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', f=True)
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
print 'grad<A =', grad < A
print 'grad>A =', grad > A
print 'grad<B =', grad < B
print 'grad>B =', grad > B
print 'grad<C =', grad < C
print 'grad>C =', grad > C
return
def basic_multivector_operations_3D():
Print_Function()
g3d = Ga('e*x|y|z')
(ex,ey,ez) = g3d.mv()
A = g3d.mv('A','mv')
A.Fmt(1,'A')
A.Fmt(2,'A')
A.Fmt(3,'A')
A.even().Fmt(1,'%A_{+}')
A.odd().Fmt(1,'%A_{-}')
X = g3d.mv('X','vector')
Y = g3d.mv('Y','vector')
print 'g_{ij} = ',g3d.g
X.Fmt(1,'X')
Y.Fmt(1,'Y')
(X*Y).Fmt(2,'X*Y')
(X^Y).Fmt(2,'X^Y')
(X|Y).Fmt(2,'X|Y')
return
def properties_of_geometric_objects():
Print_Function()
global n, nbar
g = '# # # 0 0,'+ \
'# # # 0 0,'+ \
'# # # 0 0,'+ \
'0 0 0 0 2,'+ \
'0 0 0 2 0'
c3d = Ga('p1 p2 p3 n nbar',g=g)
(p1,p2,p3,n,nbar) = c3d.mv()
print 'g_{ij} =\n',c3d.g
P1 = F(p1)
P2 = F(p2)
P3 = F(p3)
print 'Extracting direction of line from L = P1^P2^n'
L = P1^P2^n
delta = (L|n)|nbar
print '(L|n)|nbar =',delta
print 'Extracting plane of circle from C = P1^P2^P3'
C = P1^P2^P3
delta = ((C^n)|n)|nbar
print '((C^n)|n)|nbar =',delta
print '(p2-p1)^(p3-p1) =',(p2-p1)^(p3-p1)
def extracting_vectors_from_conformal_2_blade():
Print_Function()
g = '0 -1 #,'+ \
'-1 0 #,'+ \
'# # #'
e2b = Ga('P1 P2 a',g=g)
(P1,P2,a) = e2b.mv()
print 'g_{ij} =\n',e2b.g
B = P1^P2
Bsq = B*B
print 'B**2 =',Bsq
ap = a-(a^B)*B
print "a' = a-(a^B)*B =",ap
Ap = ap+ap*B
Am = ap-ap*B
print "A+ = a'+a'*B =",Ap
print "A- = a'-a'*B =",Am
print '(A+)^2 =',Ap*Ap
print '(A-)^2 =',Am*Am
aB = a|B
print 'a|B =',aB
return
def test_reciprocal_frame():
"""
Test of formula for general reciprocal frame of three vectors.
Let three independent vectors be e1, e2, and e3. The reciprocal
vectors E1, E2, and E3 obey the relations:
e_i.E_j = delta_ij*(e1^e2^e3)**2
"""
g = '1 # #,'+ \
'# 1 #,'+ \
'# # 1'
g3dn = Ga('e1 e2 e3',g=g)
(e1,e2,e3) = g3dn.mv()
E = e1^e2^e3
Esq = (E*E).scalar()
Esq_inv = 1 / Esq
E1 = (e2^e3)*E
E2 = (-1)*(e1^e3)*E
E3 = (e1^e2)*E
w = (E1|e2)
w = w.expand()
assert w.scalar() == 0
w = (E1|e3)
w = w.expand()
assert w.scalar() == 0
w = (E2|e1)
w = w.expand()
assert w.scalar() == 0
w = (E2|e3)
w = w.expand()
assert w.scalar() == 0
w = (E3|e1)
w = w.expand()
assert w.scalar() == 0
w = (E3|e2)
w = w.expand()
assert w.scalar() == 0
w = (E1|e1)
w = (w.expand()).scalar()
Esq = expand(Esq)
assert simplify(w/Esq) == 1
w = (E2|e2)
w = (w.expand()).scalar()
assert simplify(w/Esq) == 1
w = (E3|e3)
w = (w.expand()).scalar()
assert simplify(w/Esq) == 1
def basic_multivector_operations_2D():
Print_Function()
g2d = Ga('e*x|y')
(ex,ey) = g2d.mv()
print 'g_{ij} =',g2d.g
X = g2d.mv('X','vector')
A = g2d.mv('A','spinor')
X.Fmt(1,'X')
A.Fmt(1,'A')
(X|A).Fmt(2,'X|A')
(X<A).Fmt(2,'X<A')
(A>X).Fmt(2,'A>X')
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)).simplify()
print 'grad^B =',grad^B
def rounding_numerical_components():
Print_Function()
o3d = Ga('e_x e_y e_z',g=[1,1,1])
(ex,ey,ez) = o3d.mv()
X = 1.2*ex+2.34*ey+0.555*ez
Y = 0.333*ex+4*ey+5.3*ez
print 'X =',X
print 'Nga(X,2) =',Nga(X,2)
print 'X*Y =',X*Y
print 'Nga(X*Y,2) =',Nga(X*Y,2)
return
def basic_multivector_operations_2D_orthogonal():
Print_Function()
o2d = Ga('e*x|y',g=[1,1])
(ex,ey) = o2d.mv()
print 'g_{ii} =',o2d.g
X = o2d.mv('X','vector')
A = o2d.mv('A','spinor')
X.Fmt(1,'X')
A.Fmt(1,'A')
(X*A).Fmt(2,'X*A')
