are unit vectors tangent to the coordinate curves, where ˆer·ˆeθ= 0, ˆer · ˆez = 0 and
ˆeθ·ˆez= 0. The unit vector ˆez=ˆer׈eθproduces the triad system {ˆer,eˆθ,ˆez}so that
the cylindrical coordinate system is a right-handed orthogonal coordinate system.
The unit vectors are sometimes expressed in the matrix^9 form
ˆer=
cos θ
sin θ
0
, ˆeθ=
−sin θ
cos θ
0
, ˆez=
0
0
1
In the cylindrical coordinate system the element of volume is given by dV =rdrdθdz
and the element of surface area is dS =rdθdz. The direction eˆris called the radial
direction, the direction ˆeθ is called the azimuthal direction and the direction ˆezis
called the vertical direction.
Spherical Coordinates (ρ,θ,φ )
The transformation from rectangular coordinates (x, y, z )to spherical coordinates
(ρ, θ, φ )is given by the equations
x=x(ρ, θ, φ ) = ρsin θcos φ, y =y(ρ, θ, φ ) = ρsin θsin φ, z =z(ρ, θ, φ ) = ρcos θand the general position vector is given by
r =r (ρ, θ, φ ) = ρsin θcos φˆe 1 +ρsin θsin φˆe 2 +ρcos θˆe 3In this coordinate system the coordinate surfaces are
r (ρ 0 , θ, φ ), a sphere x^2 +y^2 +z^2 =ρ^20
r (ρ, θ 0 , φ ), a cone x^2 +y^2 = tan^2 θz^2
r (ρ, θ, φ 0 ), a plane through the z−axis y=xtan φ
The coordinate curves in spherical coordinates are obtained from the intersection of
the coordinate surfaces and can be represented by
r (ρ 0 , θ 0 , φ ), circles of latitude
r (ρ 0 , θ, φ 0 ), meridian curve
r (ρ, θ 0 , φ 0 ), lines through the origin
These coordinate surfaces and coordinate lines are illustrated in the figure 7-27.
(^9) See chapter 10 for a discussion of the matrix calculus.