Solution Found!
Harmonic functions A scalar-valued function ? is harmonic
Chapter 14, Problem 52AE(choose chapter or problem)
Harmonic functions A scalar-valued function \(\varphi\) is harmonic on a region D if \(\nabla^{2} \varphi=\nabla \cdot \nabla \varphi=0\) at ull points of D.
Show that the potential function \(\varphi(x, y, z)=|\mathbf{r}|^{-p}\) is harmonic provided p = 0 or p = 1, where \(\mathbf{r}=\langle x, y, z\rangle\). To what vector fields do these potentials correspond?
Text Transcription:
nabla^2varphi = nabla cdot nabla_varphi = 0
varphi(x, y, z) = |r|^-p
r = langle x, y, z rangle
Questions & Answers
QUESTION:
Harmonic functions A scalar-valued function \(\varphi\) is harmonic on a region D if \(\nabla^{2} \varphi=\nabla \cdot \nabla \varphi=0\) at ull points of D.
Show that the potential function \(\varphi(x, y, z)=|\mathbf{r}|^{-p}\) is harmonic provided p = 0 or p = 1, where \(\mathbf{r}=\langle x, y, z\rangle\). To what vector fields do these potentials correspond?
Text Transcription:
nabla^2varphi = nabla cdot nabla_varphi = 0
varphi(x, y, z) = |r|^-p
r = langle x, y, z rangle
ANSWER:Solution 52AE