∫CF·dr = lim
n→∞∑ni=1F(x∗i, y ∗i, zi∗)·∆ri
∆si∆si=∫C(
F 1dx
ds+F 2dy
ds+F 3dz
ds)
ds,=∫CF·dr =∫CF ·dr
dsds(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
∫CF·dr =∫CF·dr
dsds =∫CF·ˆetds =∫CF 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 ∫CF ×dris defined by the limiting process
∫CF ×dr = lim
n→∞∑ni=1F(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.