Multivariate Calculus - Bruce E. Shapiro

LECTURE 22. LINE INTEGRALS 187
The second integral is negative because the C 2 is oriented from A to B and not from
B to A as it would have to be to complete the closed curves.
But the integrals over C 1 and C 2 are two paths between the same two points;
hence by the path independence theorem,
∫ ∫
F · dr = F · dr
C 1 C 2
and conseequently
∮
C
F · dr = 0.
The following theorem gives an easy test to see if any given field is a gradient field.
Theorem 22.4 F is a gradient field if and only if ∇ × F = 0.
Proof. First, suppose F is a gradient field. Then there exists some scalar function
f such that F = ∇f. But
∇ × F = ∇ × ∇f = 0
because curl gradf = 0 for any function f.
The proof in the other direction, namely, that if ∇ × F = 0 then F is a gradient
function, requires the use of Stoke’s theorem, which we shall prove in the next
section. Stoke’s theorem gives us the formula
∮
F · dr = (∇ × F) · dr
C
where A is the area encloses by a closed path C. But if the curl F is zero, then
∮
F · dr = 0
C
A
Since the integral over a closed path is zero, the integral must be path-independent,
and hence the integrand must be a gradient field.
Theorem 22.5 The following statements are equivalent:
1. F is a gradient field.
2. ∇ × F = 0
3. There exists some scalar function f such that F = ∇f.
4. The line integral is path independent: for some scalar function f,
∫ B
A
F · dr = f(B) − f(A)
Math 250, Fall 2006 Revised December 6, 2006.