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Logarithmic potential Consider the potential function(x,
Chapter 14, Problem 38E(choose chapter or problem)
Logarithmic potential Consider the potential function
\(\varphi(x, y, z)=\frac{1}{2} \ln \left(x^{2}+y^{2}+z^{2}\right)=\ln |\mathbf{r}|\), where \(\mathbf{r}=\langle x, y, z\rangle\)
a Show that the gradient field associated with \(\varphi\) is
\(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|^{2}}=\frac{\langle x, y, z\rangle}{x^{2}+y^{2}+z^{2}}\)
b. Show that\(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi a\), where S is the surface of a sphere of radius a centered at the origin.
c. Compute div F.
d. Note that F is undefined at the origin. so the Divergence Theorem does not apply directly. Evaluate the volume integral as described in Exercise 37.
Text Transcription:
varphi(x, y, z)= ½ ln (x^2 + y^2 + z^2) = ln |r|
r = langle x, y, z rangle
iint_S F cdot n dS = 4pia
varphi
Questions & Answers
QUESTION:
Logarithmic potential Consider the potential function
\(\varphi(x, y, z)=\frac{1}{2} \ln \left(x^{2}+y^{2}+z^{2}\right)=\ln |\mathbf{r}|\), where \(\mathbf{r}=\langle x, y, z\rangle\)
a Show that the gradient field associated with \(\varphi\) is
\(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|^{2}}=\frac{\langle x, y, z\rangle}{x^{2}+y^{2}+z^{2}}\)
b. Show that\(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi a\), where S is the surface of a sphere of radius a centered at the origin.
c. Compute div F.
d. Note that F is undefined at the origin. so the Divergence Theorem does not apply directly. Evaluate the volume integral as described in Exercise 37.
Text Transcription:
varphi(x, y, z)= ½ ln (x^2 + y^2 + z^2) = ln |r|
r = langle x, y, z rangle
iint_S F cdot n dS = 4pia
varphi
ANSWER:Solution 38E
(a)