print '#2d general ($A,\\;B$ are linear transformations)'
A2d = g2d.lt('A')
print 'A =', A2d
print '\\f{\\det}{A} =', A2d.det()
#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
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='landscape')
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())
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('#$T$ is a linear transformation in Minkowski space')
print(r'\underline{T} =', T)
print(r'\overline{T} =', T.adj())
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='landscape', crop=True)
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} =', 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} =', g2d.g_inv
print '%gg^{-1} =', 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}} =', 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 '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()
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 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} =', 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} =', g2d.g_inv)
print('%gg^{-1} =', 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}} =', 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('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())
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