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(Solved): direction. 10. Use the Divergence Theorem to evaluate the surface integral ff, F-ds where F zi + 2yz ...
direction. 10. Use the Divergence Theorem to evaluate the surface integral ff, F-ds where F zi + 2yzj - ryk, and the closed surface S is comprised of the √4x²-y² in octents one and two and the planes part of the sphere z = y = 0, z = 0.
10. Use the Divergence Theorem to evaluate the surface integral ∬SF⋅ds where F=zi+2yzj−xyk. and the closed surface S is comprised of the part of the sphere z=4−x2−y2 in octents one and two and the planes y=0,z=0.
Step1Applying the Divergence Theorem, we have:
?
S
F?ds = ?
V
div(F)dV
where V is the region enclosed by the surface S.
First, let's find the divergence of F:
div(F) = ?
?x
(zi) + ?
?y
(2yzj) + ?
?z
(-xyk)
= 0 + 2z + 0
= 2z
Now, we need to determine the limits for the triple integral. The surface S is comprised of the following parts: