def test_differential_operators():
xyz_coords = (x, y, z) = symbols('x y z', real=True)
(o3d, ex, ey, ez) = Ga.build('e', g=[1, 1, 1], coords=xyz_coords)
f = o3d.mv('f', 'scalar', f=True)
lap = o3d.grad*o3d.grad
assert str(lap) == 'D{x}^2 + D{y}^2 + D{z}^2'
assert str(lap * f) == 'D{x}^2f + D{y}^2f + D{z}^2f'
sph_coords = (r, th, phi) = symbols('r theta phi', real=True)
(sp3d, er, eth, ephi) = Ga.build('e', g=[1, r**2, r**2 * sin(th)**2], coords=sph_coords, norm=True)
f = sp3d.mv('f', 'scalar', f=True)
lap = sp3d.grad*sp3d.grad
assert str(lap) == '2/r*D{r} + cos(theta)/(r**2*sin(theta))*D{theta} + D{r}^2 + r**(-2)*D{theta}^2 + 1/(r**2*sin(theta)**2)*D{phi}^2'
assert str(lap * f) == 'D{r}^2f + 2*D{r}f/r + D{theta}^2f/r**2 + cos(theta)*D{theta}f/(r**2*sin(theta)) + D{phi}^2f/(r**2*sin(theta)**2)'
A = o3d.mv('A','vector')
xs = o3d.mv(x)
assert o3d.grad*A == 0
assert str(A*o3d.grad) == 'A__x*D{x} + A__y*D{y} + A__z*D{z} + e_x^e_y*(-A__y*D{x} + A__x*D{y}) + e_x^e_z*(-A__z*D{x} + A__x*D{z}) + e_y^e_z*(-A__z*D{y} + A__y*D{z})'
assert o3d.grad*xs == ex
assert str(xs*o3d.grad) == 'e_x*x*D{x} + e_y*x*D{y} + e_z*x*D{z}'
assert str(o3d.grad*(o3d.grad+xs)) == 'D{x}^2 + D{y}^2 + D{z}^2 + e_x*D{}'
assert str((o3d.grad+xs)*o3d.grad) == 'D{x}^2 + D{y}^2 + D{z}^2 + e_x*x*D{x} + e_y*x*D{y} + e_z*x*D{z}'
return
def test_basic_multivector_operations():
g3d, ex, ey, ez = Ga.build('e*x|y|z')
A = g3d.mv('A', 'mv')
assert str(A) == 'A + A__x*e_x + A__y*e_y + A__z*e_z + A__xy*e_x^e_y + A__xz*e_x^e_z + A__yz*e_y^e_z + A__xyz*e_x^e_y^e_z'
X = g3d.mv('X', 'vector')
Y = g3d.mv('Y', 'vector')
assert str(X) == 'X__x*e_x + X__y*e_y + X__z*e_z'
assert str(Y) == 'Y__x*e_x + Y__y*e_y + Y__z*e_z'
assert str((X*Y)) == '(e_x.e_x)*X__x*Y__x + (e_x.e_y)*X__x*Y__y + (e_x.e_y)*X__y*Y__x + (e_x.e_z)*X__x*Y__z + (e_x.e_z)*X__z*Y__x + (e_y.e_y)*X__y*Y__y + (e_y.e_z)*X__y*Y__z + (e_y.e_z)*X__z*Y__y + (e_z.e_z)*X__z*Y__z + (X__x*Y__y - X__y*Y__x)*e_x^e_y + (X__x*Y__z - X__z*Y__x)*e_x^e_z + (X__y*Y__z - X__z*Y__y)*e_y^e_z'
assert str((X ^ Y)) == '(X__x*Y__y - X__y*Y__x)*e_x^e_y + (X__x*Y__z - X__z*Y__x)*e_x^e_z + (X__y*Y__z - X__z*Y__y)*e_y^e_z'
assert str((X | Y)) == '(e_x.e_x)*X__x*Y__x + (e_x.e_y)*X__x*Y__y + (e_x.e_y)*X__y*Y__x + (e_x.e_z)*X__x*Y__z + (e_x.e_z)*X__z*Y__x + (e_y.e_y)*X__y*Y__y + (e_y.e_z)*X__y*Y__z + (e_y.e_z)*X__z*Y__y + (e_z.e_z)*X__z*Y__z'
g2d, ex, ey = Ga.build('e*x|y')
X = g2d.mv('X', 'vector')
A = g2d.mv('A', 'spinor')
assert str(X) == 'X__x*e_x + X__y*e_y'
assert str(A) == 'A + A__xy*e_x^e_y'
assert str((X | A)) == '-A__xy*((e_x.e_y)*X__x + (e_y.e_y)*X__y)*e_x + A__xy*((e_x.e_x)*X__x + (e_x.e_y)*X__y)*e_y'
