Let S1 and S2 be, respectively, the upper hemisphere and the lower hemisphere of

Question

Let S1 and S2 be, respectively, the upper hemisphere and the lower hemisphere of the sphere x^2 + y^2 + z^2 = 4, both oriented with upward-pointing normal vector field n. Let the vector F be a smooth vector field defined on all real numbers. Show that the integral evaluated over the surface S1 of (curl F) dot dA = the integral evaluated over the surface S2 of (curl F) dot dA.

(in other words, the flux of the vector field (curl F) through S1 is the same as its flux through S2) in two ways:

(a) using Stokes' Theorem

(b) using the Divergence Theorem

Explanation / Answer

(a) Note that S? and S? share the same boundary curve C.

So, ??s? curl F

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