Mathematical Physics Vol 1

4.6 Examples

193

From here it is obvious that the unit vectors are normal to each other, namely, that the system is orthogonal.

Exercise 145 Express the vector A = z i − 2 x j + y k in terms of cylindrical coordinates. Determine A ρ , A ϕ and A z (projections of this vector on axes ρ , ϕ and z ).

Solution Vector A is expressed relative to the base i , j i k . In order to represent it in terms of cylindrical coordinates it is necessary to express the unit vectors i , j and k in terms of the base e ρ , e ϕ and e z . These relations are given in the previous example (4.218): e ρ = cos ϕ i + sin ϕ j , e ϕ = − sin ϕ i + cos ϕ j , e z = k ,

which yields

i = cos ϕ e ρ − sin ϕ e ϕ , j = sin ϕ e ρ + cos ϕ e ϕ ,

(4.219)

k = e z .

According to the definition

A ρ = A · e ρ , A ϕ = A · e ϕ ,

(4.220)

A z = A · e z . By substituting (4.219) in the expression for vector A , bearing in mind (4.220), we obtain A = z i − 2 x j + y k = z ( cos ϕ e ρ − sin ϕ e ϕ ) − 2 ρ cos ϕ ( sin ϕ e ρ + cos ϕ e ϕ )+ ρ sin ϕ e z =( z cos ϕ − 2 ρ cos ϕ sin ϕ ) e ρ − ( z sin ϕ + 2 ρ cos 2 ϕ ) e ϕ + ρ sin ϕ e z , that is A ρ = z cos ϕ − 2 ρ cos ϕ sin ϕ , A ϕ = z sin ϕ + 2 ρ cos 2 ϕ , A z = ρ sin ϕ .