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Chapter 14: Q. 34 (page 1096)

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

F(x,y)=(x2+y2,cosy)

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given Information

We have to show that the vector fields in the given exercise is not conservative.
F(x,y)=(x2+y2,cosy)

02

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

For the vector fieldF(x,y)=(x2+y2,cosy)

鈭�F1鈭�y=鈭�鈭�y(x2+y2)鈭�F1鈭�y=鈭�鈭�yx2+鈭�鈭�yy2鈭�F1鈭�y=0+2y鈭�F1鈭�y=2y

03

Step 3. Now finding ∂F2∂y

鈭�F2鈭�x=鈭�鈭�xcosy鈭�F2鈭�x=0

Hence, 鈭�F1鈭�y鈮�鈭�F2鈭�x.

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