23.4. A GENERAL GREEN’S THEOREM 443

Conservative vector fields are important because of the following theorem, sometimescalled the fundamental theorem for line integrals.

Theorem 23.4.4 Let U be a region in Rp and let F : U →Rp be a continuous vector field.Then F is conservative if and only if there exists a scalar valued function of p variables φ

such that F = ∇φ . Furthermore, if C is an oriented curve which goes from x to y in U,then ∫

CF · dR= φ (y)−φ (x) . (23.5)

Thus the line integral is path independent in this case. This function φ is called a scalarpotential for F .

Proof: To save space and fussing over things which are unimportant, denote by p(x0,x)a polygonal curve from x0 to x. Thus the orientation is such that itgoes from x0 to x. The curve p(x,x0) denotes the same set of points but in the oppositeorder. Suppose first F is conservative. Fix x0 ∈U and let

φ (x)≡∫p(x0,x)

F ·dR.

This is well defined because if q (x0,x) is another polygonal curve joining x0 to x, Thenthe curve obtained by following p(x0,x) from x0 to x and then from x to x0 alongq (x,x0) is a closed piecewise smooth curve and so by assumption, the line integral alongthis closed curve equals 0. However, this integral is just∫

p(x0,x)F ·d R+

∫q(x,x0)

F ·d R=∫p(x0,x)

F ·d R−∫q(x0,x)

F ·dR

which shows ∫p(x0,x)

F ·d R=∫q(x0,x)

F ·dR

and that φ is well defined. For small t,

φ (x + tei)−φ (x)

t=

∫p(x0,x+tei)

F ·d R−∫p(x0,x)

F ·dRt

=

∫p(x0,x)

F ·d R+∫p(x,x+tei)

F ·d R−∫p(x0,x)

F ·dRt

.

Since U is open, for small t, the ball of radius |t| centered at x is contained in U . There-fore, the line segment from x to x+ tei is also contained in U and so one can takep(x,x+ tei)(s) = x+ s(tei) for s ∈ [0,1]. Therefore, the above difference quotient re-duces to

1t

∫ 1

0F (x+ s(tei)) · tei ds =

∫ 1

0Fi (x+ s(tei)) ds

= Fi (x+ st (tei))

by the mean value theorem for integrals. Here st is some number between 0 and 1. Bycontinuity of F, this converges to Fi (x) as t→ 0. Therefore, ∇φ = F as claimed.