b) If F = (x)i + ((y^2)z + y)j − (2(y^3)z)k. Find S∫F·nˆdσ where S is the portion of z=4−x −y in the upper half space {z≥0} and nˆ is the normal in positive z-direction. c) If S1 is the upper half sphere {x^2 + y^2 + z^2 = 4,z ≥ 0} and S2 is the lower half sphere. Show that for any smooth vector field G, S1∫(∇×G)·n1ˆ dσ + S2∫(∇×G)·n2ˆ dσ = 0 (n1ˆ is the normal pointing in positive z-direction (up) and n2ˆ is the normal pointing negative z-direction (down))
b) If F = (x)i + ((y^2)z + y)j − (2(y^3)z)k. Find S∫F·nˆdσ where S is the portion of z=4−x −y in the upper half space {z≥0} and nˆ is the normal in positive z-direction. c) If S1 is the upper half sphere {x^2 + y^2 + z^2 = 4,z ≥ 0} and S2 is the lower half sphere. Show that for any smooth vector field G, S1∫(∇×G)·n1ˆ dσ + S2∫(∇×G)·n2ˆ dσ = 0 (n1ˆ is the normal pointing in positive z-direction (up) and n2ˆ is the normal pointing negative z-direction (down))
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The surface area is calculated to be: ((17)3^2 −1)π/6
b) If F = (x)i + ((y^2)z + y)j − (2(y^3)z)k. Find S∫F·nˆdσ where S is the portion of z=4−x −y in the upper half space {z≥0} and nˆ is the normal in positive z-direction.
c) If S1 is the upper half sphere {x^2 + y^2 + z^2 = 4,z ≥ 0} and S2 is the lower half sphere. Show that for any smooth vector field G, S1∫(∇×G)·n1ˆ dσ + S2∫(∇×G)·n2ˆ dσ = 0
(n1ˆ is the normal pointing in positive z-direction (up) and n2ˆ is the normal pointing negative z-direction (down))
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