def test_products():
assert TensorProduct(R2.dx, R2.dy)(
R2.e_x, R2.e_y) == R2.dx(R2.e_x) * R2.dy(R2.e_y) == 1
assert TensorProduct(R2.dx, R2.dy)(None, R2.e_y) == R2.dx
assert TensorProduct(R2.dx, R2.dy)(R2.e_x, None) == R2.dy
assert TensorProduct(R2.dx, R2.dy)(R2.e_x) == R2.dy
assert TensorProduct(R2.x, R2.dx) == R2.x * R2.dx
assert TensorProduct(R2.e_x, R2.e_y)(
R2.x, R2.y) == R2.e_x(R2.x) * R2.e_y(R2.y) == 1
assert TensorProduct(R2.e_x, R2.e_y)(None, R2.y) == R2.e_x
assert TensorProduct(R2.e_x, R2.e_y)(R2.x, None) == R2.e_y
assert TensorProduct(R2.e_x, R2.e_y)(R2.x) == R2.e_y
assert TensorProduct(R2.x, R2.e_x) == R2.x * R2.e_x
assert TensorProduct(R2.dx, R2.e_y)(
R2.e_x, R2.y) == R2.dx(R2.e_x) * R2.e_y(R2.y) == 1
assert TensorProduct(R2.dx, R2.e_y)(None, R2.y) == R2.dx
assert TensorProduct(R2.dx, R2.e_y)(R2.e_x, None) == R2.e_y
assert TensorProduct(R2.dx, R2.e_y)(R2.e_x) == R2.e_y
assert TensorProduct(R2.x, R2.e_x) == R2.x * R2.e_x
assert TensorProduct(R2.e_x, R2.dy)(
R2.x, R2.e_y) == R2.e_x(R2.x) * R2.dy(R2.e_y) == 1
assert TensorProduct(R2.e_x, R2.dy)(None, R2.e_y) == R2.e_x
assert TensorProduct(R2.e_x, R2.dy)(R2.x, None) == R2.dy
assert TensorProduct(R2.e_x, R2.dy)(R2.x) == R2.dy
assert TensorProduct(R2.e_y, R2.e_x)(R2.x**2 + R2.y**2,
R2.x**2 + R2.y**2) == 4 * R2.x * R2.y
assert WedgeProduct(R2.dx, R2.dy)(R2.e_x, R2.e_y) == 1
assert WedgeProduct(R2.e_x, R2.e_y)(R2.x, R2.y) == 1
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_products():
assert TensorProduct(
R2.dx, R2.dy)(R2.e_x, R2.e_y) == R2.dx(R2.e_x)*R2.dy(R2.e_y) == 1
assert TensorProduct(R2.dx, R2.dy)(None, R2.e_y) == R2.dx
assert TensorProduct(R2.dx, R2.dy)(R2.e_x, None) == R2.dy
assert TensorProduct(R2.dx, R2.dy)(R2.e_x) == R2.dy
assert TensorProduct(R2.x, R2.dx) == R2.x*R2.dx
assert TensorProduct(
R2.e_x, R2.e_y)(R2.x, R2.y) == R2.e_x(R2.x) * R2.e_y(R2.y) == 1
assert TensorProduct(R2.e_x, R2.e_y)(None, R2.y) == R2.e_x
assert TensorProduct(R2.e_x, R2.e_y)(R2.x, None) == R2.e_y
assert TensorProduct(R2.e_x, R2.e_y)(R2.x) == R2.e_y
assert TensorProduct(R2.x, R2.e_x) == R2.x * R2.e_x
assert TensorProduct(
R2.dx, R2.e_y)(R2.e_x, R2.y) == R2.dx(R2.e_x) * R2.e_y(R2.y) == 1
assert TensorProduct(R2.dx, R2.e_y)(None, R2.y) == R2.dx
assert TensorProduct(R2.dx, R2.e_y)(R2.e_x, None) == R2.e_y
assert TensorProduct(R2.dx, R2.e_y)(R2.e_x) == R2.e_y
assert TensorProduct(R2.x, R2.e_x) == R2.x * R2.e_x
assert TensorProduct(
R2.e_x, R2.dy)(R2.x, R2.e_y) == R2.e_x(R2.x) * R2.dy(R2.e_y) == 1
assert TensorProduct(R2.e_x, R2.dy)(None, R2.e_y) == R2.e_x
assert TensorProduct(R2.e_x, R2.dy)(R2.x, None) == R2.dy
assert TensorProduct(R2.e_x, R2.dy)(R2.x) == R2.dy
assert TensorProduct(R2.e_y,R2.e_x)(R2.x**2 + R2.y**2,R2.x**2 + R2.y**2) == 4*R2.x*R2.y
assert WedgeProduct(R2.dx, R2.dy)(R2.e_x, R2.e_y) == 1