assert str((X < A)) == '(-(e_x.e_y)*A__xy*X__x - (e_y.e_y)*A__xy*X__y + A*X__x)*e_x + ((e_x.e_x)*A__xy*X__x + (e_x.e_y)*A__xy*X__y + A*X__y)*e_y'
assert str((A > X)) == '((e_x.e_y)*A__xy*X__x + (e_y.e_y)*A__xy*X__y + A*X__x)*e_x + (-(e_x.e_x)*A__xy*X__x - (e_x.e_y)*A__xy*X__y + A*X__y)*e_y'
o2d, ex, ey = Ga.build('e*x|y', g=[1, 1])
X = o2d.mv('X', 'vector')
A = o2d.mv('A', 'spinor')
assert str(X) == 'X__x*e_x + X__y*e_y'
assert str(A) == 'A + A__xy*e_x^e_y'
assert str((X*A)) == '(A*X__x - A__xy*X__y)*e_x + (A*X__y + A__xy*X__x)*e_y'
assert str((X | A)) == '-A__xy*X__y*e_x + A__xy*X__x*e_y'
assert str((X < A)) == '(A*X__x - A__xy*X__y)*e_x + (A*X__y + A__xy*X__x)*e_y'
assert str((X > A)) == 'A*X__x*e_x + A*X__y*e_y'
assert str((A*X)) == '(A*X__x + A__xy*X__y)*e_x + (A*X__y - A__xy*X__x)*e_y'
assert str((A | X)) == 'A__xy*X__y*e_x - A__xy*X__x*e_y'
assert str((A < X)) == 'A*X__x*e_x + A*X__y*e_y'
assert str((A > X)) == '(A*X__x + A__xy*X__y)*e_x + (A*X__y - A__xy*X__x)*e_y'
return
def test_extract_plane_and_line():
"""
Show that conformal trivector encodes planes and lines. See D&L section
10.4.2
"""
metric = '# # # 0 0,' + \
'# # # 0 0,' + \
'# # # 0 0,' + \
'0 0 0 0 2,' + \
'0 0 0 2 0'
cf3d, p1, p2, p3, n, nbar = Ga.build('p1 p2 p3 n nbar', g=metric)
P1 = F(p1, n, nbar)
P2 = F(p2, n, nbar)
P3 = F(p3, n, nbar)
#Line through p1 and p2
L = P1 ^ P2 ^ n
delta = (L | n) | nbar
delta_test = 2*p1 - 2*p2
diff = delta - delta_test
assert diff == S.Zero
#Plane through p1, p2, and p3
C = P1 ^ P2 ^ P3
delta = ((C ^ n) | n) | nbar
delta_test = 2*(p1 ^ p2) - 2*(p1 ^ p3) + 2*(p2 ^ p3)
diff = delta - delta_test
assert diff == S.Zero
def main():
Eprint()
X = (x,y,z) = symbols('x y z',real=True)
(o3d,ex,ey,ez) = Ga.build('e_x e_y e_z',g=[1,1,1],coords=(x,y,z))
A = x*(ey^ez) + y*(ez^ex) + z*(ex^ey)
print 'A =', A
print 'grad^A =',(o3d.grad^A).simplify()
print
f = o3d.mv(1/sqrt(x**2 + y**2 + z**2))
print 'f =', f
print 'grad*f =',(o3d.grad*f).simplify()
print
B = f*A
print 'B =', B
print
Curl_B = o3d.grad^B
print 'grad^B =', Curl_B.simplify()
return
def basic_multivector_operations_3D():
Print_Function()
(g3d,ex,ey,ez) = Ga.build('e*x|y|z')
print 'g_{ij} =',g3d.g
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')
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 test_vector_extraction():
"""
Show that conformal bivector encodes two points. See D&L Section 10.4.1
"""
metric = '0 -1 #,' + \
'-1 0 #,' + \
'# # #'
cext, P1, P2, a = Ga.build('P1 P2 a', g=metric)
"""
P1 and P2 are null vectors and hence encode points in conformal space.
