Question

Let σ be portion of the paraboloid z = x 2 + y 2 − 9 lying on and below the xy-plane. Let σ have downward orientation. Let C be its boundary curve, oriented positively with respect to σ. Let F = x − y, x 2 z, y . (a) Evaluate R C F • T ds as a line integral. (b) Use Stokes’ theorem to evaluate R C F • T ds by evaluating an appropriate surface integral.

Answer 1

See explaination

Explanation:

Stoke theorem proposes that the surface integral of the curl of a function over any surface bounded by a closed path is equal to the line integral of a particular vector function round that path.

The Stoke Theorem can be used if you see a two dimensional region bounded by a closed curve, or if you see a single integral ie a line integral.

See attached file for further solution.