def test_functional_diffgeom_ch2():
x0, y0, r0, theta0 = symbols('x0, y0, r0, theta0', real=True)
x, y = symbols('x, y', real=True)
f = Function('f')
assert (R2_p.point_to_coords(R2_r.point([x0, y0])) ==
Matrix([sqrt(x0**2 + y0**2), atan2(y0, x0)]))
assert (R2_r.point_to_coords(R2_p.point([r0, theta0])) ==
Matrix([r0*cos(theta0), r0*sin(theta0)]))
assert R2_p.jacobian(R2_r, [r0, theta0]) == Matrix(
[[cos(theta0), -r0*sin(theta0)], [sin(theta0), r0*cos(theta0)]])
field = f(R2.x, R2.y)
p1_in_rect = R2_r.point([x0, y0])
p1_in_polar = R2_p.point([sqrt(x0**2 + y0**2), atan2(y0, x0)])
assert field.rcall(p1_in_rect) == f(x0, y0)
assert field.rcall(p1_in_polar) == f(x0, y0)
p_r = R2_r.point([x0, y0])
p_p = R2_p.point([r0, theta0])
assert R2.x(p_r) == x0
assert R2.x(p_p) == r0*cos(theta0)
assert R2.r(p_p) == r0
assert R2.r(p_r) == sqrt(x0**2 + y0**2)
assert R2.theta(p_r) == atan2(y0, x0)
h = R2.x*R2.r**2 + R2.y**3
assert h.rcall(p_r) == x0*(x0**2 + y0**2) + y0**3
assert h.rcall(p_p) == r0**3*sin(theta0)**3 + r0**3*cos(theta0)
def test_functional_diffgeom_ch2():
x0, y0, r0, theta0 = symbols('x0, y0, r0, theta0', real=True)
x, y = symbols('x, y', real=True)
f = Function('f')
assert (R2_p.point_to_coords(R2_r.point([x0, y0])) ==
Matrix([sqrt(x0**2 + y0**2), atan2(y0, x0)]))
assert (R2_r.point_to_coords(R2_p.point([r0, theta0])) ==
Matrix([r0*cos(theta0), r0*sin(theta0)]))
assert R2_p.jacobian(R2_r, [r0, theta0]) == Matrix(
[[cos(theta0), -r0*sin(theta0)], [sin(theta0), r0*cos(theta0)]])
field = f(R2.x, R2.y)
p1_in_rect = R2_r.point([x0, y0])
p1_in_polar = R2_p.point([sqrt(x0**2 + y0**2), atan2(y0, x0)])
assert field.rcall(p1_in_rect) == f(x0, y0)
assert field.rcall(p1_in_polar) == f(x0, y0)
p_r = R2_r.point([x0, y0])
p_p = R2_p.point([r0, theta0])
assert R2.x(p_r) == x0
assert R2.x(p_p) == r0*cos(theta0)
assert R2.r(p_p) == r0
assert R2.r(p_r) == sqrt(x0**2 + y0**2)
assert R2.theta(p_r) == atan2(y0, x0)
h = R2.x*R2.r**2 + R2.y**3
assert h.rcall(p_r) == x0*(x0**2 + y0**2) + y0**3
assert h.rcall(p_p) == r0**3*sin(theta0)**3 + r0**3*cos(theta0)
def test_R2():
x0, y0, r0, theta0 = symbols('x0, y0, r0, theta0', real=True)
point_r = R2_r.point([x0, y0])
point_p = R2_p.point([r0, theta0])
# r**2 = x**2 + y**2
assert (R2.r**2 - R2.x**2 - R2.y**2)(point_r) == 0
assert trigsimp( (R2.r**2 - R2.x**2 - R2.y**2)(point_p) ) == 0
assert trigsimp(R2.e_r(R2.x**2+R2.y**2)(point_p).doit()) == 2*r0
# polar->rect->polar == Id
a, b = symbols('a b', positive=True)
m = Matrix([[a], [b]])
#TODO assert m == R2_r.coord_tuple_transform_to(R2_p, R2_p.coord_tuple_transform_to(R2_r, [a, b])).applyfunc(simplify)
assert m == R2_p.coord_tuple_transform_to(R2_r, R2_r.coord_tuple_transform_to(R2_p, m)).applyfunc(simplify)
def test_functional_diffgeom_ch4():
x0, y0, theta0 = symbols('x0, y0, theta0', real=True)
x, y, r, theta = symbols('x, y, r, theta', real=True)
r0 = symbols('r0', positive=True)
f = Function('f')
b1 = Function('b1')
b2 = Function('b2')
p_r = R2_r.point([x0, y0])
p_p = R2_p.point([r0, theta0])
f_field = b1(R2.x,R2.y)*R2.dx + b2(R2.x,R2.y)*R2.dy
assert f_field(R2.e_x)(p_r) == b1(x0, y0)
assert f_field(R2.e_y)(p_r) == b2(x0, y0)
