Question

Use stokes' theorem to evaluate ∫ c f · dr where f = (x²+z) i (4x + y) j (9y − z) k and c is the curve of intersection of the plane x 3y z = 6 with the coordinate planes. (assume that c is oriented counterclockwise as viewed from above.)

Answer 1

The question involves using Stokes' theorem to convert a line integral of a vector field over a closed curve into an equivalent surface integral over the surface bounded by the curve.

Step-by-step explanation:

The subject of the question is the application of Stokes' theorem to evaluate a line integral over a vector field f = (x²+z) i + (4x + y) j + (9y − z) k. Stokes' theorem relates a line integral around a closed curve C, to a surface integral over a surface bounded by C.

In this case, C is the intersection of the plane x + 3y + z = 6 with the coordinate planes. The orientation of C is counterclockwise when viewed from above, which dictates the direction in which the surface integral should be computed. To use Stokes' theorem, we first need to find a vector field whose curl is f and then evaluate the surface integral of this curl over a surface bounded by C.