Transform vector to cylindrical coordinates

Given: A = y a_x + x a_y + z a_z at P(ρ = 2, φ = 45°, z = 5). Find A_ρ, A_φ, A_z.

Method

Cartesian to cylindrical transformation:

A_ρ = A_x cosφ + A_y sinφ
A_φ = −A_x sinφ + A_y cosφ
A_z = A_z

Here A_x = y = ρ sinφ, A_y = x = ρ cosφ, A_z = z.

Components in terms of ρ, φ, z

A_ρ = ρ sinφ cosφ + ρ cosφ sinφ = 2ρ sinφ cosφ = ρ sin2φ
A_φ = −ρ sin²φ + ρ cos²φ = ρ cos2φ
A_z = z

A = ρ sin2φ a_ρ + ρ cos2φ a_φ + z a_z

Value at P (ρ = 2, φ = 45°, z = 5)

A_ρ = 2 sin90° = 2
A_φ = 2 cos90° = 0
A_z = 5

(Check with Cartesian values: x = y = 2 cos45° = 1.414; A = 1.414 a_x + 1.414 a_y + 5 a_z. Then A_ρ = 1.414(0.7071) + 1.414(0.7071) = 2, A_φ = −1.414(0.7071) + 1.414(0.7071) = 0. ✓)

Answer: A = 2 a_ρ + 5 a_z at P