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_intcurve_diffequ():
t = symbols('t')
start_point = R2_r.point([1, 0])
vector_field = -R2.y*R2.d_dx + R2.x*R2.d_dy
equations, init_cond = intcurve_diffequ(vector_field, t, start_point)
assert str(equations) == '[f_1(t) + Derivative(f_0(t), t), -f_0(t) + Derivative(f_1(t), t)]'
assert str(init_cond) == '[f_0(0) - 1, f_1(0)]'
equations, init_cond = intcurve_diffequ(vector_field, t, start_point, R2_p)
#TODO correct but too complicated: equations
assert str(init_cond) == '[f_0(0) - 1, f_1(0)]'
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
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)])