Mathematical Methods for Physics and Engineering : A Comprehensive Guide

VECTOR CALCULUS

ρ, φ, z,where

x=ρcosφ, y=ρsinφ, z=z, (10.44)

andρ≥0, 0≤φ< 2 πand−∞<z<∞. The position vector ofPmay therefore

be written

r=ρcosφi+ρsinφj+zk. (10.45)

If we take the partial derivatives ofrwith respect toρ,φandzrespectively then

we obtain the three vectors

eρ=

∂r

∂ρ

=cosφi+sinφj, (10.46)

eφ=

∂r

∂φ

=−ρsinφi+ρcosφj, (10.47)

ez=

∂r

∂z

=k. (10.48)

These vectors lie in the directions of increasingρ,φandzrespectively but are

not all of unit length. Althougheρ,eφandezform a useful set of basis vectors

in their own right (we will see in section 10.10 that such a basis is sometimes the

mostuseful), it is usual to work with the correspondingunitvectors, which are

obtained by dividing each vector by its modulus to give

eˆρ=eρ=cosφi+sinφj, (10.49)

ˆeφ=

1

ρ

eφ=−sinφi+cosφj, (10.50)

ˆez=ez=k. (10.51)

These three unit vectors, like the Cartesian unit vectorsi,jandk,forman

orthonormal triad at each point in space, i.e. the basis vectors are mutually

orthogonal and of unit length (see figure 10.7). Unlike the fixed vectorsi,jandk,

however,ˆeρandeˆφchange direction asPmoves.

The expression for a general infinitesimal vector displacementdrin the position

ofPis given, from (10.19), by

dr=

∂r

∂ρ

dρ+

∂r

∂φ

dφ+

∂r

∂z

dz

=dρeρ+dφeφ+dzez

=dρeˆρ+ρdφeˆφ+dzeˆz. (10.52)

This expression illustrates an important difference between Cartesian and cylin-

drical polar coordinates (or non-Cartesian coordinates in general). In Cartesian

coordinates, the distance moved in going fromxtox+dx, withyandzheld

constant, is simplyds=dx. However, in cylindrical polars, ifφchanges bydφ,

withρandzheld constant, then the distance moved isnotdφ, butds=ρdφ.