Radial fields Consider the radial vector field (x, y, z) F Let S be the sphere of radius a |r|P¯ (x² + y² + z²)p/2" centered at the origin. a. Use a surface integral to show that the outward flux of F across S is 4ra P. Recall that the unit normal to the sphere is r/[r|. b. For what values of p does F satisfy the conditions of the Divergence Theorem? For these values of p, use the fact (Theorem 17.10) that V · F = - 3 - P - to compute the flux |r|" across S using the Divergence Theorem. THEOREM 17.10 Divergence of Radial Vector Fields For a real number p, the divergence of the radial vector field (x, y, z) 3 — Р F = |r| (x² + y? + z?)r/2 is V.F = %3D |r|"
Radial fields Consider the radial vector field (x, y, z) F Let S be the sphere of radius a |r|P¯ (x² + y² + z²)p/2" centered at the origin. a. Use a surface integral to show that the outward flux of F across S is 4ra P. Recall that the unit normal to the sphere is r/[r|. b. For what values of p does F satisfy the conditions of the Divergence Theorem? For these values of p, use the fact (Theorem 17.10) that V · F = - 3 - P - to compute the flux |r|" across S using the Divergence Theorem. THEOREM 17.10 Divergence of Radial Vector Fields For a real number p, the divergence of the radial vector field (x, y, z) 3 — Р F = |r| (x² + y? + z?)r/2 is V.F = %3D |r|"
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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Transcribed Image Text:Radial fields Consider the radial vector field
(x, y, z)
F
Let S be the sphere of radius a
|r|P¯ (x² + y² + z²)p/2"
centered at the origin.
a. Use a surface integral to show that the outward flux of F
across S is 4ra P. Recall that the unit normal to the sphere
is r/[r|.
b. For what values of p does F satisfy the conditions of the
Divergence Theorem? For these values of p, use the fact
(Theorem 17.10) that V · F = -
3 - P
- to compute the flux
|r|"
across S using the Divergence Theorem.
Transcribed Image Text:THEOREM 17.10 Divergence of Radial Vector Fields
For a real number p, the divergence of the radial vector field
(x, y, z)
3 — Р
F =
|r| (x² + y? + z?)r/2
is V.F =
%3D
|r|"
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