rect = CoordSystem('rect', patch)
polar = CoordSystem('polar', patch)
print(rect in patch.coord_systems)
polar.connect_to(rect, [r, theta], [r*cos(theta), r*sin(theta)])
print(polar.coord_tuple_transform_to(rect, [0, 2]))
print(polar.coord_tuple_transform_to(rect, [2, pi/2]))
print(rect.coord_tuple_transform_to(polar, [1, 1]).applyfunc(simplify))
print(polar.jacobian(rect, [r, theta]))
p = polar.point([1, 3*pi/4])
print(rect.point_to_coords(p))
print(rect.coord_function(0)(p))
print(rect.coord_function(1)(p))
v_x = rect.base_vector(0)
x = rect.coord_function(0)
print(v_x(x))
print(v_x(v_x(x)))
v_r = polar.base_vector(0)
print(v_r(x))
# ^ ????
dx = rect.base_oneform(0)
print(dx(v_x))
# ????
from sympy import symbols, sin, cos, pi
from sympy.diffgeom import Manifold, Patch, CoordSystem
r, theta = symbols('r, theta')
m = Manifold('M', 2)
patch = Patch('P', m)
rect = CoordSystem('rect', patch)
polar = CoordSystem('polar', patch)
rect in patch.coord_systems
polar.connect_to(rect, [r, theta], [r*cos(theta), r*sin(theta)])
polar.coord_tuple_transform_to(rect, [0, 2])
polar.coord_tuple_transform_to(rect, [2, pi/2])
rect.coord_tuple_transform_to(polar, [1, 1])
polar.jacobian(rect, [r, theta])
p = polar.point([1, 3*pi/4])
rect.point_to_coords(p)
#Define a basis scalar field (i.e. a coordinate function), that takes a point and returns its coordinates. It is an instance of BaseScalarField.
rect.coord_function(0)(p)
rect.coord_function(1)(p)
#Define a basis vector field (i.e. a unit vector field along the coordinate line). Vectors are also differential operators on scalar fields. It is an instance of BaseVectorField.
v_x = rect.base_vector(0)
x = rect.coord_function(0)
v_x(x)
v_x(v_x(x))
#Define a basis oneform field:
dx = rect.base_oneform(0)
dx(v_x)
#If you provide a list of names the fields will print nicely: - without provided names:
x, v_x, dx
#with provided names
rect = CoordSystem('rect', patch, ['x', 'y'])
rect.coord_function(0), rect.base_vector(0), rect.base_oneform(0)