Show that P1 and P2 can be extracted from the bivector B = P1^P2. a is a
third vector in the conformal space with a.B not 0.
"""
B = P1 ^ P2
Bsq = B*B
ap = a - (a ^ B)*B
Ap = ap + ap*B
Am = ap - ap*B
P1dota = cext.g[0,2]
P2dota = cext.g[1,2]
Ap_test = (-2*P2dota)*P1
Am_test = (-2*P1dota)*P2
assert Ap == Ap_test
assert Am == Am_test
Ap2 = Ap*Ap
Am2 = Am*Am
assert Ap2 == S.Zero
assert Am2 == S.Zero
def Maxwells_Equations_in_Geom_Calculus():
Print_Function()
X = symbols('t x y z',real=True)
(st4d,g0,g1,g2,g3) = Ga.build('gamma*t|x|y|z',g=[1,-1,-1,-1],coords=X)
I = st4d.i
B = st4d.mv('B','vector',f=True)
E = st4d.mv('E','vector',f=True)
B.set_coef(1,0,0)
E.set_coef(1,0,0)
B *= g0
E *= g0
J = st4d.mv('J','vector',f=True)
F = E+I*B
print r'\text{Pseudo Scalar\;\;}I =',I
print '\\text{Magnetic Field Bi-Vector\\;\\;} B = \\bm{B\\gamma_{t}} =',B
print '\\text{Electric Field Bi-Vector\\;\\;} E = \\bm{E\\gamma_{t}} =',E
print '\\text{Electromagnetic Field Bi-Vector\\;\\;} F = E+IB =',F
print '%\\text{Four Current Density\\;\\;} J =',J
gradF = st4d.grad*F
print '#Geom Derivative of Electomagnetic Field Bi-Vector'
gradF.Fmt(3,'grad*F')
print '#Maxwell Equations'
print 'grad*F = J'
print '#Div $E$ and Curl $H$ Equations'
(gradF.get_grade(1)-J).Fmt(3,'%\\grade{\\nabla F}_{1} -J = 0')
print '#Curl $E$ and Div $B$ equations'
(gradF.get_grade(3)).Fmt(3,'%\\grade{\\nabla F}_{3} = 0')
return
def test_extracting_vectors_from_conformal_2_blade():
metric = '0 -1 #,' + \
'-1 0 #,' + \
'# # #'
(cf1d, P1, P2, a) = Ga.build('P1 P2 a', g=metric)
B = P1 ^ P2
Bsq = B*B
assert str(Bsq) == '1'
ap = a - (a ^ B)*B
assert str(ap) == '-(P2.a)*P1 - (P1.a)*P2'
Ap = ap + ap*B
Am = ap - ap*B
assert str(Ap) == '-2*(P2.a)*P1'
assert str(Am) == '-2*(P1.a)*P2'
assert str(Ap*Ap) == '0'
assert str(Am*Am) == '0'
aB = a | B
assert str(aB) == '-(P2.a)*P1 + (P1.a)*P2'
return
def Lorentz_Tranformation_in_Geog_Algebra():
Print_Function()
(alpha,beta,gamma) = symbols('alpha beta gamma')
(x,t,xp,tp) = symbols("x t x' t'",real=True)
(st2d,g0,g1) = Ga.build('gamma*t|x',g=[1,-1])
from sympy import sinh,cosh
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()
return
def test_conformal_representations_of_circles_lines_spheres_and_planes():
global n, nbar
metric = '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'
