Here the quantities A 1 , A 2 , A 3 represent the components of the vector field Ain rect-
angular coordinates, and Ar, A θ, A z represent the components of the same vector
field A when referenced with respect to cylindrical coordinates. The unit vectors
ˆer,ˆeθ,eˆz are orthogonal unit vectors and hence
A·ˆer=A 1 ˆe 1 ·ˆer+A 2 ˆe 2 ·ˆer+A 3 ˆe 3 ·ˆer=Ar
=Component of A in the ˆerdirection
A·ˆeθ=A 1 ˆe 1 ·ˆeθ+A 2 ˆe 2 ·ˆeθ+A 3 ˆe 3 ·ˆeθ=Aθ
=Component of A in the ˆeθdirection
A·eˆz=A 1 eˆ 1 ·ˆez+A 2 eˆ 2 ·ˆez+A 3 ˆe 3 ·eˆz=Az
=Component of A in the ˆezdirection.
These equations can be expressed in the matrix form as follows:
Ar
Aθ
Az
=
eˆ 1 ·ˆer ˆe 2 ·ˆer eˆ 3 ·ˆer
eˆ 1 ·ˆeθ ˆe 2 ·ˆeθ eˆ 3 ·ˆeθ
eˆ 1 ·ˆez ˆe 2 ·ˆez eˆ 3 ·ˆez
A 1
A 2
A 3
. (8 .85)
For example, it is known that the unit vectors in cylindrical coordinates are
ˆer= cos θˆe 1 + sin θˆe 2
ˆeθ=−sin θˆe 1 + cos θˆe 2
eˆz=ˆe 3 ,
and consequently the matrix (8.85) can be expressed as
Ar
Aθ
Az
=
cos θ sin θ 0
−sin θ cos θ 0
0 0 1
A 1
A 2
A 3
. (8 .86)
Equation (8.86) illustrates how to represent the vector field components Ar, A θ, A z
of cylindrical coordinates, in terms of the components A 1 , A 2 , A 3 of rectangular coor-
dinates. In using the above transformation equation, be sure to convert all x, y, z co-
ordinates to r, θ, z cylindrical coordinates using the transformation equations (8.83).
Note also that the coefficient matrix in equation (8.83) is an orthonormal matrix.
