   Chapter 16.9, Problem 26E

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.26. V ( E )   =   1 3 ∬ S F   ⋅   d S , where F(x, y, z) = x i + y j + z k

To determine

To prove: The expression V(E)=13SFdS .

Explanation

Given data:

The vector function is given as follows.

F(x,y,z)=xi+yj+zk

Formula used:

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

SFdS=EdivFdV (1)

Here,

E is the solid region.

Write the expression to find divergence of vector field F(x,y,z)=Pi+Qj+Rk .

divF=xP+yQ+zR (2)

Calculation of divF :

Substitute x for P , y for Q , and z for R

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