   Chapter 16.9, Problem 29E

Chapter
Section
Textbook Problem

Prove each identity, assuming that S and E satisfy the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second-order partial derivatives.29. ∬ S ( f ∇ g )   ⋅   n   d S   =   ∭ E ( f ∇ 2 g   +   ∇ f ⋅   ∇ g   ) d V

To determine

To prove: The expression S(fg)ndS=E(f2g+fg)dV .

Explanation

Formula used:

Write the expression for SFndS .

SFndS=Ediv(F)dV (1)

Here,

E is the solid region.

Write the expression for div(fF) .

div(fF)=fdivF+Ff (2)

Calculation of S(fg)ndS :

Substitute (fg) for F in equation (1),

S(fg)ndS=Ediv(fg)dV (3)

Calculation of div(fg) :

Substitute g for F in equation (2),

div(fg)=fdiv(g)+(g)f

Substitute fdiv(g)+<

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