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Chapter 14: Q. 39 (page 1096)

Show that the vector fields in Exercises 33鈥�40 are not conservative.

G(x,y,z)=(3,yz,z+12)

Short Answer

Expert verified

The vector fields is not conservative because 鈭�F1鈭�z鈮�鈭�F2鈭�y..

Step by step solution

01

Step 1. Given Information

We have to show that the vector fields in the given exercise is not conservative.
G(x,y,z)=(3,yz,z+12)

02

Step 2. A vector field G(x,y)=(G1(x,y),G2(x,y)) is not conservative if and only if ∂F1∂z≠∂F2∂y.

For the vector fieldG(x,y,z)=(3,yz,z+12)

鈭�F1鈭�z=鈭�鈭�zyz鈭�F1鈭�z=y鈭�鈭�zz鈭�F1鈭�z=y

03

Step 3. Now finding ∂F2∂y

鈭�F2鈭�y=鈭�鈭�y(z+12)鈭�F2鈭�y=鈭�鈭�yz+鈭�鈭�y12鈭�F2鈭�y=0

Hence, 鈭�F1鈭�z鈮�鈭�F2鈭�y.

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