def test_deprecations(self):
base = Ga('a b', g=[1, 1], coords=symbols('x, y', real=True))
l = base.lt([[1, 2], [3, 4]])
with pytest.warns(DeprecationWarning):
assert l.X == l.Ga.coord_vec
with pytest.warns(DeprecationWarning):
assert l.coords == l.Ga.coords
l = base.lt('L', mode='a')
with pytest.warns(DeprecationWarning):
assert l.mode == 'a'
with pytest.warns(DeprecationWarning):
assert not l.fct_flg
l = base.lt('L', mode='s', f=True)
with pytest.warns(DeprecationWarning):
assert l.mode == 's'
with pytest.warns(DeprecationWarning):
assert l.fct_flg
def test_lt_function(self):
""" Test construction from a function """
base = Ga('a b', g=[1, 1], coords=symbols('x, y', real=True))
a, b = base.mv()
def not_linear(x):
return x * x
with pytest.raises(ValueError, match='linear'):
base.lt(not_linear)
def not_vector(x):
return x + S.One
with pytest.raises(ValueError, match='vector'):
base.lt(not_vector)
def ok(x):
return (x | b) * a + 2*x
f = base.lt(ok)
x = base.mv('x', 'vector')
y = base.mv('y', 'vector')
assert f(x) == ok(x)
assert f(x^y) == ok(x)^ok(y)
assert f(1 + 2*(x^y)) == 1 + 2*(ok(x)^ok(y))
def main():
# Print_Function()
(x, y, z) = xyz = symbols('x,y,z',real=True)
(o3d, ex, ey, ez) = Ga.build('e_x e_y e_z', g=[1, 1, 1], coords=xyz)
grad = o3d.grad
(u, v) = uv = symbols('u,v',real=True)
(g2d, eu, ev) = Ga.build('e_u e_v', coords=uv)
grad_uv = g2d.grad
v_xyz = o3d.mv('v','vector')
A_xyz = o3d.mv('A','vector',f=True)
A_uv = g2d.mv('A','vector',f=True)
print('#3d orthogonal ($A$ is vector function)')
print('A =', A_xyz)
print('%A^{2} =', A_xyz * A_xyz)
print('grad|A =', grad | A_xyz)
print('grad*A =', grad * A_xyz)
print('v|(grad*A) =',v_xyz|(grad*A_xyz))
print('#2d general ($A$ is vector function)')
print('A =', A_uv)
print('%A^{2} =', A_uv * A_uv)
print('grad|A =', grad_uv | A_uv)
print('grad*A =', grad_uv * A_uv)
A = o3d.lt('A')
print('#3d orthogonal ($A,\\;B$ are linear transformations)')
print('A =', A)
print(r'\f{mat}{A} =', A.matrix())
print('\\f{\\det}{A} =', A.det())
print('\\overline{A} =', A.adj())
print('\\f{\\Tr}{A} =', A.tr())
print('\\f{A}{e_x^e_y} =', A(ex^ey))
print('\\f{A}{e_x}^\\f{A}{e_y} =', A(ex)^A(ey))
B = o3d.lt('B')
print('g =', o3d.g)
print('%g^{-1} =', latex(o3d.g_inv))
print('A + B =', A + B)
print('AB =', A * B)
print('A - B =', A - B)
print('General Symmetric Linear Transformation')
Asym = o3d.lt('A',mode='s')
print('A =', Asym)
print('General Antisymmetric Linear Transformation')
Aasym = o3d.lt('A',mode='a')
print('A =', Aasym)
print('#2d general ($A,\\;B$ are linear transformations)')
A2d = g2d.lt('A')
print('g =', g2d.g)
print('%g^{-1} =', latex(g2d.g_inv))
print('%gg^{-1} =', latex(simplify(g2d.g * g2d.g_inv)))
print('A =', A2d)
print(r'\f{mat}{A} =', A2d.matrix())
print('\\f{\\det}{A} =', A2d.det())
A2d_adj = A2d.adj()
print('\\overline{A} =', A2d_adj)
print('\\f{mat}{\\overline{A}} =', latex(simplify(A2d_adj.matrix())))
print('\\f{\\Tr}{A} =', A2d.tr())
