Mathematical Methods for Physics and Engineering : A Comprehensive Guide

10.1 DIFFERENTIATION OF VECTORS

in terms ofiandj. From figure 10.2, we see that

ˆeρ=cosφi+sinφj,

eˆφ=−sinφi+cosφj.

Sinceiandjare constant vectors, we find that the derivatives of the basis vectors

ˆeρandˆeφwith respect totare given by

dˆeρ

dt

=−sinφ

dφ

dt

i+cosφ

dφ

dt

j=φ ̇eˆφ, (10.2)

dˆeφ

dt

=−cosφ

dφ

dt

i−sinφ

dφ

dt

j=−φ ̇eˆρ, (10.3)

where the overdot is the conventional notation for differentiation with respect to

time.

The position vector of a particle in plane polar coordinates isr(t)=ρ(t)eˆρ.Findexpres-

sions for the velocity and acceleration of the particle in these coordinates.

Using result (10.4) below, the velocity of the particle is given by

v(t)= ̇r(t)=ρ ̇eˆρ+ρ ̇eˆρ= ̇ρeˆρ+ρφ ̇eˆφ,

where we have used (10.2). In a similar way its acceleration is given by

a(t)=

d

dt

( ̇ρeˆρ+ρφ ̇ˆeφ)

= ̈ρeˆρ+ρ ̇e ̇ˆρ+ρ ̇φ ̇eˆφ+ρ ̈φˆeφ+ ̇ρφ ̇ˆeφ

= ̈ρeˆρ+ ̇ρ(φ ̇ˆeφ)+ρ ̇φ(− ̇φeˆρ)+ρφ ̈ˆeφ+ ̇ρφ ̇ˆeφ

=( ̈ρ−ρφ ̇^2 )ˆeρ+(ρ ̈φ+2ρ ̇ ̇φ)ˆeφ.

Here we have used (10.2) and (10.3).

10.1.1 Differentiation of composite vector expressions

In composite vector expressions each of the vectors or scalars involved may be

a function of some scalar variableu, as we have seen. The derivatives of such

expressions are easily found using the definition (10.1) and the rules of ordinary

differential calculus. They may be summarised by the following, in which we

assume thataandbare differentiable vector functions of a scalaruand thatφ

is a differentiable scalar function ofu:

d

du

(φa)=φ

da

du

+

dφ

du

a, (10.4)

d

du

(a·b)=a·

db

du

+

da

du

·b, (10.5)

d

du

(a×b)=a×

db

du

+

da

du

×b. (10.6)