Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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 timedtis

dr=

dr

dt

dt=vdt,

wherevis 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 scalarucanberegardedasthe

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, ifa(u)=d[A(u)]/duthen the indefinite integral ofa(u) is given by

∫

a(u)du=A(u)+b,

wherebis a constant vector. The definite integral ofa(u)fromu=u 1 tou=u 2

is given by
∫u 2

u 1

a(u)du=A(u 2 )−A(u 1 ).

A small particle of massmorbits a much larger massMcentred at the originO. According

to Newton’s law of gravitation, the position vectorrof the small mass obeys the differential

equation

m

d^2 r

dt^2

=−

GMm

r^2

ˆr.

Show that the vectorr×dr/dtis a constant of the motion.

Forming the vector product of the differential equation withr,weobtain

r×

d^2 r

dt^2

=−

GM

r^2

r׈r.

Sincerandˆrare collinear,r×rˆ= 0 and therefore we have

r×

d^2 r

dt^2

= 0. (10.10)

However,

d

dt

(

r×

dr

dt

)

=r×

d^2 r

dt^2

+

dr

dt

×

dr

dt

= 0 ,