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Express a_x field in cylindrical, spherical

Short answer

In cylindrical: F=5 cosφ a_ρ −5 sinφ a_φ. In spherical: F=5 sinθ cosφ a_r +5 cosθ cosφ a_θ −5 sinφ a_φ.

Express the uniform vector field F = 5a_x in (a) cylindrical components (b) spherical components.

Given: F = 5 a_x (uniform field).

(a) Cylindrical components

Use the projections of F on the cylindrical unit vectors.

F_ρ = F · a_ρ = 5 a_x · a_ρ = 5 cosφ
F_φ = F · a_φ = 5 a_x · a_φ = −5 sinφ
F_z = F · a_z = 0

F = 5 cosφ a_ρ − 5 sinφ a_φ

(b) Spherical components

F_r = F · a_r = 5 sinθ cosφ
F_θ = F · a_θ = 5 cosθ cosφ
F_φ = F · a_φ = −5 sinφ

F = 5 sinθ cosφ a_r + 5 cosθ cosφ a_θ − 5 sinφ a_φ

(The dot products used: a_x · a_ρ = cosφ, a_x · a_φ = −sinφ; a_x · a_r = sinθ cosφ, a_x · a_θ = cosθ cosφ.)

Check: the magnitude must stay 5. Cylindrical: 25cos²φ + 25sin²φ = 25 ✓. Spherical: 25[sin²θ cos²φ + cos²θ cos²φ + sin²φ] = 25[cos²φ + sin²φ] = 25 ✓.

✓ Answer checked by Claude Opus 5.5 (high effort)

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