(cf3d, ex, ey, ez, n, nbar) = Ga.build('e_x e_y e_z n nbar', g=metric)
x, y, z = symbols('x y z', real=True)
e = n + nbar
#conformal representation of points
A = make_vector(ex, cf3d)
B = make_vector(ey, cf3d)
C = make_vector(-ex, cf3d)
D = make_vector(ez, cf3d)
X = make_vector(x*ex + y*ey +z*ez, cf3d)
assert A == ex + (n - nbar)/S(2)
assert B == ey + (n - nbar)/S(2)
assert C == -ex + (n - nbar)/S(2)
assert D == ez + (n - nbar)/S(2)
assert X == x*ex + y*ey + z*ez + (x**2/2 + y**2/2 + z**2/2)*n - nbar/2
assert A ^ B ^ C ^ X == -z*(ex^ey^ez^n) + z*(ex^ey^ez^nbar) + ((x**2 + y**2 + z**2 - S(1))/2)*(ex^ey^n^nbar)
assert A ^ B ^ n ^ X == -z*(ex^ey^ez^n) + ((x + y - S(1))/2)*(ex^ey^n^nbar) + (z/2)*(ex^ez^n^nbar) - (z/2)*(ey^ez^n^nbar)
assert A ^ B ^ C ^ D ^ X == ((-x**2 - y**2 - z**2 + S(1))/2)*ex^ey^ez^n^nbar
assert A ^ B ^ n ^ D ^ X == ((-x - y - z + S(1))/2)*(ex^ey^ez^n^nbar)
L = (A ^ B ^ e) ^ X
assert L == -z*(ex^ey^ez^n) - z*(ex^ey^ez^nbar) + (-x**2/2 + x - y**2/2 + y - z**2/2 - S(1)/2)*(ex^ey^n^nbar) + z*(ex^ez^n^nbar) - z*(ey^ez^n^nbar)
return
def derivatives_in_spherical_coordinates():
Print_Function()
coords = (r,th,phi) = symbols('r theta phi', real=True)
(sp3d,er,eth,ephi) = Ga.build('e_r e_theta e_phi',g=[1,r**2,r**2*sin(th)**2],coords=coords,norm=True)
grad = sp3d.grad
f = sp3d.mv('f','scalar',f=True)
A = sp3d.mv('A','vector',f=True)
B = sp3d.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 'grad\\times A = -I*(grad^A) =',-sp3d.i*(grad^A)
print '%\\nabla^{2}f =',grad|(grad*f)
print 'grad^B =',grad^B
"""
print '( \\nabla\\W\\nabla )\\bm{e}_{r} =',((grad^grad)*er).trigsimp()
print '( \\nabla\\W\\nabla )\\bm{e}_{\\theta} =',((grad^grad)*eth).trigsimp()
print '( \\nabla\\W\\nabla )\\bm{e}_{\\phi} =',((grad^grad)*ephi).trigsimp()
"""
return
def test_derivatives_in_rectangular_coordinates():
X = (x, y, z) = symbols('x y z')
o3d, ex, ey, ez = Ga.build('e_x e_y e_z', g=[1,1,1], coords=X)
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)
assert str(f) == 'f'
assert str(A) == 'A__x*e_x + A__y*e_y + A__z*e_z'
assert str(B) == 'B__xy*e_x^e_y + B__xz*e_x^e_z + B__yz*e_y^e_z'
assert str(C) == 'C + C__x*e_x + C__y*e_y + C__z*e_z + C__xy*e_x^e_y + C__xz*e_x^e_z + C__yz*e_y^e_z + C__xyz*e_x^e_y^e_z'
