which is determined by the vector force field. Newton’s second law of motion is
expressed
F =ma =md(^2) r
dt^2 =m
dv
dt.
The work done in moving along the curve Cbetween two points Aand Bcan then
be expressed as
WAB =∫BAF·dr =∫BAF·dr
dtdt =∫BAF·v dt =∫BAmdv
dt·v dt =∫BAmv ·dv
dtdt.Now utilize the vector identity
1
2d
dt(
v^2)
=^1
2d
dt(v ·v ) = v ·dv
dt,so that the above line integral can be expressed in the form
∫BAF·dr
dt dt =∫BAF·v dt =∫BAm
2d
dt(
v^2)
dt,which is easily integrated. One finds
WAB =∫BAF·dr =m
2 v2 B
A=m
2(
vB^2 −v^2 A)
=Ek(vB)−Ek(vA).In this equation the line integral WAB =
∫BAF·dr is called the work done in moving
the particle from Ato Bthrough the force field F . The quantity Ek(v) = m 2 v^2 is called
the kinetic energy of the particle. The above equation tells us that the work done in
moving a particle from Ato Bin a force field F must equal the change in the kinetic
energy of the particle between the points Aand B.
Representation of Line Integrals
The line integral
∫F·dr can be expressed in many different forms:
1. ∫B
AF·dr =∫tBtAF·dr
dtdt =∫tBtAF·v dtIntegrals of this form are used if F =F(t)and v =V(t)are known func-
tions of the parameter t.
2. ∫B
AF·dr =∫BAF·dr
ds ds =∫ sBsAF·eˆtds