text: """Predefined R^n manifolds together with common coord. systems.
  
  Coordinate systems are predefined as well as the transformation laws between
  them.
  
  Coordinate functions can be accessed as attributes of the manifold (eg `R2.x`),
  as attributes of the coordinate systems (eg `R2_r.x` and `R2_p.theta`), or by
  using the usual `coord_sys.coord_function(index, name)` interface.
  """
  
  from diffgeom import Manifold, Patch, CoordSystem
  from sympy import sqrt, atan2, acos, sin, cos, Dummy
  
  ###############################################################################
  # R2
  ###############################################################################
  R2 = Manifold('R^2', 2)
  # Patch and coordinate systems.
  R2_origin = Patch('origin', R2)
  R2_r = CoordSystem('rectangular', R2_origin, ['x', 'y'])
  R2_p = CoordSystem('polar', R2_origin, ['r', 'theta'])
  
  # Connecting the coordinate charts.
  x, y, r, theta = [Dummy(s) for s in ['x', 'y', 'r', 'theta']]
  R2_r.connect_to(R2_p, [x, y],
                        [sqrt(x**2 + y**2), atan2(y, x)],
                  inverse=False, fill_in_gaps=False)
  R2_p.connect_to(R2_r, [r, theta],
                        [r*cos(theta), r*sin(theta)],
                  inverse=False, fill_in_gaps=False)
  del x, y, r, theta
  
  # Defining the basis coordinate functions and adding shortcuts for them to the
  # manifold and the patch.
  R2.x, R2.y = R2_origin.x, R2_origin.y = R2_r.x, R2_r.y = R2_r.coord_functions()
  R2.r, R2.theta = R2_origin.r, R2_origin.theta = R2_p.r, R2_p.theta = R2_p.coord_functions()
  
  # Defining the basis vector fields and adding shortcuts for them to the
  # manifold and the patch.
  R2.e_x, R2.e_y = R2_origin.e_x, R2_origin.e_y = R2_r.e_x, R2_r.e_y = R2_r.base_vectors()
  R2.e_r, R2.e_theta = R2_origin.e_r, R2_origin.e_theta = R2_p.e_r, R2_p.e_theta = R2_p.base_vectors()
  
  # Defining the basis oneform fields and adding shortcuts for them to the
  # manifold and the patch.
  R2.dx, R2.dy = R2_origin.dx, R2_origin.dy = R2_r.dx, R2_r.dy = R2_r.base_oneforms()
  R2.dr, R2.dtheta = R2_origin.dr, R2_origin.dtheta = R2_p.dr, R2_p.dtheta = R2_p.base_oneforms()
  
  ###############################################################################
  # R3
  ###############################################################################
  R3 = Manifold('R^3', 3)
  # Patch and coordinate systems.
  R3_origin = Patch('origin', R3)
  R3_r = CoordSystem('rectangular', R3_origin, ['x', 'y', 'z'])
  R3_c = CoordSystem('cylindrical', R3_origin, ['rho', 'psi', 'z'])
  R3_s = CoordSystem('spherical', R3_origin, ['r', 'theta', 'phi'])
  
  # Connecting the coordinate charts.
  x, y, z, rho, psi, r, theta, phi = [Dummy(s) for s in ['x', 'y', 'z',
                                            'rho', 'psi', 'r', 'theta', 'phi']]
  ## rectangular <-> cylindrical
  R3_r.connect_to(R3_c, [x, y, z],
                        [sqrt(x**2 + y**2), atan2(y, x), z],
                  inverse=False, fill_in_gaps=False)
  R3_c.connect_to(R3_r, [rho, psi, z],
                        [rho*cos(psi), rho*sin(psi), z],
                  inverse=False, fill_in_gaps=False)
  ## rectangular <-> spherical
  R3_r.connect_to(R3_s, [x, y, z],
                        [sqrt(x**2 + y**2 + z**2), acos(z/
                              sqrt(x**2 + y**2 + z**2)), atan2(y, x)],
                  inverse=False, fill_in_gaps=False)
  R3_s.connect_to(R3_r, [r, theta, phi],
                        [r*sin(theta)*cos(phi), r*sin(
                            theta)*sin(phi), r*cos(theta)],
                  inverse=False, fill_in_gaps=False)
  ## cylindrical <-> spherical
  R3_c.connect_to(R3_s, [rho, psi, z],
                        [sqrt(rho**2 + z**2), acos(z/sqrt(rho**2 + z**2)), psi],
                  inverse=False, fill_in_gaps=False)
  R3_s.connect_to(R3_c, [r, theta, phi],
                        [r*sin(theta), phi, r*cos(theta)],
                  inverse=False, fill_in_gaps=False)
  del x, y, z, rho, psi, r, theta, phi
  
  # Defining the basis coordinate functions.
  R3_r.x, R3_r.y, R3_r.z = R3_r.coord_functions()
  R3_c.rho, R3_c.psi, R3_c.z = R3_c.coord_functions()
  R3_s.r, R3_s.theta, R3_s.phi = R3_s.coord_functions()
  
  # Defining the basis vector fields.
  R3_r.e_x, R3_r.e_y, R3_r.e_z = R3_r.base_vectors()
  R3_c.e_rho, R3_c.e_psi, R3_c.e_z = R3_c.base_vectors()
  R3_s.e_r, R3_s.e_theta, R3_s.e_phi = R3_s.base_vectors()
  
  # Defining the basis oneform fields.
  R3_r.dx, R3_r.dy, R3_r.dz = R3_r.base_oneforms()
  R3_c.drho, R3_c.dpsi, R3_c.dz = R3_c.base_oneforms()
  R3_s.dr, R3_s.dtheta, R3_s.dphi = R3_s.base_oneforms()
  