Here F·ˆetis the tangential component of the force F along the given
curve C. This form of the line integral is used if F =F(s)and ˆet are
known functions of the arc length s.
3. For a force field given by
F =F(x, y, z) = F 1 (x, y, z )ˆe 1 +F 2 (x, y, z)ˆe 2 +F 3 (x, y, z)ˆe 3
and the position vector of a point (x, y, z )on a curve Cgiven by
r =xˆe 1 +yeˆ 2 +zˆe 3 with dr =dx ˆe 1 +dy ˆe 2 +dz ˆe 3 ,
Here the work done is represented in the form
∫B
A
F·dr =
∫B
A
F 1 dx +F 2 dy +F 3 dz.
Line integrals are written in this form when a parametric representation
of the curve is known. In the special case where r =xˆe 1 + 0 ˆe 2 + 0 ˆe 3 , the
above line integral reduces to an ordinary integral.
4. The line integral
∫
CF·dr may be broken up into a sum of line integrals
along different portions of the curve C. If the curve Cis comprised of n
separate curves C 1 , C 2 ,... , C n,one can write
∫
C
F·dr =
∫
C 1
F·dr +
∫
C 2
F·dr +···+
∫
Cn
F·dr.
5. When the curve C is a simple closed curve (i.e., the curve does not
intersect itself), the line integral is represented by
...............................................................
∫
............................
C
F ·dr or ...............................................................
∫
..............................
C
F·dr (6 .98)
where the direction of integration is either in the counterclockwise sense
or clockwise sense. Whenever the line integral is represented in the form∫
C
©F·dr then it is to be understood that the integration direction is in
the counterclockwise sense which is known as the positive sense. Note
that when the curve is a simple closed curve, there is no need to specify
a beginning and end point for the integration. One need only specify a
direction to the integration. The integration is said to be in the positive
sense if the integration is in a counterclockwise direction or it is said
