Question

. Let C be the curve x2+y2=1 lying in the plane z = 1. Let ?=(?−?)?̂+??...

. Let C be the curve x2+y2=1 lying in the plane z = 1. Let ?=(?−?)?̂+?? =

(a) Calculate ∇×?

(b) Calculate ∫?∙?? F · ds using a parametrization of C and a chosen orientation for C.

(c) Write C = ∂S for a suitably chosen surface S and, applying Stokes’ theorem, verify your answer in (b)

(d) Consider the sphere with radius √22 and center the origin. Let S’ be the part of the sphere that is above the curve (i.e., lies in the region z ≥ 1), and has C as boundary. Evaluate the surface integral of ∇ × F over S′. Specify the orientation you are using for S′.

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