(X|A).Fmt(2,'X|A')
(X<A).Fmt(2,'X<A')
(X>A).Fmt(2,'X>A')
(A*X).Fmt(2,'A*X')
(A|X).Fmt(2,'A|X')
(A<X).Fmt(2,'A<X')
(A>X).Fmt(2,'A>X')
return
class LieAlgebra(object):
def __init__(self,n):
self.n = n
e = ''
ebar = ''
for i in range(1,n+1):
e += ' e_' + str(i)
if GaLatexPrinter.latex_flg:
ebar += r' \bar{e}_' + str(i)
else:
ebar += ' ebar_' + str(i)
g = n * [1] + n * [-1]
basis = e[1:] + ebar
self.Ga = Ga(basis, g=g)
self.basis = self.Ga.mv()
self.e = self.basis[:n]
self.ebar = self.basis[n:]
self.w = []
self.wstar = []
for i in range(n):
self.w.append(self.e[i] + self.ebar[i])
self.wstar.append(self.e[i] - self.ebar[i])
self.Nu_bais = self.w + self.wstar
self.Eij = []
self.Fij = []
self.Ki = []
for i in range(n):
self.Ki.append(self.e[i] * self.ebar[i])
print r'%F_{'+str(i)+'} =',self.Ki[-1] * self.Ki[-1].rev()
for j in range(i):
self.Eij.append(self.e[i] * self.e[j] - self.ebar[i] * self.ebar[j])
self.Fij.append(self.e[i] * self.ebar[j] - self.ebar[i] * self.e[j])
print r'%E_{'+str(i)+str(j)+'} =',self.Eij[-1] * self.Eij[-1].rev()
print r'%F_{'+str(i)+str(j)+'} =',self.Fij[-1] * self.Fij[-1].rev()
print 'K_{i} =',self.Ki
print 'E_{ij} =',self.Eij
print 'F_{ij} =',self.Fij
E = self.Eij[0]/2
for i in range(2*n):
print E
E *= self.Eij[0]/2
def check_generalized_BAC_CAB_formulas():
Print_Function()
g4d = Ga('a b c d')
(a,b,c,d) = g4d.mv()
print 'g_{ij} =',g4d.g
print '\\bm{a|(b*c)} =',a|(b*c)
print '\\bm{a|(b^c)} =',a|(b^c)
print '\\bm{a|(b^c^d)} =',a|(b^c^d)
print '\\bm{a|(b^c)+c|(a^b)+b|(c^a)} =',(a|(b^c))+(c|(a^b))+(b|(c^a))
print '\\bm{a*(b^c)-b*(a^c)+c*(a^b)} =',a*(b^c)-b*(a^c)+c*(a^b)
print '\\bm{a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)} =',a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)
print '\\bm{(a^b)|(c^d)} =',(a^b)|(c^d)
print '\\bm{((a^b)|c)|d} =',((a^b)|c)|d
print '\\bm{(a^b)\\times (c^d)} =',Com(a^b,c^d)
return
def LieBasis(n):
g = n * [1] + n * [-1]
basis = ''
for i in range(1,n+1):
basis += 'e_' +str(i) + ' '
for i in range(1,n+1):
basis += r'\bar{e}_' +str(i) + ' '
basis = basis[:-1]
LieGA = Ga(basis,g=g)
bases = LieGA.mv()
e = bases[:n]
ebar = bases[n:]
print e
print ebar
print LieGA.g
E = []
F = []
K = []
indexes = []
for i in range(n):
K.append(e[i]*ebar[i])
for j in range(n):
if i < j:
indexes.append((i+1,j+1))
E.append(e[i]*e[j]-ebar[i]*ebar[j])
F.append(e[i]*ebar[j]-ebar[i]*e[j])
print indexes
print 'E =',E
print 'F =',F
print 'K =',K,'\n'
for k in range(len(indexes)):
k_i = indexes[k][0]
k_j = indexes[k][1]
for l in range(len(indexes)):
l_i = indexes[l][0]
l_j = indexes[l][1]
print 'E_'+str(k_i)+str(k_j)+' x F_'+str(l_i)+str(l_j)+' = '+str(Com(E[k],F[l]))
return
def conformal_representations_of_circles_lines_spheres_and_planes():
global n,nbar
Print_Function()
g = '1 0 0 0 0,0 1 0 0 0,0 0 1 0 0,0 0 0 0 2,0 0 0 2 0'
cnfml3d = Ga('e_1 e_2 e_3 n nbar',g=g)
(e1,e2,e3,n,nbar) = cnfml3d.mv()
print 'g_{ij} =\n',cnfml3d.g
e = n+nbar
#conformal representation of points
A = make_vector(e1,ga=cnfml3d) # point a = (1,0,0) A = F(a)
B = make_vector(e2,ga=cnfml3d) # point b = (0,1,0) B = F(b)
C = make_vector(-e1,ga=cnfml3d) # point c = (-1,0,0) C = F(c)
D = make_vector(e3,ga=cnfml3d) # point d = (0,0,1) D = F(d)
X = make_vector('x',3,ga=cnfml3d)
print 'F(a) =',A
print 'F(b) =',B
print 'F(c) =',C
print 'F(d) =',D
print 'F(x) =',X
print 'a = e1, b = e2, c = -e1, and d = e3'
print 'A = F(a) = 1/2*(a*a*n+2*a-nbar), etc.'
