   Chapter 16.9, Problem 28E

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.28. ∬ S D n   f   d S   =   ∭ E ∇ 2 f   d V

To determine

To prove: The expression SDnfdS=E2fdV .

Explanation

Formula used:

Write the expression for SDnadS .

Write the expression for SFndS .

SFndS=Ediv(F)dV (2)

Here,

E is the solid region.

Calculation of SDnfdS :

Substitute f for a in equation (1),

SDnfdS=S(fn)dS (3)

Substitute f for F in equation (2),

S(fn)dS=Ediv(f)dV

As is a divergence operator, rewrite the expression as follows

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