assert str(grad*f) == 'D{x}f*e_x + D{y}f*e_y + D{z}f*e_z'
assert str(grad | A) == 'D{x}A__x + D{y}A__y + D{z}A__z'
assert str(grad*A) == 'D{x}A__x + D{y}A__y + D{z}A__z + (-D{y}A__x + D{x}A__y)*e_x^e_y + (-D{z}A__x + D{x}A__z)*e_x^e_z + (-D{z}A__y + D{y}A__z)*e_y^e_z'
assert str(-o3d.I()*(grad ^ A)) == '(-D{z}A__y + D{y}A__z)*e_x + (D{z}A__x - D{x}A__z)*e_y + (-D{y}A__x + D{x}A__y)*e_z'
assert str(grad*B) == '(-D{y}B__xy - D{z}B__xz)*e_x + (D{x}B__xy - D{z}B__yz)*e_y + (D{x}B__xz + D{y}B__yz)*e_z + (D{z}B__xy - D{y}B__xz + D{x}B__yz)*e_x^e_y^e_z'
assert str(grad ^ B) == '(D{z}B__xy - D{y}B__xz + D{x}B__yz)*e_x^e_y^e_z'
assert str(grad | B) == '(-D{y}B__xy - D{z}B__xz)*e_x + (D{x}B__xy - D{z}B__yz)*e_y + (D{x}B__xz + D{y}B__yz)*e_z'
assert str(grad < A) == 'D{x}A__x + D{y}A__y + D{z}A__z'
assert str(grad > A) == 'D{x}A__x + D{y}A__y + D{z}A__z'
assert str(grad < B) == '(-D{y}B__xy - D{z}B__xz)*e_x + (D{x}B__xy - D{z}B__yz)*e_y + (D{x}B__xz + D{y}B__yz)*e_z'
assert str(grad > B) == '0'
assert str(grad < C) == 'D{x}C__x + D{y}C__y + D{z}C__z + (D{x}C - D{y}C__xy - D{z}C__xz)*e_x + (D{y}C + D{x}C__xy - D{z}C__yz)*e_y + (D{z}C + D{x}C__xz + D{y}C__yz)*e_z + D{z}C__xyz*e_x^e_y - D{y}C__xyz*e_x^e_z + D{x}C__xyz*e_y^e_z'
assert str(grad > C) == 'D{x}C__x + D{y}C__y + D{z}C__z + D{x}C*e_x + D{y}C*e_y + D{z}C*e_z'
return
def main():
Print_Function()
(a, b, c) = abc = symbols('a,b,c',real=True)
(o3d, ea, eb, ec) = Ga.build('e_a e_b e_c', g=[1, 1, 1], coords=abc)
grad = o3d.grad
x = symbols('x',real=True)
A = o3d.lt([[x*a*c**2,x**2*a*b*c,x**2*a**3*b**5],\
[x**3*a**2*b*c,x**4*a*b**2*c**5,5*x**4*a*b**2*c],\
[x**4*a*b**2*c**4,4*x**4*a*b**2*c**2,4*x**4*a**5*b**2*c]])
print 'A =',A
v = a*ea+b*eb+c*ec
print 'v =',v
f = v|A(v)
print r'%f = v\cdot \f{A}{v} =',f
(grad * f).Fmt(3,r'%\nabla f')
Av = A(v)
print r'%\f{A}{v} =', Av
(grad * Av).Fmt(3,r'%\nabla \f{A}{v}')
return
def test_substitution():
o3d, e_x, e_y, e_z = Ga.build('e_x e_y e_z', g='1 0 0, 0 1 0, 0 0 1')
x, y, z = symbols('x y z')
X = x*e_x + y*e_y + z*e_z
Y = X.subs([(x, 2), (y, 3), (z, 4)])
assert Y == 2*e_x + 3*e_y + 4*e_z
def test_rmul():
"""
Test for commutative scalar multiplication. Leftover from when sympy and
numpy were not working together and __mul__ and __rmul__ would not give the
same answer.