print 'Circle through a, b, and c'
print 'Circle: A^B^C^X = 0 =',(A^B^C^X)
print 'Line through a and b'
print 'Line : A^B^n^X = 0 =',(A^B^n^X)
print 'Sphere through a, b, c, and d'
print 'Sphere: A^B^C^D^X = 0 =',(((A^B)^C)^D)^X
print 'Plane through a, b, and d'
print 'Plane : A^B^n^D^X = 0 =',(A^B^n^D^X)
L = (A^B^e)^X
L.Fmt(3,'Hyperbolic Circle: (A^B^e)^X = 0 =')
return
def check_generalized_BAC_CAB_formulas():
Print_Function()
g5d = Ga('a b c d e')
(a, b, c, d, e) = g5d.mv()
print 'g_{ij} =\n', g5d.g
print 'a|(b*c) =', a | (b * c)
print 'a|(b^c) =', a | (b ^ c)
print 'a|(b^c^d) =', a | (b ^ c ^ d)
print 'a|(b^c)+c|(a^b)+b|(c^a) =', (a | ( b ^ c)) + (c | (a ^ b)) + (b | (c ^ a))
print 'a*(b^c)-b*(a^c)+c*(a^b) =',a*(b^c)-b*(a^c)+c*(a^b)
print 'a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c) =',a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)
print '(a^b)|(c^d) =',(a^b)|(c^d)
print '((a^b)|c)|d =',((a^b)|c)|d
print '(a^b)x(c^d) =',Com(a^b,c^d)
print '(a|(b^c))|(d^e) =',(a|(b^c))|(d^e)
return
def basic_multivector_operations():
Print_Function()
g3d = Ga('e*x|y|z')
(ex, ey, ez) = g3d.mv()
A = g3d.mv('A', 'mv')
A.Fmt(1, 'A')
A.Fmt(2, 'A')
A.Fmt(3, 'A')
X = g3d.mv('X', 'vector')
Y = g3d.mv('Y', 'vector')
print 'g_{ij} =\n', g3d.g
X.Fmt(1, 'X')
Y.Fmt(1, 'Y')
(X * Y).Fmt(2, 'X*Y')
(X ^ Y).Fmt(2, 'X^Y')
(X | Y).Fmt(2, 'X|Y')
g2d = Ga('e*x|y')
(ex, ey) = g2d.mv()
print 'g_{ij} =\n', g2d.g
X = g2d.mv('X', 'vector')
A = g2d.mv('A', 'spinor')
X.Fmt(1, 'X')
A.Fmt(1, 'A')
(X | A).Fmt(2, 'X|A')
(X < A).Fmt(2, 'X<A')
(A > X).Fmt(2, 'A>X')
o2d = Ga('e*x|y', g=[1, 1])
(ex, ey) = o2d.mv()
print 'g_{ii} =\n', o2d.g
X = o2d.mv('X', 'vector')
A = o2d.mv('A', 'spinor')
X.Fmt(1, 'X')
A.Fmt(1, 'A')
(X * A).Fmt(2, 'X*A')
(X | A).Fmt(2, 'X|A')
(X < A).Fmt(2, 'X<A')
(X > A).Fmt(2, 'X>A')
(A * X).Fmt(2, 'A*X')
(A | X).Fmt(2, 'A|X')
(A < X).Fmt(2, 'A<X')
(A > X).Fmt(2, 'A>X')
return
def reciprocal_frame_test():
Print_Function()
g = '1 # #,'+ \
'# 1 #,'+ \
'# # 1'
g3dn = Ga('e1 e2 e3',g=g)
(e1,e2,e3) = g3dn.mv()
print 'g_{ij} =\n',g3dn.g
E = e1^e2^e3
Esq = (E*E).scalar()
print 'E =',E
print 'E**2 =',Esq
Esq_inv = 1 / Esq
E1 = (e2^e3)*E
E2 = (-1)*(e1^e3)*E
E3 = (e1^e2)*E
print 'E1 = (e2^e3)*E =',E1
print 'E2 =-(e1^e3)*E =',E2
print 'E3 = (e1^e2)*E =',E3
w = (E1|e2)
w = w.expand()
print 'E1|e2 =',w
w = (E1|e3)
w = w.expand()
print 'E1|e3 =',w
w = (E2|e1)
w = w.expand()
print 'E2|e1 =',w
w = (E2|e3)
w = w.expand()
print 'E2|e3 =',w
w = (E3|e1)
w = w.expand()
print 'E3|e1 =',w
w = (E3|e2)
w = w.expand()
print 'E3|e2 =',w
w = (E1|e1)
w = (w.expand()).scalar()
Esq = expand(Esq)
print '(E1|e1)/E**2 =',simplify(w/Esq)
w = (E2|e2)
w = (w.expand()).scalar()
print '(E2|e2)/E**2 =',simplify(w/Esq)
w = (E3|e3)
w = (w.expand()).scalar()
print '(E3|e3)/E**2 =',simplify(w/Esq)
return
#ALGEBRA & DEFINITIONS
########################################################################
#Clifford(1,4)
#Flat space, no metric, just signature
#All constants = 1
metric=[1
,-1
,-1
,-1
,-1]
#Dimensions
variables = (t, x, y, z, w) = symbols('t x y z w', real=True)
myBasis='gamma_t gamma_x gamma_y gamma_z gamma_w'
#Algebra
sp5d = Ga(myBasis, g=metric, coords=variables,norm=True)
(gamma_t, gamma_x, gamma_y, gamma_z, gamma_w) = sp5d.mv()
(grad, rgrad) = sp5d.grads()
#Imaginary unit
