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

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