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Green’s First Identity Prove Green’s First Identity for
Chapter 14, Problem 50AE(choose chapter or problem)
Green's First Identity Prove Green's First Identity for twice differentiable scalar-valued functions u and v defined on a region D:
\(\iiint_{D}\left(u \nabla^{2} v+\nabla u \cdot \nabla v\right) d V=\iint_{S} u \nabla v \cdot \mathbf{n} d S\)
where \(\nabla^{2} v=\nabla \cdot \nabla v\). You may apply Gauss' Formula in Exercise 48 to \(\mathbf{F}=\nabla v\) or apply the Divergence Theorem to \(\mathbf{F}=u \nabla v\).
Text Transcription:
nabla^2v = nabla cdot nabla v
F = nabla v
F = u nabla v
iiint_D(u nabla^2 v + nabla u cdot nabla v) dV = iint_S u nabla v cdot n dS
Questions & Answers
QUESTION:
Green's First Identity Prove Green's First Identity for twice differentiable scalar-valued functions u and v defined on a region D:
\(\iiint_{D}\left(u \nabla^{2} v+\nabla u \cdot \nabla v\right) d V=\iint_{S} u \nabla v \cdot \mathbf{n} d S\)
where \(\nabla^{2} v=\nabla \cdot \nabla v\). You may apply Gauss' Formula in Exercise 48 to \(\mathbf{F}=\nabla v\) or apply the Divergence Theorem to \(\mathbf{F}=u \nabla v\).
Text Transcription:
nabla^2v = nabla cdot nabla v
F = nabla v
F = u nabla v
iiint_D(u nabla^2 v + nabla u cdot nabla v) dV = iint_S u nabla v cdot n dS
ANSWER:Soluti