imag=gamma_w
imag.texLabel='i'
#Associative Hyperbolic Quaternions
ihquat=gamma_t
jhquat=gamma_t*gamma_x*gamma_y*gamma_z*gamma_w
khquat=gamma_x*gamma_y*gamma_z*gamma_w
ihquat.texLabel='\\mathbf{i}'
jhquat.texLabel='\\mathbf{j}'
khquat.texLabel='\\mathbf{k}'
#Quaternions
iquat=gamma_y*gamma_z
jquat=gamma_z*gamma_x
from sympy import symbols, sin, latex, diff, Function, expand
from sympy.galgebra.ga import Ga
from sympy.galgebra.lt import Mlt
from sympy.galgebra.printer import Eprint, Format, xpdf
Format()
#Define spherical coordinate system in 3-d
coords = (r, th, phi) = symbols('r,theta,phi', real=True)
sp3d = Ga('e_r e_th e_ph', g=[1, r**2, r**2*sin(th)**2], coords=coords)
(er, eth, ephi) = sp3d.mv()
#Define coordinates for 2-d (u,v) and 1-d (s) manifolds
u,v,s,alpha = symbols('u v s alpha',real=True)
sub_coords = (u,v)
smap = [1, u, v] # Coordinate map for sphere of r = 1 in 3-d
print r'(u,v)\rightarrow (r,\theta,\phi) = ',smap
#Define unit sphere manifold
sph2d = sp3d.sm(smap,sub_coords)
print '#Unit Sphere Manifold:'
from sympy.galgebra.printer import Format, xpdf
from sympy.galgebra.ga import Ga
Format()
g3d = Ga('e*x|y|z')
A = g3d.mv('A','mv')
print r'\bm{A} =',A
A.Fmt(2,r'\bm{A}')
A.Fmt(3,r'\bm{A}')
xpdf(paper='letter')
def main():
Format()
(g3d,ex,ey,ez) = Ga.build('e*x|y|z')
A = g3d.mv('A','mv')
print r'\bm{A} =',A
A.Fmt(2,r'\bm{A}')
A.Fmt(3,r'\bm{A}')
X = (x,y,z) = symbols('x y z',real=True)
o3d = Ga('e_x e_y e_z',g=[1,1,1],coords=X)
(ex,ey,ez) = o3d.mv()
f = o3d.mv('f','scalar',f=True)
A = o3d.mv('A','vector',f=True)
B = o3d.mv('B','bivector',f=True)
print r'\bm{A} =',A
print r'\bm{B} =',B
print 'grad*f =',o3d.grad*f
print r'grad|\bm{A} =',o3d.grad|A
print r'grad*\bm{A} =',o3d.grad*A
print r'-I*(grad^\bm{A}) =',-o3d.i*(o3d.grad^A)
print r'grad*\bm{B} =',o3d.grad*B
print r'grad^\bm{B} =',o3d.grad^B
print r'grad|\bm{B} =',o3d.grad|B
g4d = Ga('a b c d')
(a,b,c,d) = g4d.mv()
print 'g_{ij} =',g4d.g
print '\\bm{a|(b*c)} =',a|(b*c)
print '\\bm{a|(b^c)} =',a|(b^c)
print '\\bm{a|(b^c^d)} =',a|(b^c^d)
print '\\bm{a|(b^c)+c|(a^b)+b|(c^a)} =',(a|(b^c))+(c|(a^b))+(b|(c^a))
print '\\bm{a*(b^c)-b*(a^c)+c*(a^b)} =',a*(b^c)-b*(a^c)+c*(a^b)
print '\\bm{a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)} =',a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)
print '\\bm{(a^b)|(c^d)} =',(a^b)|(c^d)
print '\\bm{((a^b)|c)|d} =',((a^b)|c)|d
print '\\bm{(a^b)\\times (c^d)} =',Com(a^b,c^d)
g = '1 # #,'+ \
'# 1 #,'+ \
'# # 1'
ng3d = Ga('e1 e2 e3',g=g)
(e1,e2,e3) = ng3d.mv()
E = e1^e2^e3
Esq = (E*E).scalar()
print 'E =',E
print '%E^{2} =',Esq
Esq_inv = 1/Esq
E1 = (e2^e3)*E
E2 = (-1)*(e1^e3)*E
E3 = (e1^e2)*E
print 'E1 = (e2^e3)*E =',E1
print 'E2 =-(e1^e3)*E =',E2
print 'E3 = (e1^e2)*E =',E3
print 'E1|e2 =',(E1|e2).expand()
print 'E1|e3 =',(E1|e3).expand()
print 'E2|e1 =',(E2|e1).expand()
print 'E2|e3 =',(E2|e3).expand()
print 'E3|e1 =',(E3|e1).expand()
print 'E3|e2 =',(E3|e2).expand()
w = ((E1|e1).expand()).scalar()
Esq = expand(Esq)
print '%(E1\\cdot e1)/E^{2} =',simplify(w/Esq)
w = ((E2|e2).expand()).scalar()
print '%(E2\\cdot e2)/E^{2} =',simplify(w/Esq)
w = ((E3|e3).expand()).scalar()
print '%(E3\\cdot e3)/E^{2} =',simplify(w/Esq)
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()
f = s3d.mv('f','scalar',f=True)
A = s3d.mv('A','vector',f=True)
