∫
C
F·dr = lim
n→∞
∑n
i=1
F(x∗i, y ∗i, zi∗)·
∆ri
∆si
∆si
=
∫
C
(
F 1
dx
ds
+F 2
dy
ds
+F 3
dz
ds
)
ds,
=
∫
C
F·dr =
∫
C
F ·dr
ds
ds
(6 .95)
where (x∗i, y ∗i, zi∗)is a point inside the ith subinterval of the arc length ∆si.
In the above definition the dot product F·∆∆srii represents the projection of the
vector F or component of F in the direction of the tangent vector to the curve C. The
line integral of the vector function may be thought of as representing a summation
of the tangential components of the vector F along the curve C between the points
P 0 and P 1. Line integrals of this type arise in the calculation of the work done in
moving through a force field along a curve. Here the work is given by a summation
of force times distance traveled.
In particular, the above line integral can be expressed in the form
∫
C
F·dr =
∫
C
F·dr
ds
ds =
∫
C
F·ˆetds =
∫
C
F 1 dx +F 2 dy +F 3 dz, (6 .96)
where at each point on the curve C, the dot product F ·ˆet is a scalar function of
position and represents the projection of F on the unit tangent vector to the curve.
Summations of cross products along a curve produce another type of line integral.
Definition (Line integral along a curve Cinvolving cross products.)
The line integral ∫
C
F ×dr
is defined by the limiting process
∫
C
F ×dr = lim
n→∞
∑n
i=1
F(x∗i, y i∗, zi∗)×∆r i, (6 .97)
where F =F(x∗i, y i∗, z ∗i)is the value of F at a point (x∗i, y ∗i, zi∗)in
the ith subinterval of arc length on the curve C.