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Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN: 9781305658004
Author: Ron Larson
Publisher: Cengage Learning
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
Transcribed Image Text:Consider the vector field
F(x, y, z)=(3xy, 4x2, -3yz+6)
Consider also the 3-dimensional region D bounded by the surface S
S₁US₂ where
1
=
• S₁ = {(x, y, 0) | x² + y² ≤ 1}, the unit disc in the plane z = 0, with
boundary circle C = {(x, y, 0) | x² + y² = 1}.
S₂ = {(x, y, 1x2 − y²) | x² + y² ≤ 1}, an upside down paraboloid
with the same boundary circle C.
To visualize all this, think of a bullet standing on its flat end.
Let n denote the outward pointing unit normal vector on S. (Note that ñ is
only piecewise continuous: it is discontinuous along the common boundary
circle C of S₁ and S2; but piecewise continuity is just fine, as it is in Green's
Theorem).
(a) Verify that divF = 0.
(b) Use the divergence theorem to calculate JJ FdS where n is the
outward pointing unit normal vector on the surface S.
(c) Calculate the surface integral ffs, F.ñ dS using a double integral
(Hint: What are the values of F(x, y, z) and of n on the plane z = 0?)
(d) Use your previous results to write down the value of the surface
integral f√s, F. ñ ds.
(Hint: In this problem, there are almost no actual integrals that you have
to calculate. If you find yourself buried deep in the calculation of some
multiple integral, you are doing it wrong).
Expert Solution
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