B = s3d.mv('B','bivector',f=True)
print 'A =',A
print 'B =',B
print 'grad*f =',s3d.grad*f
print 'grad|A =',s3d.grad|A
print '-I*(grad^A) =',-s3d.i*(s3d.grad^A)
print 'grad^B =',s3d.grad^B
coords = symbols('t x y z')
m4d = Ga('gamma*t|x|y|z',g=[1,-1,-1,-1],coords=coords)
(g0,g1,g2,g3) = m4d.mv()
I = m4d.i
B = m4d.mv('B','vector',f=True)
E = m4d.mv('E','vector',f=True)
B.set_coef(1,0,0)
E.set_coef(1,0,0)
B *= g0
E *= g0
J = m4d.mv('J','vector',f=True)
F = E+I*B
print 'B = \\bm{B\\gamma_{t}} =',B
print 'E = \\bm{E\\gamma_{t}} =',E
print 'F = E+IB =',F
print 'J =',J
gradF = m4d.grad*F
gradF.Fmt(3,'grad*F')
print 'grad*F = J'
(gradF.get_grade(1)-J).Fmt(3,'%\\grade{\\nabla F}_{1} -J = 0')
(gradF.get_grade(3)).Fmt(3,'%\\grade{\\nabla F}_{3} = 0')
(alpha,beta,gamma) = symbols('alpha beta gamma')
(x,t,xp,tp) = symbols("x t x' t'")
m2d = Ga('gamma*t|x',g=[1,-1])
(g0,g1) = m2d.mv()
R = cosh(alpha/2)+sinh(alpha/2)*(g0^g1)
X = t*g0+x*g1
Xp = tp*g0+xp*g1
print 'R =',R
print r"#%t\bm{\gamma_{t}}+x\bm{\gamma_{x}} = t'\bm{\gamma'_{t}}+x'\bm{\gamma'_{x}} = R\lp t'\bm{\gamma_{t}}+x'\bm{\gamma_{x}}\rp R^{\dagger}"
Xpp = R*Xp*R.rev()
Xpp = Xpp.collect()
Xpp = Xpp.trigsimp()
print r"%t\bm{\gamma_{t}}+x\bm{\gamma_{x}} =",Xpp
Xpp = Xpp.subs({sinh(alpha):gamma*beta,cosh(alpha):gamma})
print r'%\f{\sinh}{\alpha} = \gamma\beta'
print r'%\f{\cosh}{\alpha} = \gamma'
print r"%t\bm{\gamma_{t}}+x\bm{\gamma_{x}} =",Xpp.collect()
coords = symbols('t x y z')
m4d = Ga('gamma*t|x|y|z',g=[1,-1,-1,-1],coords=coords)
(g0,g1,g2,g3) = m4d.mv()
I = m4d.i
(m,e) = symbols('m e')
psi = m4d.mv('psi','spinor',f=True)
A = m4d.mv('A','vector',f=True)
sig_z = g3*g0
print '\\bm{A} =',A
print '\\bm{\\psi} =',psi
dirac_eq = (m4d.grad*psi)*I*sig_z-e*A*psi-m*psi*g0
dirac_eq.simplify()
dirac_eq.Fmt(3,r'\nabla \bm{\psi} I \sigma_{z}-e\bm{A}\bm{\psi}-m\bm{\psi}\gamma_{t} = 0')
xpdf()
return
from sympy import symbols, sin
from sympy.galgebra.printer import Format, xpdf
from sympy.galgebra.ga import Ga
Format()
print r'#\newline Normalized Spherical Coordinates:'
sph_coords = (r, th, phi) = symbols('r theta phi', real=True)
sp3d = Ga('e', g=[1, r**2, r**2 * sin(th)**2], coords=sph_coords, norm=True)
f = sp3d.mv('f', 'scalar', f=True)
F = sp3d.mv('F', 'vector', f=True)
lap = sp3d.grad*sp3d.grad
print 'g =', sp3d.g
print r'%\nabla =', sp3d.grad
print r'%\nabla^{2} =', lap
print r'%\lp\nabla^{2}\rp f =', lap*f
print r'%\nabla\cdot\lp\nabla f\rp =', sp3d.grad | (sp3d.grad * f)
print 'F =', F
print r'%\nabla\cdot F =', sp3d.grad | F
print '#Unnormalized Spherical Coordinates:'
sp3du = Ga('e', g=[1, r**2, r**2 * sin(th)**2], coords=sph_coords)
f = sp3du.mv('f', 'scalar', f=True)
F = sp3du.mv('F', 'vector', f=True)
lap = sp3du.grad*sp3du.grad
print 'g =', sp3du.g
print r'%\nabla =', sp3du.grad
print r'%\nabla^{2} =', lap
from sympy import symbols, sin
from sympy.galgebra.printer import Format, xpdf
from sympy.galgebra.ga import Ga
Format()
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)).simplify()
print 'grad^B =',grad^B
xpdf(paper='letter')
def test_submanifolds():
#Define spherical coordinate system in 3-d
coords = (r, th, phi) = symbols('r,theta,phi', real=True)
sp3d = Ga('e_r e_th e_ph', g=[1, r**2, r**2*sin(th)**2], coords=coords)
(er, eth, ephi) = sp3d.mv()
