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Singular radial field Consider the radial field .a.
Chapter 14, Problem 37E(choose chapter or problem)
Singular radial field Consider the radial field
\(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|}=\frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{1 / 2}}\)
a. Evaluate a surface integral to show that \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi a^{2}\). 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 t are undefined at the origin. so the Divergence Theorem does not apply directly. Nevertheless Lhe flux across the sphere as computed in pan (a) is finite. Evaluate the triple integral of the Divergence Theorem as an improper integral 35 follows. Integrate div F over the region between two spheres of radius a and \(0<\varepsilon<a\). Then let \(\varepsilon \rightarrow 0^{+}\) to obtain the flux computed in pan (a).
Text Transcription:
F = r / |r| = langle x, y, z rangle / (x^2 + y^2 + z^2)^½
iint_S F cdot ndS =4pia^2
0 < varepsilon < a
varepsilon rightarrow 0^+
Questions & Answers
QUESTION:
Singular radial field Consider the radial field
\(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|}=\frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{1 / 2}}\)
a. Evaluate a surface integral to show that \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi a^{2}\). 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 t are undefined at the origin. so the Divergence Theorem does not apply directly. Nevertheless Lhe flux across the sphere as computed in pan (a) is finite. Evaluate the triple integral of the Divergence Theorem as an improper integral 35 follows. Integrate div F over the region between two spheres of radius a and \(0<\varepsilon<a\). Then let \(\varepsilon \rightarrow 0^{+}\) to obtain the flux computed in pan (a).
Text Transcription:
F = r / |r| = langle x, y, z rangle / (x^2 + y^2 + z^2)^½
iint_S F cdot ndS =4pia^2
0 < varepsilon < a
varepsilon rightarrow 0^+
ANSWER:Solution 37E1. The