Page 111 - Vector Analysis

§4.2 Conservative Vector Fields 107

respectively. Therefore,

ż ż ż ż1 ż1

F ¨ dr = F ¨ dr + F ¨ dr = (t2, 0) ¨ (0, 1) dt + (1, 2t) ¨ (1, 0) dt = 1 .

C C1 C2 0 0

We note that in this example the line integrals of F over three different paths joining (0, 0)
and (1, 1) are identical.

ż

Example 4.28. Let F(x, y) = (y, ´x). Evaluate the line integral F ¨ dr from (1, 0) to

(0, ´1) along C

1. the straight line segment joining these points, and

2. three-quarters of the circle of unit radius centered at the origin and traversed counter-
clockwise.

For the first case, we parameterize the path by γ(t) = (1 ´ t, ´t) for t P [0, 1]. Then

ż ż1 ż1

F ¨ dr = (´t, t ´ 1) ¨ (´1, ´1) dt = 1 dt = 1 .

C0 0

For the second case, we parameterize the path by γ(t) = (cos t, sin t) for t P [ 3π ]
0, .
2
Then
3π 3π
ż ż 2ż 2 3π
F ¨ dr = (sin t, ´ cos t) ¨ (´ sin t, cos t) dt = (´1) dt = ´ .
02
C0

We note that in this example the line integrals of F over different paths joining (1, 0) and

(0, ´1) might be different.

4.2 Conservative Vector Fields

In the previous section, we define the line integral of a force along a curve in a given
orientation. In Example 4.27, we see that the line integrals along three different paths
connecting two given points are the same, while in Example 4.28 the line integrals along
two different paths (connecting two given points) are different. In this section, we are
interested in the rule of judging whether the line integral is path independent or not.

Definition 4.29 (Conservative Fields). A vector field F : D Ď Rn Ñ Rn is said to be
conservative if F = ∇ϕ for some scalar function φ : D Ñ R. Such a ϕ is called a (scalar)
potential for F on D.