Singular radial field Consider the radial field (x, y, z) F |r| (x² + y² + z²)!/2 a. Evaluate a surface integral to show that Slg F ·n dS = 4ra², where S is the surface of a sphere of radius a centered at the origin. b. Note that the first partial derivatives of the components of F are undefined at the origin, so the Divergence Theorem does not apply directly. Nevertheless, the flux across the sphere as computed in part (a) is finite. Evaluate the triple integral of the Divergence Theorem as an improper integral as follows. Integrate div F over the region between two spheres of radius a and 0 < e < a. Then let e →0* to obtain the flux computed in part (a).
Singular radial field Consider the radial field (x, y, z) F |r| (x² + y² + z²)!/2 a. Evaluate a surface integral to show that Slg F ·n dS = 4ra², where S is the surface of a sphere of radius a centered at the origin. b. Note that the first partial derivatives of the components of F are undefined at the origin, so the Divergence Theorem does not apply directly. Nevertheless, the flux across the sphere as computed in part (a) is finite. Evaluate the triple integral of the Divergence Theorem as an improper integral as follows. Integrate div F over the region between two spheres of radius a and 0 < e < a. Then let e →0* to obtain the flux computed in part (a).
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:Singular radial field Consider the radial field
(x, y, z)
F
|r| (x² + y² + z²)!/2
a. Evaluate a surface integral to show that Slg F ·n dS = 4ra²,
where S is the surface of a sphere of radius a centered at the
origin.
b. Note that the first partial derivatives of the components of F
are undefined at the origin, so the Divergence Theorem does
not apply directly. Nevertheless, the flux across the sphere as
computed in part (a) is finite. Evaluate the triple integral of
the Divergence Theorem as an improper integral as follows.
Integrate div F over the region between two spheres of radius a
and 0 < e < a. Then let e →0* to obtain the flux computed
in part (a).
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