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Chapter 14: Q. 64 (page 1108)

Prove that if F is a conservative vector field, then the line integral of F along any smooth closed curve C is zero.

Short Answer

Expert verified

Hence, we prove that鈭�C1+C2f路dr=0.

Step by step solution

01

Step 1. Given Information

Prove that if F is a conservative vector field, then the line integral of F along any smooth closed curve C is zero.

02

Step 2. As we know that F is a conservative vector field.

Sof(x,y)=鈭�F=鈭�f鈭�xi+鈭�f鈭�yj

The closed path is

03

Step 3. The field is closed, so

鈭�C1f路dr=鈭�C2f路dr鈭�C1f路dr-鈭�C2f路dr=0

If we change the direction of the curve, then

鈭�C1f路dr=-鈭�-C2f路dr-鈭�C1f路dr=鈭�-C2f路dr

04

Step 4. Now putting the value of -∫C1f·dr

鈭�C1f路dr+鈭�-C2f路dr=0鈭�C1+C2f路dr=0

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