Integration of vectors

10.2 INTEGRATION OF VECTORS
Note that the differential of a vector is also a vector. As an example, the
infinitesimal change in the position vector of a particle in an infinitesimal time dt is
dr =
dr
dt = v dt,
dt
where v is the particle’s velocity.
10.2 Integration of vectors
The integration of a vector (or of an expression involving vectors that may itself
be either a vector or scalar) with respect to a scalar u can be regarded as the
inverse of differentiation. We must remember, however, that
(i) the integral has the same nature (vector or scalar) as the integrand,
(ii) the constant of integration for indefinite integrals must be of the same
nature as the integral.
For example, if a(u) = d[A(u)]/du then the indefinite integral of a(u) is given by
a(u) du = A(u) + b,
where b is a constant vector. The definite integral of a(u) from u = u1 to u = u2
is given by
u2
a(u) du = A(u2 ) − A(u1 ).
u1
A small particle of mass m orbits a much larger mass M centred at the origin O. According
to Newton’s law of gravitation, the position vector r of the small mass obeys the differential
equation
d2 r
GMm
m 2 = − 2 r̂.
dt
r
Show that the vector r × dr/dt is a constant of the motion.
Forming the vector product of the differential equation with r, we obtain
r×
d2 r
GM
= − 2 r × r̂.
dt2
r
Since r and r̂ are collinear, r × r̂ = 0 and therefore we have
r×
However,
d
dt
r×
dr
dt
d2 r
= 0.
dt2
=r×
d2 r dr dr
+
×
= 0,
dt2
dt
dt
339
(10.10)