbf{i} + x^2 \mathbf{j} + y^2 \mathbf{k}) \cdot \mathbf{T} \, ds$
(in a counterclockwise direction as seen from above) by first turning it into a surface
integral.
(b) For comparison (and to review), find the circulation integral directly as a line integral,
but just for one of the line segments (your choice).
(The line segments are all basically the same, so doing more would just be busy work,
though you can do it if you like.)
