Scalar and Vector Line
Integrals
February 8, 2006
Scalar Line Integrals
Let x : [a, b] → R3 be a path of class C 1. And
f : X ⊆ R3 → R be a continuous function
such that the domain X contains the image
of x. The scalar line integral of f along x
is
Z b
a
f (x(t))kx0(t)kdt
NOTATION: This integral is usually written
R
x f ds.
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Vector Line Integral
The vector line integral of F along x :
[a, b] → Rn is
Z
x
F · ds =
Z b
a
F(x(t)) · x0(t) dt
2
Physical Interpretation of Line Integrals
If F is a force field in space, then the vector line integral is the work done by F on a
particle as the particle moves along the path
x.
3
Tangent and Vector Line Integral
Let x : [a, b] → Rn with x0(t) 6= 0 in [a, b].
Unit tangent vector T:
x0(t)
T(t) = 0
.
kx (t)k
Then
Z
x
F · ds =
Z
x
(F · T) ds
4
Reparametrizations of Paths
We say that y : [c, d] → Rn is a reparametrization of the path x : [a, b] → Rn if both describe the same curve.
Mathematically, this means that there is a
one-to-one and onto C 1 function u : [a, b] →
[c, d] such that x(u(t)) = y(t).
5
Scalar Line Integrals
Do NOT depend on parametrization
Theorem:
then
If y is a reparametrization of x
Z
y
f ds =
Z
x
f ds
6
Scalar Line Integrals and parametrizations
If y is a reparametrization of x. Then
• If y is orientation-preserving, then
R
R
F
·
d
s
=
y
x F · ds
• If y is orientation-reversing, then
R
R
y F · ds = − x F · ds
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