def test_functional_differential_geometry_ch2():
# From "Functional Differential Geometry" as of 2011
# by Sussman and Wisdom.
x0, y0, r0, theta0 = symbols('x0, y0, r0, theta0', real=True)
x, y, r, theta = symbols('x, y, r, theta', 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)]))
#TODO jacobian page 12 - 32
field = ScalarField(R2_r, [x, y], f(x, 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(p1_in_rect) == f(x0, y0)
# TODO better simplification for the next one
#print simplify(field(p1_in_polar))
#assert simplify(field(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(p_r) == x0*(x0**2 + y0**2) + y0**3
assert h(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 simplify( (R2.r**2 - R2.x**2 - R2.y**2)(point_p) ) == 0
assert simplify( R2.d_dr(R2.x**2+R2.y**2)(point_p) ) == 2*r0