   Chapter 16, Problem 19RE

Chapter
Section
Textbook Problem

Show that there is no vector field G such that curl G = 2x i + 3yz j − xz2 k

To determine

To Show: There is no vector field G such that curlG=2xi+3yjxz2k .

Explanation

Given data:

curlG=2xi+3yjxz2k

Formula used:

If f has continuous partial derivatives of all orders on 3 , then div(curl f)=0 .

div(curl f)=0 (1)

Write the expression for F .

F=f

Substitute F for f in equation (1),

div(curl F)=0 (2)

Write the expression for divergence.

Write the expression for divF .

divF=Px+Qy+Rz (3)

Consider the expression of curlG .

curlG=2xi+3yjxz2k (4)

Consider the expression for F(x,y,z) as follows.

F(x,y,z)=Pi+Qj+Rk (5)

Compare equation (4) and (5).

F(x,y,z)=curlGP=2xQ=3yR=xz2

Find the value of div(curl G) using equation (3)

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