   Chapter 16, Problem 4RQ

Chapter
Section
Textbook Problem

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f has continuous partial derivatives on ℝ3 and C is any circle, then ∫c ∇f · dr = 0.

To determine

Whether the statement “If f has continuous partial derivatives on 3 and C is any circle, then Cfdr=0 ” is true or false.

Explanation

Formula used:

F=f

Draw the closed curve as shown in Figure 1.

Consider two points located on closed curve.

Modify Figure 1 as shown in Figure 2.

C is composed of path C1 from point A to B and path C2 from point B to A.

Write the expression for line integral.

CFdr=C1Fdr+C2Fdr

Substitute f for F .

Cfdr=C1fdr+C2fdr (1)

Since C1 and C2 have the same initial and terminal points.

C1=C2

Modify equation (2) as follows

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