def derivatives_in_rectangular_coordinates():
Print_Function()
X = (x, y, z) = symbols('x y z')
o3d = Ga('e_x e_y e_z', g=[1, 1, 1], coords=X)
(ex, ey, ez) = o3d.mv()
grad = o3d.grad
f = o3d.mv('f', 'scalar', f=True)
A = o3d.mv('A', 'vector', f=True)
B = o3d.mv('B', 'bivector', f=True)
C = o3d.mv('C', 'mv')
print 'f =', f
print 'A =', A
print 'B =', B
print 'C =', C
print 'grad*f =', grad * f
print 'grad|A =', grad | A
print 'grad*A =', grad * A
print '-I*(grad^A) =', -o3d.I() * (grad ^ A)
print 'grad*B =', grad * B
print 'grad^B =', grad ^ B
print 'grad|B =', grad | B
return
def derivatives_in_spherical_coordinates():
Print_Function()
X = (r, th, phi) = symbols('r theta phi')
s3d = Ga('e_r e_theta e_phi',
g=[1, r**2, r**2 * sin(th)**2],
coords=X,
norm=True)
(er, eth, ephi) = s3d.mv()
grad = s3d.grad
f = s3d.mv('f', 'scalar', f=True)
A = s3d.mv('A', 'vector', f=True)
B = s3d.mv('B', 'bivector', f=True)
print 'f =', f
print 'A =', A
print 'B =', B
print 'grad*f =', grad * f
print 'grad|A =', grad | A
print '-I*(grad^A) =', -s3d.I() * (grad ^ A)
print 'grad^B =', grad ^ B
return
def signature_test():
Print_Function()
e3d = Ga('e1 e2 e3', g=[1, 1, 1])
print 'e3d.g =', e3d.g
print 'Signature = (3,0) I =', e3d.I(), ' I**2 =', e3d.I() * e3d.I()
e3d = Ga('e1 e2 e3', g=[2, 2, 2])
print 'e3d.g =', e3d.g
print 'Signature = (3,0) I =', e3d.I(), ' I**2 =', e3d.I() * e3d.I()
sp4d = Ga('e1 e2 e3 e4', g=[1, -1, -1, -1])
print 'e3d.g =', sp4d.g
print 'Signature = (1,3) I =', sp4d.I(), ' I**2 =', sp4d.I() * sp4d.I()
sp4d = Ga('e1 e2 e3 e4', g=[2, -2, -2, -2])
print 'e3d.g =', sp4d.g
print 'Signature = (1,3) I =', sp4d.I(), ' I**2 =', sp4d.I() * sp4d.I()
e4d = Ga('e1 e2 e3 e4', g=[1, 1, 1, 1])
print 'e4d.g =', e4d.g
print 'Signature = (4,0) I =', e4d.I(), ' I**2 =', e4d.I() * e4d.I()
cf3d = Ga('e1 e2 e3 e4 e5', g=[1, 1, 1, 1, -1])
print 'cf4d.g =', cf3d.g
print 'Signature = (4,1) I =', cf3d.I(), ' I**2 =', cf3d.I() * cf3d.I()
cf3d = Ga('e1 e2 e3 e4 e5', g=[2, 2, 2, 2, -2])
print 'cf4d.g =', cf3d.g
print 'Signature = (4,1) I =', cf3d.I(), ' I**2 =', cf3d.I() * cf3d.I()
return
def signature_test():
Print_Function()
e3d = Ga('e1 e2 e3',g=[1,1,1])
print 'g =', e3d.g
print r'%Signature = (3,0)\: I =', e3d.I(),'\: I^{2} =', e3d.I()*e3d.I()
e3d = Ga('e1 e2 e3',g=[2,2,2])
print 'g =', e3d.g
print r'%Signature = (3,0)\: I =', e3d.I(),'|; I^{2} =', e3d.I()*e3d.I()
sp4d = Ga('e1 e2 e3 e4',g=[1,-1,-1,-1])
print 'g =', sp4d.g
print r'%Signature = (1,3)\: I =', sp4d.I(),'\: I^{2} =', sp4d.I()*sp4d.I()
sp4d = Ga('e1 e2 e3 e4',g=[2,-2,-2,-2])
print 'g =', sp4d.g
print r'%Signature = (1,3)\: I =', sp4d.I(),'\: I^{2} =', sp4d.I()*sp4d.I()
e4d = Ga('e1 e2 e3 e4',g=[1,1,1,1])
print 'g =', e4d.g
print r'%Signature = (4,0)\: I =', e4d.I(),'\: I^{2} =', e4d.I()*e4d.I()
cf3d = Ga('e1 e2 e3 e4 e5',g=[1,1,1,1,-1])
print 'g =', cf3d.g
print r'%Signature = (4,1)\: I =', cf3d.I(),'\: I^{2} =', cf3d.I()*cf3d.I()
cf3d = Ga('e1 e2 e3 e4 e5',g=[2,2,2,2,-2])
print 'g =', cf3d.g
print r'%Signature = (4,1)\: I =', cf3d.I(),'\: I^{2} =', cf3d.I()*cf3d.I()
return