Example 8-18. Express the vector
A= 2yˆe 1 +zˆe 2 + 2xˆe 3
in cylindrical coordinates.
Solution The rectangular components of A are A 1 = 2y, A 2 =z, A 3 = 2 x, and from
equation (8.86) the cylindrical components are
Ar= 2ycos θ+zsin θ
Aθ=− 2 ysin θ+zcos θ
Az= 2x,
where the variables x, y, z must be expressed in terms of the variables r, θ, z. From the
transformation equations from rectangular to cylindrical coordinates one finds
x=rcos θ, y =rsin θ, z =z
so that
Ar= 2rsin θcos θ+zsin θ
Aθ=− 2 rsin^2 θ+zcos θ
Az= 2rcos θ
and the vector A in cylindrical coordinates can be represented as
A=A(r, θ, z) = Arˆer+Aθˆeθ+Azˆez.
General Coordinate Transformations
In general, a vector in rectangular coordinates
A=A 1 ˆe 1 +A 2 ˆe 2 +A 3 ˆe 3
can be expressed in terms of the orthogonal unit vectors ˆeu, ˆev,ˆew associated with
a set of orthogonal curvilinear coordinates defined by the transformation equations
given in equation (8.68). Let the representation of this vector in the orthogonal
curvilinear coordinates system be denoted by
A=A(u, v, w ) = Auˆeu+Avˆev+Aweˆw,