print('\\f{A}{e_u^e_v} =', A2d(eu^ev))
print('\\f{A}{e_u}^\\f{A}{e_v} =', A2d(eu)^A2d(ev))
B2d = g2d.lt('B')
print('B =', B2d)
print('A + B =', A2d + B2d)
print('A - B =', A2d - B2d)
# TODO: add this back when we drop Sympy 1.3. The 64kB of output is far too
# printer-dependent
if False:
print('AB =', A2d * B2d)
a = g2d.mv('a','vector')
b = g2d.mv('b','vector')
print(r'a|\f{\overline{A}}{b}-b|\f{\underline{A}}{a} =',((a|A2d.adj()(b))-(b|A2d(a))).simplify())
m4d = Ga('e_t e_x e_y e_z', g=[1, -1, -1, -1],coords=symbols('t,x,y,z',real=True))
T = m4d.lt('T')
print('g =', m4d.g)
print(r'\underline{T} =',T)
print(r'\overline{T} =',T.adj())
print(r'\f{\det}{\underline{T}} =',T.det())
print(r'\f{\mbox{tr}}{\underline{T}} =',T.tr())
a = m4d.mv('a','vector')
b = m4d.mv('b','vector')
print(r'a|\f{\overline{T}}{b}-b|\f{\underline{T}}{a} =',((a|T.adj()(b))-(b|T(a))).simplify())
coords = (r, th, phi) = symbols('r,theta,phi', real=True)
(sp3d, er, eth, ephi) = Ga.build('e_r e_th e_ph', g=[1, r**2, r**2*sin(th)**2], coords=coords)
grad = sp3d.grad
sm_coords = (u, v) = symbols('u,v', real=True)
smap = [1, u, v] # Coordinate map for sphere of r = 1
sph2d = sp3d.sm(smap,sm_coords,norm=True)
(eu, ev) = sph2d.mv()
grad_uv = sph2d.grad
F = sph2d.mv('F','vector',f=True)
f = sph2d.mv('f','scalar',f=True)
print('f =',f)
print('grad*f =',grad_uv * f)
print('F =',F)
print('grad*F =',grad_uv * F)
tp = (th,phi) = symbols('theta,phi',real=True)
smap = [sin(th)*cos(phi),sin(th)*sin(phi),cos(th)]
sph2dr = o3d.sm(smap,tp,norm=True)
(eth, ephi) = sph2dr.mv()
grad_tp = sph2dr.grad
F = sph2dr.mv('F','vector',f=True)
f = sph2dr.mv('f','scalar',f=True)
print('f =',f)
print('grad*f =',grad_tp * f)
print('F =',F)
print('grad*F =',grad_tp * F)
return
def test_lt_matrix(self):
base = Ga('a b', g=[1, 1], coords=symbols('x, y', real=True))
a, b = base.mv()
A = base.lt([a+b, 2*a-b])
assert str(A) == 'Lt(a) = a + b\nLt(b) = 2*a - b'
assert str(A.matrix()) == 'Matrix([[1, 2], [1, -1]])'
# A2d.adj().Fmt(4,'\\overline{A}')
print('\\f{\\Tr}{A} =', A2d.tr())
print('\\f{A}{e_u^e_v} =', A2d(eu ^ ev))
print('\\f{A}{e_u}^\\f{A}{e_v} =', A2d(eu) ^ A2d(ev))
B2d = g2d.lt('B')
print('B =', B2d)
print('A + B =', A2d + B2d)
print('AB =', A2d * B2d)
print('A - B =', A2d - B2d)
a = g2d.mv('a', 'vector')
b = g2d.mv('b', 'vector')
print(r'a|\f{\overline{A}}{b}-b|\f{\underline{A}}{a} =',
((a | A2d.adj()(b)) - (b | A2d(a))).simplify())
print('#4d Minkowski spaqce (Space Time)')
m4d = Ga('e_t e_x e_y e_z',
g=[1, -1, -1, -1],
coords=symbols('t,x,y,z', real=True))
T = m4d.lt('T')
print('g =', m4d.g)
# FIXME incorrect sign for T and T.adj()
print(r'\underline{T} =', T)
print(r'\overline{T} =', T.adj())
# m4d.mv(T.det()).Fmt(4,r'\f{\det}{\underline{T}}')
print(r'\f{\mbox{tr}}{\underline{T}} =', T.tr())
a = m4d.mv('a', 'vector')
b = m4d.mv('b', 'vector')
print(r'a|\f{\overline{T}}{b}-b|\f{\underline{T}}{a} =',
((a | T.adj()(b)) - (b | T(a))).simplify())
xpdf(paper=(10, 12), debug=True)