#Define coordinates for 2-d (u,v) and 1-d (s) manifolds
u,v,s,alpha = symbols('u v s alpha',real=True)
sub_coords = (u,v)
smap = [1, u, v] # Coordinate map for sphere of r = 1 in 3-d
#Define unit sphere manifold
sph2d = sp3d.sm(smap,sub_coords)
(eu,ev) = sph2d.mv()
#Define vector and vector field on unit sphere tangent space
a = sph2d.mv('a','vector')
b = sph2d.mv('b','vector')
c = sph2d.mv('c','vector')
f = sph2d.mv('f','vector',f=True)
#Define directional derivative in direction a for unit sphere manifold
dd = a|sph2d.grad
assert str(dd) == 'a__u*D{u} + a__v*D{v}'
assert str(dd * eu) == 'a__v*e_v/tan(u)'
assert str(dd * ev) == '-a__v*sin(2*u)*e_u/2 + a__u*e_v/tan(u)'
assert str(dd * f) == '(a__u*D{u}f__u - a__v*f__v*sin(2*u)/2 + a__v*D{v}f__u)*e_u + (a__u*f__v/tan(u) + a__u*D{u}f__v + a__v*f__u/tan(u) + a__v*D{v}f__v)*e_v'
V = Mlt('V',sph2d,nargs=1,fct=True)
T = Mlt('T',sph2d,nargs=2,fct=True)
assert str(T.contract(1,2)) == 'a_1__u**2*D{u}^2T_uu + a_1__u**2*D{v}^2T_uu/sin(u)**2 + a_1__u*a_1__v*D{u}^2T_uv + a_1__u*a_1__v*D{u}^2T_vu + a_1__u*a_1__v*D{v}^2T_uv/sin(u)**2 + a_1__u*a_1__v*D{v}^2T_vu/sin(u)**2 + a_1__v**2*D{u}^2T_vv + a_1__v**2*D{v}^2T_vv/sin(u)**2'
#Tensor Evaluation
assert str(T(a,b)) == 'a__u*b__u*T_uu + a__u*b__v*T_uv + a__v*b__u*T_vu + a__v*b__v*T_vv'
assert str(T(a,b+c).expand()) == 'a__u*b__u*T_uu + a__u*b__v*T_uv + a__u*c__u*T_uu + a__u*c__v*T_uv + a__v*b__u*T_vu + a__v*b__v*T_vv + a__v*c__u*T_vu + a__v*c__v*T_vv'
assert str(T(a,alpha*b)) == 'a__u*alpha*b__u*T_uu + a__u*alpha*b__v*T_uv + a__v*alpha*b__u*T_vu + a__v*alpha*b__v*T_vv'
#Geometric Derivative With Respect To Slot
assert str(T.pdiff(1)) == '(a_1__u*a_2__u*D{u}T_uu + a_1__u*a_2__v*D{u}T_uv + a_1__v*a_2__u*D{u}T_vu + a_1__v*a_2__v*D{u}T_vv)*e_u + (a_1__u*a_2__u*D{v}T_uu + a_1__u*a_2__v*D{v}T_uv + a_1__v*a_2__u*D{v}T_vu + a_1__v*a_2__v*D{v}T_vv)*e_v/sin(u)**2'
assert str(T.pdiff(2)) == '(a_1__u*a_2__u*D{u}T_uu + a_1__u*a_2__v*D{u}T_uv + a_1__v*a_2__u*D{u}T_vu + a_1__v*a_2__v*D{u}T_vv)*e_u + (a_1__u*a_2__u*D{v}T_uu + a_1__u*a_2__v*D{v}T_uv + a_1__v*a_2__u*D{v}T_vu + a_1__v*a_2__v*D{v}T_vv)*e_v/sin(u)**2'
#Covariant Derivatives
assert str(V.cderiv()) == 'a_2__u*(a_1__u*D{u}V_u + a_1__v*D{u}V_v) + a_2__v*(a_1__u*D{v}V_u + a_1__v*D{v}V_v)'
DT = T.cderiv()
assert str(DT) == 'a_3__u*(a_1__u*a_2__u*D{u}T_uu + a_1__u*a_2__v*D{u}T_uv + a_1__v*a_2__u*D{u}T_vu + a_1__v*a_2__v*D{u}T_vv) + a_3__v*(a_1__u*a_2__u*D{v}T_uu + a_1__u*a_2__v*D{v}T_uv + a_1__v*a_2__u*D{v}T_vu + a_1__v*a_2__v*D{v}T_vv)'
#Define curve on unit sphere manifold
us = Function('u__s')(s)
vs = Function('v__s')(s)
#Define 1-d submanifold on unit shpere manifold
crv1d = sph2d.sm([us,vs],[s])
(es,) = crv1d.mv()
#1-D Manifold On Unit Sphere:
assert str(crv1d.grad) == 'e_s*1/(sin(u__s)**2*D{s}v__s**2 + D{s}u__s**2)*D{s}'
#Define scalar and vector fields on 1-d manifold tangent space
g = crv1d.mv('g','scalar',f=True)
h = crv1d.mv('h','vector',f=True)
assert str(crv1d.grad * g) == 'D{s}g*e_s/(sin(u__s)**2*D{s}v__s**2 + D{s}u__s**2)'
assert str(crv1d.grad | h) == 'D{s}h__s + h__s*sin(u__s)**2*D{s}v__s*D{s}^2v__s/(sin(u__s)**2*D{s}v__s**2 + D{s}u__s**2) + h__s*sin(2*u__s)*D{s}u__s*D{s}v__s**2/(2*(sin(u__s)**2*D{s}v__s**2 + D{s}u__s**2)) + h__s*D{s}u__s*D{s}^2u__s/(sin(u__s)**2*D{s}v__s**2 + D{s}u__s**2)'
return
