Multivariate Calculus - Bruce E. Shapiro

184 LECTURE 22. LINE INTEGRALS
Recall from integral calculus the following statement of the fundamental theorem
of calculus: If f is differentiable and f ′ (t) is integrable on some interval (a, b) then
∫ b
a
f ′ (t)dt = f(b) − f(a) (22.20)
In the generalization to line integrals, the derivative becomes a directional derivative,
but otherwise the statement of the theorem remains almost unchanged.
Theorem 22.1 Fundamental Theorem of Calculus for Line Integrals Suppose
that C is a piecewise smooth curve that can be parameterized as
for some open set S ⊂ R 3 , and let
C = {r(t), a ≤ t ≤ b} ⊂ S (22.21)
A = r(a) (22.22)
B = r(b) (22.23)
Then if f : S ↦→ R 3 is continuously differentiable on S,
∫
∇f(r) · dr = f(B) − f(A) (22.24)
C
Since the directional derivative in the direction of v(t) is
D v f(r) = ∇f(r(t)) · v(t)
= ∇f(r(t)) · dr(t)
dt
then the left-hand side of equation 22.24 can be rewritten to give
∫
∫ b
∇f(r) · r = D v f(r(t))dt (22.25)
C
and the fundamental theorem of calculus for line integrals becomes
∫ b
Proof. Expanding the directional derivative,
a
a
D v f(r(t))dt = f(B) − f(A) (22.26)
D v f(r(t)) = ∇f(r(t)) · v
( ∂f(r(t))
= , ∂f(r(t)) , ∂f(r(t))
∂x ∂y ∂z
= ∂f(r(t)) dx
∂x dt + ∂f(r(t)) dy
∂y dt + ∂f(r(t))
∂z
= df(r(t))
dt
) ( dx
·
dt , dy
dt , dz
dt
Revised December 6, 2006. Math 250, Fall 2006
dz
dt
)