assert WedgeProduct(R2.e_x, R2.e_y)(R2.x, R2.y) == 1
def test_correct_arguments():
raises(ValueError, lambda: R2.e_x(R2.e_x))
raises(ValueError, lambda: R2.e_x(R2.dx))
raises(ValueError, lambda: Commutator(R2.e_x, R2.x))
raises(ValueError, lambda: Commutator(R2.dx, R2.e_x))
raises(ValueError, lambda: Differential(Differential(R2.e_x)))
raises(ValueError, lambda: R2.dx(R2.x))
raises(ValueError, lambda: LieDerivative(R2.dx, R2.dx))
raises(ValueError, lambda: LieDerivative(R2.x, R2.dx))
raises(ValueError, lambda: CovarDerivativeOp(R2.dx, []))
raises(ValueError, lambda: CovarDerivativeOp(R2.x, []))
a = Symbol('a')
raises(ValueError, lambda: intcurve_series(R2.dx, a, R2_r.point([1, 2])))
raises(ValueError, lambda: intcurve_series(R2.x, a, R2_r.point([1, 2])))
raises(ValueError, lambda: intcurve_diffequ(R2.dx, a, R2_r.point([1, 2])))
raises(ValueError, lambda: intcurve_diffequ(R2.x, a, R2_r.point([1, 2])))
raises(ValueError, lambda: contravariant_order(R2.e_x + R2.dx))
raises(ValueError, lambda: covariant_order(R2.e_x + R2.dx))
raises(ValueError, lambda: contravariant_order(R2.e_x * R2.e_y))
raises(ValueError, lambda: covariant_order(R2.dx * R2.dy))
def test_products():
assert TensorProduct(R2.dx, R2.dy)(R2.e_x, R2.e_y) == R2.dx(R2.e_x)*R2.dy(R2.e_y) == 1
assert WedgeProduct(R2.dx, R2.dy)(R2.e_x, R2.e_y) == 1
assert TensorProduct(R2.dx, R2.dy)(None, R2.e_y) == R2.dx
assert TensorProduct(R2.dx, R2.dy)(R2.e_x, None) == R2.dy
assert TensorProduct(R2.dx, R2.dy)(R2.e_x) == R2.dy
assert TensorProduct(R2.x, R2.dx) == R2.x*R2.dx
def test_correct_arguments():
raises(ValueError, lambda : R2.e_x(R2.e_x))
raises(ValueError, lambda : R2.e_x(R2.dx))
raises(ValueError, lambda : Commutator(R2.e_x, R2.x))
raises(ValueError, lambda : Commutator(R2.dx, R2.e_x))
raises(ValueError, lambda : Differential(Differential(R2.e_x)))
raises(ValueError, lambda : R2.dx(R2.x))
raises(ValueError, lambda : TensorProduct(R2.e_x, R2.dx))
raises(ValueError, lambda : LieDerivative(R2.dx, R2.dx))
raises(ValueError, lambda : LieDerivative(R2.x, R2.dx))
raises(ValueError, lambda : CovarDerivativeOp(R2.dx, []))
raises(ValueError, lambda : CovarDerivativeOp(R2.x, []))
a = Symbol('a')
raises(ValueError, lambda : intcurve_series(R2.dx, a, R2_r.point([1,2])))
raises(ValueError, lambda : intcurve_series(R2.x, a, R2_r.point([1,2])))
raises(ValueError, lambda : intcurve_diffequ(R2.dx, a, R2_r.point([1,2])))
raises(ValueError, lambda : intcurve_diffequ(R2.x, a, R2_r.point([1,2])))
raises(ValueError, lambda : contravariant_order(R2.e_x + R2.dx))
raises(ValueError, lambda : covariant_order(R2.e_x + R2.dx))
raises(ValueError, lambda : contravariant_order(R2.e_x*R2.e_y))
raises(ValueError, lambda : covariant_order(R2.dx*R2.dy))
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
