   Chapter 16.9, Problem 31E

Chapter
Section
Textbook Problem

Suppose S and E satisfy the conditions of the Divergence Theorem and f is a scalar function with continuous partial derivatives. Prove that ∬ S f n   d S   =   ∭ E ∇ f   d V These surface and triple integrals of vector functions are vectors defined by integrating each component function. [Hint: Start by applying the Divergence Theorem to F = fc, where c is an arbitrary constant vector.]

To determine

To prove: The expression SfndS=EfdV.

Explanation

Given Data:

The regions S and E satisfy the conditions of the Divergence Theorem and f is a scalar function with continuous partial derivatives.

The given expression that has to be proved is SfndS=EfdV.

Formula used:

The expression to find flux of the vector field F(x,y,z) across the surface S is,

SFndS=EdivFdV (1)

The expression to find divergence of vector field F(x,y,z)=Pi+Qj+Rk is,

divF=xP+yQ+zR (2)

Calculation:

Consider an arbitrary constant vector c as follows.

c=c1i+c2j+c3k

Consider the vector field F as follows.

F=fc

Substitute c1i+c2j+c3k for c.

F=f(c1i+c2j+c3k)=fc1i+fc2j+fc3k

Substitute fc1 for P, fc2 for Q, and fc3 for R in equation (2) to calculate the divF.

divF=x(fc1)+y(fc2)+z(fc3)=fxc1+fxc2+fxc3=divf(c1i+c2j+c3k)=divfc

Substitute (divfc) for divF in equation (1).

SFndS=E(divfc)dV (3)

If c=i, then the expression is written as follows

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