Solution Found!
Answer: Harmonic functions A scalar-valued function is
Chapter 14, Problem 54AE(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 if u is harmonic on a region D enclosed by a surface S. then \(\iint_{S} u \nabla u \cdot \mathbf{n} d S=\iiint_{D}|\nabla u|^{2} d V\).
Text Transcription:
nabla^2varphi = nabla cdot nabla_varphi = 0
iint_S u nabla u cdo n dS = iiint_D |nabla u|^2 dV
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 if u is harmonic on a region D enclosed by a surface S. then \(\iint_{S} u \nabla u \cdot \mathbf{n} d S=\iiint_{D}|\nabla u|^{2} d V\).
Text Transcription:
nabla^2varphi = nabla cdot nabla_varphi = 0
iint_S u nabla u cdo n dS = iiint_D |nabla u|^2 dV
ANSWER:Solution 54AE
If is harmonic on a region
enclosed by a surface
Apply Green’s first identity to and