Let f = ay i + bz j + cx k where a, b, and c are positive constants. let c be the triangle obtained by tracing out the path from (7, 0, 0) to (7, 0, 2) to (7, 6, 2) to (7, 0, 0). find f · dr c.

You can either compute three line integrals, or use Stokes' theorem and compute one surface integral. I prefer the latter.

The curl of the given vector field is



Parameterize the triangular surface bounded by - I'll call it - by



with and . By Stokes' theorem, we have



where is positively oriented; that is, every vector normal to is pointed in the positive direction. The surface element is given by



So our integral is

Many things that seem threatening in the dark become welcoming when we shine light on them