s_field_r = f(R2.x,R2.y)
df = Differential(s_field_r)
assert df(R2.e_x)(p_r).doit() == Derivative(f(x0, y0), x0)
assert df(R2.e_y)(p_r).doit() == Derivative(f(x0, y0), y0)
s_field_p = f(R2.r,R2.theta)
df = Differential(s_field_p)
assert trigsimp(df(R2.e_x)(p_p).doit()) == cos(theta0)*Derivative(f(r0, theta0), r0) - sin(theta0)*Derivative(f(r0, theta0), theta0)/r0
assert trigsimp(df(R2.e_y)(p_p).doit()) == sin(theta0)*Derivative(f(r0, theta0), r0) + cos(theta0)*Derivative(f(r0, theta0), theta0)/r0
assert R2.dx(R2.e_x)(p_r) == 1
assert R2.dx(R2.e_y)(p_r) == 0
circ = -R2.y*R2.e_x + R2.x*R2.e_y
assert R2.dx(circ)(p_r).doit() == -y0
assert R2.dy(circ)(p_r) == x0
assert R2.dr(circ)(p_r) == 0
assert simplify(R2.dtheta(circ)(p_r)) == 1
assert (circ - R2.e_theta)(s_field_r)(p_r) == 0
def test_functional_diffgeom_ch4():
x0, y0, theta0 = symbols('x0, y0, theta0', real=True)
x, y, r, theta = symbols('x, y, r, theta', real=True)
r0 = symbols('r0', positive=True)
f = Function('f')
b1 = Function('b1')
b2 = Function('b2')
p_r = R2_r.point([x0, y0])
p_p = R2_p.point([r0, theta0])
f_field = b1(R2.x, R2.y)*R2.dx + b2(R2.x, R2.y)*R2.dy
assert f_field.rcall(R2.e_x).rcall(p_r) == b1(x0, y0)
assert f_field.rcall(R2.e_y).rcall(p_r) == b2(x0, y0)
s_field_r = f(R2.x, R2.y)
df = Differential(s_field_r)
assert df(R2.e_x).rcall(p_r).doit() == Derivative(f(x0, y0), x0)
assert df(R2.e_y).rcall(p_r).doit() == Derivative(f(x0, y0), y0)
s_field_p = f(R2.r, R2.theta)
df = Differential(s_field_p)
assert trigsimp(df(R2.e_x).rcall(p_p).doit()) == (
cos(theta0)*Derivative(f(r0, theta0), r0) -
sin(theta0)*Derivative(f(r0, theta0), theta0)/r0)
assert trigsimp(df(R2.e_y).rcall(p_p).doit()) == (
sin(theta0)*Derivative(f(r0, theta0), r0) +
cos(theta0)*Derivative(f(r0, theta0), theta0)/r0)
assert R2.dx(R2.e_x).rcall(p_r) == 1
assert R2.dx(R2.e_x) == 1
assert R2.dx(R2.e_y).rcall(p_r) == 0
assert R2.dx(R2.e_y) == 0
circ = -R2.y*R2.e_x + R2.x*R2.e_y
assert R2.dx(circ).rcall(p_r).doit() == -y0
assert R2.dy(circ).rcall(p_r) == x0
assert R2.dr(circ).rcall(p_r) == 0
assert simplify(R2.dtheta(circ).rcall(p_r)) == 1
assert (circ - R2.e_theta).rcall(s_field_r).rcall(p_r) == 0
p = Patch('P', m)
rect = CoordSystem('rect', p)
polar = CoordSystem('polar', p)
polar.connect_to(rect, [r, theta], [r*cos(theta), r*sin(theta)])
# Define a point using coordinates from one of the coordinate systems:
p = Point(polar, [r, 3*pi/4])
p.coords()
p.coords(rect)
# Use the predefined R2 manifold, setup some boilerplate.
from sympy import symbols, pi, Function
from sympy.diffgeom.rn import R2, R2_p, R2_r
from sympy.diffgeom import BaseVectorField
from sympy import pprint
x0, y0, r0, theta0 = symbols('x0, y0, r0, theta0')
# Points to be used as arguments for the field:
point_p = R2_p.point([r0, theta0])
point_r = R2_r.point([x0, y0])
#Scalar field to operate on:
g = Function('g')
s_field = g(R2.x, R2.y)
s_field.rcall(point_r)
s_field.rcall(point_p)
#Vector field:
v = BaseVectorField(R2_r, 1)
pprint(v(s_field))
pprint(v(s_field).rcall(point_r).doit())
pprint(v(s_field).rcall(point_p).doit())
# Differential Examples
# scalar 0 forms
from sympy import Function
from sympy.diffgeom.rn import R2