• S₁ = {(x, y, 0) | x² + y² ≤ 1}, the unit disc in the plane z = boundary circle C = {(x, y, 0) | x² + y² = 1}. - - 0, with S₂ = {(x, y, 1 x² − 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 ñ 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 divỄ = 0. (b) Use the divergence theorem to calculate JJs F.ñ dS where n is the outward pointing unit normal vector on the surface S. (c) Calculate the surface integral √√s, F. ñ dS using a double integral (Hint: What are the values of F(x, y, z) and of ñ on the plane z = 0?) (d) Use your previous results to write down the value of the surface. integral √√s, F. ñ ds. S2 (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). Consider the vector field F(x, y, z) = (3xy, 4xz, -3yz+6) Consider also the 3-dimensional region D bounded by the surface S S₁ US2 where
• S₁ = {(x, y, 0) | x² + y² ≤ 1}, the unit disc in the plane z = boundary circle C = {(x, y, 0) | x² + y² = 1}. - - 0, with S₂ = {(x, y, 1 x² − 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 ñ 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 divỄ = 0. (b) Use the divergence theorem to calculate JJs F.ñ dS where n is the outward pointing unit normal vector on the surface S. (c) Calculate the surface integral √√s, F. ñ dS using a double integral (Hint: What are the values of F(x, y, z) and of ñ on the plane z = 0?) (d) Use your previous results to write down the value of the surface. integral √√s, F. ñ ds. S2 (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). Consider the vector field F(x, y, z) = (3xy, 4xz, -3yz+6) Consider also the 3-dimensional region D bounded by the surface S S₁ US2 where
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:• S₁ = {(x, y, 0) | x² + y² ≤ 1}, the unit disc in the plane z =
boundary circle C = {(x, y, 0) | x² + y² = 1}.
-
-
0, with
S₂ = {(x, y, 1 x² − 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 ñ 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 divỄ
= 0.
(b) Use the divergence theorem to calculate JJs F.ñ dS where n is the
outward pointing unit normal vector on the surface S.
(c) Calculate the surface integral √√s, F. ñ dS using a double integral
(Hint: What are the values of F(x, y, z) and of ñ on the plane z = 0?)
(d) Use your previous results to write down the value of the surface.
integral √√s, F. ñ ds.
S2
(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).
Transcribed Image Text:Consider the vector field
F(x, y, z) = (3xy, 4xz, -3yz+6)
Consider also the 3-dimensional region D bounded by the surface S
S₁ US2 where
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