from sympy import expand, simplify
from sympy.galgebra.ga import Ga
from sympy.galgebra.printer import Format, xpdf
Format()
g = "1 # #," + "# 1 #," + "# # 1"
ng3d = Ga("e1 e2 e3", g=g)
(e1, e2, e3) = ng3d.mv()
print "g_{ij} =", ng3d.g
E = e1 ^ e2 ^ e3
Esq = (E * E).scalar()
print "E =", E
print "%E^{2} =", Esq
Esq_inv = 1 / Esq
E1 = (e2 ^ e3) * E
E2 = (-1) * (e1 ^ e3) * E
E3 = (e1 ^ e2) * E
print "E1 = (e2^e3)*E =", E1
print "E2 =-(e1^e3)*E =", E2
print "E3 = (e1^e2)*E =", E3
w = E1 | e2
w = w.expand()
print "E1|e2 =", w
return [a1,a2]
def norm(X):
Y=sqrt((X*X).scalar())
return Y
Get_Program(True)
Eprint()
g='1 0 0 0, \
0 1 0 0, \
0 0 0 2, \
0 0 2 0'
c2d = Ga('e_1 e_2 n \\bar{n}',g=g)
(e1,e2,n,nbar) = c2d.mv()
global n,nbar,I
def F(x):
global n,nbar
Fx = ((x*x)*n+2*x-nbar) / 2
return(Fx)
e = (n+nbar)/2
ebar = n - e
I=e1*e2*e*ebar
def intersect_lines(L1,L2):
global I
'''
def noneuclidian_distance_calculation():
from sympy import solve,sqrt
Print_Function()
g = '0 # #,# 0 #,# # 1'
necl = Ga('X Y e',g=g)
(X,Y,e) = necl.mv()
print 'g_{ij} =',necl.g
print '(X^Y)**2 =',(X^Y)*(X^Y)
L = X^Y^e
B = (L*e).expand().blade_rep() # D&L 10.152
print 'B =',B
Bsq = B*B
print 'B**2 =',Bsq.obj
Bsq = Bsq.scalar()
print '#L = X^Y^e is a non-euclidian line'
print 'B = L*e =',B
BeBr =B*e*B.rev()
print 'B*e*B.rev() =',BeBr
print 'B**2 =',B*B
print 'L**2 =',L*L # D&L 10.153
(s,c,Binv,M,S,C,alpha) = symbols('s c (1/B) M S C alpha')
XdotY = necl.g[0,1]
Xdote = necl.g[0,2]
Ydote = necl.g[1,2]
Bhat = Binv*B # D&L 10.154
R = c+s*Bhat # Rotor R = exp(alpha*Bhat/2)
print 's = sinh(alpha/2) and c = cosh(alpha/2)'
print 'exp(alpha*B/(2*|B|)) =',R
Z = R*X*R.rev() # D&L 10.155
Z.obj = expand(Z.obj)
Z.obj = Z.obj.collect([Binv,s,c,XdotY])
Z.Fmt(3,'R*X*R.rev()')
W = Z|Y # Extract scalar part of multivector
# From this point forward all calculations are with sympy scalars
print 'Objective is to determine value of C = cosh(alpha) such that W = 0'
W = W.scalar()
print 'Z|Y =',W
W = expand(W)
W = simplify(W)
W = W.collect([s*Binv])
M = 1/Bsq
W = W.subs(Binv**2,M)
W = simplify(W)
Bmag = sqrt(XdotY**2-2*XdotY*Xdote*Ydote)
W = W.collect([Binv*c*s,XdotY])
#Double angle substitutions
W = W.subs(2*XdotY**2-4*XdotY*Xdote*Ydote,2/(Binv**2))
W = W.subs(2*c*s,S)
W = W.subs(c**2,(C+1)/2)
W = W.subs(s**2,(C-1)/2)
W = simplify(W)
W = W.subs(1/Binv,Bmag)
W = expand(W)
print 'S = sinh(alpha) and C = cosh(alpha)'
print 'W =',W
Wd = collect(W,[C,S],exact=True,evaluate=False)
print 'Wd =', Wd
Wd_1 = Wd[one]
Wd_C = Wd[C]
Wd_S = Wd[S]
print 'Scalar Coefficient =',Wd_1
print 'Cosh Coefficient =',Wd_C
print 'Sinh Coefficient =',Wd_S
print '|B| =',Bmag
Wd_1 = Wd_1.subs(Bmag,1/Binv)
Wd_C = Wd_C.subs(Bmag,1/Binv)
Wd_S = Wd_S.subs(Bmag,1/Binv)
lhs = Wd_1+Wd_C*C
rhs = -Wd_S*S
lhs = lhs**2
rhs = rhs**2
W = expand(lhs-rhs)
W = expand(W.subs(1/Binv**2,Bmag**2))
W = expand(W.subs(S**2,C**2-1))
W = W.collect([C,C**2],evaluate=False)
a = simplify(W[C**2])
b = simplify(W[C])
c = simplify(W[one])
print 'Require a*C**2+b*C+c = 0'
print 'a =',a
print 'b =',b
print 'c =',c
x = Symbol('x')
C = solve(a*x**2+b*x+c,x)[0]
print 'cosh(alpha) = C = -b/(2*a) =',expand(simplify(expand(C)))
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 '\\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 'A + B =', A + B
print 'AB =', A * B