"""
g3d, x, y, z = Ga.build('x y z')
a, b, c = symbols('a b c')
assert 5*x == x*5
assert Rational(1, 2)*x == x*Rational(1, 2)
assert a*x == x*a
def test_derivative():
coords = x, y, z = symbols('x y z')
o3d, e_x, e_y, e_z = Ga.build('e', g=[1, 1, 1], coords=coords)
grad = o3d.grad
X = x*e_x + y*e_y + z*e_z
a = o3d.mv('a', 'vector')
assert (grad * (X | a)) == a
assert (grad * (X*X)) == 2*X
assert grad * (X*X*X) == 5*X*X
assert (grad *X).scalar() == 3
def test_constructor():
"""
Test various multivector constructors
"""
o3d, e_1, e_2, e_3 = Ga.build('e_1 e_2 e_3', g=[1,1,1])
assert str(o3d.mv('a', 'scalar')) == 'a'
assert str(o3d.mv('a', 'vector')) == 'a__1*e_1 + a__2*e_2 + a__3*e_3'
assert str(o3d.mv('a', 'pseudo')) == 'a__123*e_1^e_2^e_3'
assert str(o3d.mv('a', 'spinor')) == 'a + a__12*e_1^e_2 + a__13*e_1^e_3 + a__23*e_2^e_3'
assert str(o3d.mv('a', 'mv')) == 'a + a__1*e_1 + a__2*e_2 + a__3*e_3 + a__12*e_1^e_2 + a__13*e_1^e_3 + a__23*e_2^e_3 + a__123*e_1^e_2^e_3'
assert str(o3d.mv('a', 'bivector')) == 'a__12*e_1^e_2 + a__13*e_1^e_3 + a__23*e_2^e_3'
def Mv_setup_options():
Print_Function()
(o3d,e1,e2,e3) = Ga.build('e_1 e_2 e_3',g=[1,1,1])
v = o3d.mv('v', 'vector')
print v
(o3d,e1,e2,e3) = Ga.build('e*1|2|3',g=[1,1,1])
v = o3d.mv('v', 'vector')
print v
(o3d,e1,e2,e3) = Ga.build('e*x|y|z',g=[1,1,1])
v = o3d.mv('v', 'vector')
print v
coords = symbols('x y z',real=True)
(o3d,e1,e2,e3) = Ga.build('e',g=[1,1,1],coords=coords)
v = o3d.mv('v', 'vector')
print v
return
def make_cf3d():
if Cf3d.ga_flg:
return
Cf3d.ga_flg = True
Cf3d.basis_sups = ['__x','__y','__z']
Cf3d.basis = 'e_x e_y e_z e ebar'
(Cf3d.cf3d,Cf3d.ex,Cf3d.ey,Cf3d.ez,Cf3d.e,Cf3d.ebar) = Ga.build(Cf3d.basis,g=[1,1,1,1,-1])
Cf3d.xyz = (Cf3d.ex,Cf3d.ey,Cf3d.ez)
Cf3d.n = Cf3d.e + Cf3d.ebar
Cf3d.nbar = Cf3d.e - Cf3d.ebar
return
def test_contraction():
"""
Test for inner product and left and right contraction
"""
o3d, e_1, e_2, e_3 = Ga.build('e_1 e_2 e_3', g='1 0 0, 0 1 0, 0 0 1')
assert ((e_1 ^ e_3) | e_1) == -e_3
assert ((e_1 ^ e_3) > e_1) == -e_3
assert (e_1 | (e_1 ^ e_3)) == e_3
assert (e_1 < (e_1 ^ e_3)) == e_3
assert ((e_1 ^ e_3) < e_1) == 0
assert (e_1 > (e_1 ^ e_3)) == 0
def test_rounding_numerical_components():
(o3d, ex, ey, ez) = Ga.build('e_x e_y e_z', g=[1,1,1])
X = 1.2*ex + 2.34*ey + 0.555*ez
Y = 0.333*ex + 4*ey + 5.3*ez
assert str(X) == '1.2*e_x + 2.34*e_y + 0.555*e_z'
assert str(Nga(X, 2)) == '1.2*e_x + 2.3*e_y + 0.55*e_z'
assert str(X*Y) == '12.7011000000000 + 4.02078*e_x^e_y + 6.175185*e_x^e_z + 10.182*e_y^e_z'
assert str(Nga(X*Y, 2)) == '13. + 4.0*e_x^e_y + 6.2*e_x^e_z + 10.0*e_y^e_z'
return
def test_metrics():
"""
Test specific metrics (diagpq, arbitrary_metric, arbitrary_metric_conformal)
"""
o3d, p1, p2, p3 = Ga.build('p1 p2 p3', g=[1,1,1], debug=0)
x1, y1, z1 = symbols('x1 y1 z1')
x2, y2, z2 = symbols('x2 y2 z2')
v1 = x1*p1 + y1*p2 + z1*p3
v2 = x2*p1 + y2*p2 + z2*p3
prod1 = v1*v2
prod2 = (v1|v2) + (v1^v2)
diff = prod1 - prod2
assert diff == o3d.mv(S.Zero)
g3d, p1, p2, p3 = Ga.build('p1 p2 p3', g='# # #, # # #, # # #', debug=0)
v1 = x1*p1 + y1*p2 + z1*p3
v2 = x2*p1 + y2*p2 + z2*p3
prod1 = v1*v2
prod2 = (v1|v2) + (v1^v2)
diff = prod1 - prod2
assert diff == g3d.mv(S.Zero)
def test_geometry():
"""
Test conformal geometric description of circles, lines, spheres, and planes.
