The distributions of the axial symmetry magnetic vector potential and the magnetic induction lines are studied by using Mathematica 10.3. The calculation formula of the magnetic vector potential A of the current carrying circular coil is expressed as the elliptic integral，the magnetic induction intensity B="×A and the magnetic induction line equation are deduced，the distribution maps of the magnetic vector potential and the magnetic induction lines are drawn， some defects of the distribution maps of the magnetic induction lines in the current literature are corrected. Then，the present method is applied to the current carrying tightly wound finite-length cylindrical solenoid and ellipsoidal coil.