def test_functional_differential_geometry_ch3():
# 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, t = symbols('x, y, r, theta, t', real=True)
f = Function('f')
b1 = Function('b1')
b2 = Function('b2')
p_r = R2_r.point([x0, y0])
s_field = ScalarField(R2_r, [x, y], f(x,y))
v_field = VectorField(R2_r, [x, y], [b1(x), b2(y)])
assert v_field(s_field)(p_r) == b1(x0)*Derivative(f(x0, y0), x0) + b2(y0)*Derivative(f(x0, y0), y0)
assert R2.d_dx(R2.r**2)(p_r) == 2*x0
v = R2.d_dx + 2*R2.d_dy
s = R2.r**2 + 3*R2.x
assert v(s)(p_r) == 2*x0 + 4*y0 + 3
circ = -R2.y*R2.d_dx + R2.x*R2.d_dy
series = intcurve_series(circ, t, R2_r.point([1, 0]))
series_x, series_y = zip(*series)
assert all([term == cos(t).taylor_term(i,t) for i, term in enumerate(series_x)])
assert all([term == sin(t).taylor_term(i,t) for i, term in enumerate(series_y)])