"""
metric = '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'
cf3d, e0, e1, e2, n, nbar = Ga.build('e0 e1 e2 n nbar', g=metric)
e = n + nbar
#conformal representation of points
A = F(e0, n, nbar) # point a = (1,0,0) A = F(a)
B = F(e1, n, nbar) # point b = (0,1,0) B = F(b)
C = F(-1*e0, n, nbar) # point c = (-1,0,0) C = F(c)
D = F(e2, n, nbar) # point d = (0,0,1) D = F(d)
x0, x1, x2 = symbols('x0 x1 x2')
x = x0 * e0 + x1 * e1 + x2 * e2
X = F(x, n, nbar)
Circle = A ^ B ^ C ^ X
Line = A ^ B ^ n ^ X
Sphere = A ^ B ^ C ^ D ^ X
Plane = A ^ B ^ n ^ D ^ X
#Circle through a, b, and c
Circle_test = -x2*(e0 ^ e1 ^ e2 ^ n) + x2*(
e0 ^ e1 ^ e2 ^ nbar) + Rational(1, 2)*(-1 + x0**2 + x1**2 + x2**2)*(e0 ^ e1 ^ n ^ nbar)
diff = Circle - Circle_test
assert diff == S.Zero
#Line through a and b
Line_test = -x2*(e0 ^ e1 ^ e2 ^ n) + \
Rational(1, 2)*(-1 + x0 + x1)*(e0 ^ e1 ^ n ^ nbar) + \
(Rational(1, 2)*x2)*(e0 ^ e2 ^ n ^ nbar) + \
(-Rational(1, 2)*x2)*(e1 ^ e2 ^ n ^ nbar)
diff = Line - Line_test
assert diff == S.Zero
#Sphere through a, b, c, and d
Sphere_test = Rational(1, 2)*(1 - x0**2 - x1**2 - x2**2)*(e0 ^ e1 ^ e2 ^ n ^ nbar)
diff = Sphere - Sphere_test
assert diff == S.Zero
#Plane through a, b, and d
Plane_test = Rational(1, 2)*(1 - x0 - x1 - x2)*(e0 ^ e1 ^ e2 ^ n ^ nbar)
diff = Plane - Plane_test
assert diff == S.Zero
def test_check_generalized_BAC_CAB_formulas():
g5d, a, b, c, d, e = Ga.build('a b c d e')
assert str(a | (b*c)) == '-(a.c)*b + (a.b)*c'
assert str(a | (b ^ c)) == '-(a.c)*b + (a.b)*c'
assert str(a | (b ^ c ^ d)) == '(a.d)*b^c - (a.c)*b^d + (a.b)*c^d'
assert str((a | (b ^ c)) + (c | (a ^ b)) + (b | (c ^ a))) == '0'
assert str(a*(b ^ c) - b*(a ^ c) + c*(a ^ b)) == '3*a^b^c'
assert str(a*(b ^ c ^ d) - b*(a ^ c ^ d) + c*(a ^ b ^ d) - d*(a ^ b ^ c)) == '4*a^b^c^d'
assert str((a ^ b) | (c ^ d)) == '-(a.c)*(b.d) + (a.d)*(b.c)'
assert str(((a ^ b) | c) | d) == '-(a.c)*(b.d) + (a.d)*(b.c)'
assert str(Com(a ^ b, c ^ d)) == '-(b.d)*a^c + (b.c)*a^d + (a.d)*b^c - (a.c)*b^d'
assert str((a | (b ^ c)) | (d ^ e)) == '(-(a.b)*(c.e) + (a.c)*(b.e))*d + ((a.b)*(c.d) - (a.c)*(b.d))*e'
return
def basic_multivector_operations_2D():
Print_Function()
(g2d,ex,ey) = Ga.build('e*x|y')
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 main():
Eprint()
(o3d,ex,ey,ez) = Ga.build('e*x|y|z',g=[1,1,1])
u = o3d.mv('u','vector')
v = o3d.mv('v','vector')
w = o3d.mv('w','vector')
print u
print v
uv = u^v
print uv
print uv.is_blade()
exp_uv = uv.exp()
print 'exp(uv) =', exp_uv
