Transform vector B to cylindrical

Given: B = y a_x − x a_y + z a_z.

Transformation formulas

B_ρ = B_x cosφ + B_y sinφ
B_φ = −B_x sinφ + B_y cosφ
B_z = B_z

Here B_x = y, B_y = −x, B_z = z. Substitute x = ρ cosφ and y = ρ sinφ:

B_ρ = y cosφ − x sinφ = ρ sinφ cosφ − ρ cosφ sinφ = 0
B_φ = −y sinφ − x cosφ = −ρ sin²φ − ρ cos²φ = −ρ
B_z = z

Answer: B = −ρ a_φ + z a_z

Check: |B|² in Cartesian = y² + x² + z² = ρ² + z², and in cylindrical = ρ² + z² ✓. The field circulates around the z-axis in the −φ (clockwise) sense, with magnitude proportional to ρ.