   Chapter 16.9, Problem 30E

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.30. ∬ S ( f ∇ g   −   g ∇ f )   ⋅   n   d S   =   ∭ E ( f ∇ 2 g   −   g ∇ 2 f   ) d V

To determine

To prove: The expression S(fggf)ndS=E(f2gg2f)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(fggf)ndS :

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

S(fggf)ndS=Ediv(fggf)dV

S(fggf)ndS=Ediv(fg)dVEdiv(gf)dV . (3)

Calculation of div(fg) :

Substitute g for F in equation (2),

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

As is a divergence operator, rewrite the expression as follows.

div(fg)=f2g+gf

Calculation of div(gf) :

Substitute f for F in equation (2),

div(gf)=gdiv(f)+(f)g

As is a divergence operator, rewrite the expression as follows

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