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Radial fields Consider the radial vector field . Let S be
Chapter 14, Problem 36E(choose chapter or problem)
Radial fields Consider the radial vector field
\(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}}=\frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{p / 2}}\), let S be the sphere of radius a centered at the origin.
a. Use a surface integral to show that the outward flux of F across S is \(4 \pi a^{3-p}\). Recall that the unit normal lo 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 \(\nabla \cdot \mathbf{F}=\frac{3-p}{|\mathbf{r}|^{p}}\) across S using the Divergence Theorem.
Text Transcription:
F = r / |r|^p = langle x, y, z rangle / (x^2 + y^2 + z^2)^p/2
4pia^3-p
Nabla cdot F = 3 - p / |r|^p
Questions & Answers
QUESTION:
Radial fields Consider the radial vector field
\(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}}=\frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{p / 2}}\), let S be the sphere of radius a centered at the origin.
a. Use a surface integral to show that the outward flux of F across S is \(4 \pi a^{3-p}\). Recall that the unit normal lo 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 \(\nabla \cdot \mathbf{F}=\frac{3-p}{|\mathbf{r}|^{p}}\) across S using the Divergence Theorem.
Text Transcription:
F = r / |r|^p = langle x, y, z rangle / (x^2 + y^2 + z^2)^p/2
4pia^3-p
Nabla cdot F = 3 - p / |r|^p
ANSWER: