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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∮C1+C2f·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)=∇F=∂f∂xi+∂f∂yj

The closed path is

03

Step 3. The field is closed, so

∫C1f·dr=∫C2f·dr∫C1f·dr-∫C2f·dr=0

If we change the direction of the curve, then

∫C1f·dr=-∫-C2f·dr-∫C1f·dr=∫-C2f·dr

04

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

∫C1f·dr+∫-C2f·dr=0∮C1+C2f·dr=0

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Most popular questions from this chapter

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The result of integrating a vector field over a surface is a vector.

(b) True or False: The result of integrating a function over a surface is a scalar.

(c) True or False: For a region R in thexy-plane,dS=dA.

(d) True or False: In computing ∫Sf(x,y,z)dS, the direction of the normal vector is irrelevant.

(e) True or False: If f (x, y, z) is defined on an open region containing a smooth surface S, then ∫Sf(x,y,z)dSmeasures the flow through S in the positive z direction determined by f (x, y, z).

(f) True or False: If F(x, y, z) is defined on an open region containing a smooth surface S , then ∫SF(x,y,z).ndSmeasures the flow through S in the direction of n determined by the field F(x, y, z).

(g) True or False: In computing ∫SF(x,y,z).ndS,the direction of the normal vector is irrelevant.

(h) True or False: In computing ∫SF(x,y,z).ndS,with n pointing in the correct direction, we could use a scalar multiple of n, since the length will cancel in the dSterm.

Let a, b, and c be nonzero constants. Find a general formula for the area of the portion of the plane with equation ax+by+cz=kthat lies above a rectangle [α,β]×[γ,δ] in thexy-plane.

Find the areas of the given surfaces in Exercises 21–26.

Sis the portion of the surface determined by x=9-y2-z2 that lies on the positive side of the yzplane (i.e., where x≥0)

What are the outputs of a vector field in the Cartesian plane?

Given a smooth parametrization for a “generalized cylinder” S, given by extending the curve y = x2 upwards and downwards from z =−2 to z = 3.
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