print 'A - B =', A - B
print '#2d general ($A,\\;B$ are linear transformations)'
A2d = g2d.lt('A')
print 'A =', A2d
print '\\f{\\det}{A} =', A2d.det()
print '\\overline{A} =', A2d.adj()
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 noneuclidian_distance_calculation():
Print_Function()
from sympy import solve,sqrt
g = '0 # #,# 0 #,# # 1'
nel = Ga('X Y e',g=g)
(X,Y,e) = nel.mv()
print 'g_{ij} =',nel.g
print '%(X\\W Y)^{2} =',(X^Y)*(X^Y)
L = X^Y^e
B = L*e # D&L 10.152
Bsq = (B*B).scalar()
print '#%L = X\\W Y\\W e \\text{ is a non-euclidian line}'
print 'B = L*e =',B
BeBr =B*e*B.rev()
print '%BeB^{\\dagger} =',BeBr
print '%B^{2} =',B*B
print '%L^{2} =',L*L # D&L 10.153
(s,c,Binv,M,S,C,alpha) = symbols('s c (1/B) M S C alpha')
XdotY = nel.g[0,1]
Xdote = nel.g[0,2]
Ydote = nel.g[1,2]
Bhat = Binv*B # D&L 10.154
R = c+s*Bhat # Rotor R = exp(alpha*Bhat/2)
print '#%s = \\f{\\sinh}{\\alpha/2} \\text{ and } c = \\f{\\cosh}{\\alpha/2}'
print '%e^{\\alpha B/{2\\abs{B}}} =',R
Z = R*X*R.rev() # D&L 10.155
Z.obj = expand(Z.obj)
Z.obj = Z.obj.collect([Binv,s,c,XdotY])
Z.Fmt(3,'%RXR^{\\dagger}')
W = Z|Y # Extract scalar part of multivector
# From this point forward all calculations are with sympy scalars
#print '#Objective is to determine value of C = cosh(alpha) such that W = 0'
W = W.scalar()
print '%W = Z\\cdot Y =',W
W = expand(W)
W = simplify(W)
W = W.collect([s*Binv])
M = 1/Bsq
W = W.subs(Binv**2,M)
W = simplify(W)
Bmag = sqrt(XdotY**2-2*XdotY*Xdote*Ydote)
W = W.collect([Binv*c*s,XdotY])
#Double angle substitutions
W = W.subs(2*XdotY**2-4*XdotY*Xdote*Ydote,2/(Binv**2))
W = W.subs(2*c*s,S)
W = W.subs(c**2,(C+1)/2)
W = W.subs(s**2,(C-1)/2)
W = simplify(W)
W = W.subs(1/Binv,Bmag)
W = expand(W)
print '#%S = \\f{\\sinh}{\\alpha} \\text{ and } C = \\f{\\cosh}{\\alpha}'
print 'W =',W
Wd = collect(W,[C,S],exact=True,evaluate=False)
Wd_1 = Wd[one]
Wd_C = Wd[C]
Wd_S = Wd[S]
print '%\\text{Scalar Coefficient} =',Wd_1
print '%\\text{Cosh Coefficient} =',Wd_C
print '%\\text{Sinh Coefficient} =',Wd_S
print '%\\abs{B} =',Bmag
Wd_1 = Wd_1.subs(Bmag,1/Binv)
Wd_C = Wd_C.subs(Bmag,1/Binv)
Wd_S = Wd_S.subs(Bmag,1/Binv)
lhs = Wd_1+Wd_C*C
rhs = -Wd_S*S
lhs = lhs**2
rhs = rhs**2
W = expand(lhs-rhs)
W = expand(W.subs(1/Binv**2,Bmag**2))
W = expand(W.subs(S**2,C**2-1))
W = W.collect([C,C**2],evaluate=False)
a = simplify(W[C**2])
b = simplify(W[C])
c = simplify(W[one])
print '#%\\text{Require } aC^{2}+bC+c = 0'
print 'a =',a
print 'b =',b
print 'c =',c
x = Symbol('x')
C = solve(a*x**2+b*x+c,x)[0]
print '%b^{2}-4ac =',simplify(b**2-4*a*c)
print '%\\f{\\cosh}{\\alpha} = C = -b/(2a) =',expand(simplify(expand(C)))
return
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')
from sympy.galgebra.printer import Format, xpdf
from sympy.galgebra.ga import Ga
from sympy import symbols
Format()
X = (x,y,z) = symbols('x y z')
o3d = Ga('e_x e_y e_z',g=[1,1,1],coords=X)
f = o3d.mv('f','scalar',f=True)
A = o3d.mv('A','vector',f=True)
B = o3d.mv('B','bivector',f=True)
print r'\bm{A} =',A
print r'\bm{B} =',B
print 'grad*f =',o3d.grad*f
print r'grad|\bm{A} =',o3d.grad|A
(o3d.grad*A).Fmt(2,r'grad*\bm{A}')
print r'-I*(grad^\bm{A}) =',-o3d.mv_I*(o3d.grad^A)
(o3d.grad*B).Fmt(2,r'grad*\bm{B}')
print r'grad^\bm{B} =',o3d.grad^B
print r'grad|\bm{B} =',o3d.grad|B
xpdf(paper='letter')
