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Problem 10. Let F = <y, z − x, 0> and let S be the surface...

Problem 10. Let F = <y, z − x, 0> and let S be the surface z = 4 − x^2 − y^2 for z ≥ 0, oriented by outward-pointing normal vectors.

a. Calculate curl(F).

b. Calculate Z Z S curl(F) · dS directly, i.e., evaluate it as a surface integral.

c. Calculate Z Z S curl(F) · dS using Stokes’ Theorem, i.e., evaluate instead the line integral I ∂S F · ds.

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