The cylindrical coordinate formulation of the Biot-Savart law is derived by coordinate transformation between Cartesian and cylindrical systems. The derived formulation is applied to calculate the magnetic field distributions generated by infinitely long current-carrying solenoids and cylindrical conductors. The results demonstrate that the cylindrical coordinate representation of the Biot-Savart law significantly simplifies magnetic field calculations for current-carrying conductors with axial symmetry. Furthermore, the magnetic field distribution along the central axis of a current-carrying elliptic coil is also calculated. It is specifically noted that the coordinate scaling method proposed in some literature for calculating the magnetic field at arbitrary spatial points around such coils is unfeasible.