return
def test_derivatives_in_spherical_coordinates():
X = (r, th, phi) = symbols('r theta phi')
sph3d, er, eth, ephi = Ga.build('e_r e_theta e_phi', g=[1,r**2,r**2*sin(th)**2], coords=X, norm=True)
grad = sph3d.grad
f = sph3d.mv('f', 'scalar', f=True)
A = sph3d.mv('A', 'vector', f=True)
B = sph3d.mv('B', 'bivector', f=True)
assert str(f) == 'f'
assert str(A) == 'A__r*e_r + A__theta*e_theta + A__phi*e_phi'
assert str(B) == 'B__rtheta*e_r^e_theta + B__rphi*e_r^e_phi + B__thetaphi*e_theta^e_phi'
assert str(grad*f) == 'D{r}f*e_r + D{theta}f*e_theta/r + D{phi}f*e_phi/(r*sin(theta))'
assert str(grad | A) == 'D{r}A__r + (A__r + D{theta}A__theta)/r + (A__r*sin(theta) + A__theta*cos(theta) + D{phi}A__phi)/(r*sin(theta))'
assert str(-sph3d.I()*(grad ^ A)) == '(A__phi/tan(theta) + D{theta}A__phi - D{phi}A__theta/sin(theta))*e_r/r + (-r*D{r}A__phi - A__phi + D{phi}A__r/sin(theta))*e_theta/r + (r*D{r}A__theta + A__theta - D{theta}A__r)*e_phi/r'
assert str(grad ^ B) == '(r*D{r}B__thetaphi - B__rphi/tan(theta) + 2*B__thetaphi - D{theta}B__rphi + D{phi}B__rtheta/sin(theta))*e_r^e_theta^e_phi/r'
return
def derivatives_in_elliptic_cylindrical_coordinates():
Print_Function()
a = symbols('a', real=True)
coords = (u,v,z) = symbols('u v z', real=True)
(elip3d,er,eth,ephi) = Ga.build('e_u e_v e_z',X=[a*cosh(u)*cos(v),a*sinh(u)*sin(v),z],coords=coords,norm=True)
grad = elip3d.grad
f = elip3d.mv('f','scalar',f=True)
A = elip3d.mv('A','vector',f=True)
B = elip3d.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) =',-elip3d.i*(grad^A)
print 'grad^B =',grad^B
return
def derivatives_in_prolate_spheroidal_coordinates():
Print_Function()
a = symbols('a', real=True)
coords = (xi,eta,phi) = symbols('xi eta phi', real=True)
(ps3d,er,eth,ephi) = Ga.build('e_xi e_eta e_phi',X=[a*sinh(xi)*sin(eta)*cos(phi),a*sinh(xi)*sin(eta)*sin(phi),
a*cosh(xi)*cos(eta)],coords=coords,norm=True)
grad = ps3d.grad
f = ps3d.mv('f','scalar',f=True)
A = ps3d.mv('A','vector',f=True)
B = ps3d.mv('B','bivector',f=True)
print 'f =',f
print 'A =',A
print 'B =',B
print 'grad*f =',grad*f
print 'grad|A =',grad|A
(-ps3d.i*(grad^A)).Fmt(3,'-I*(grad^A)')
(grad^B).Fmt(3,'grad^B')
return
def Dirac_Equation_in_Geog_Calculus():
Print_Function()
coords = symbols('t x y z',real=True)
(st4d,g0,g1,g2,g3) = Ga.build('gamma*t|x|y|z',g=[1,-1,-1,-1],coords=coords)
I = st4d.i
(m,e) = symbols('m e')
psi = st4d.mv('psi','spinor',f=True)
A = st4d.mv('A','vector',f=True)
sig_z = g3*g0
print '\\text{4-Vector Potential\\;\\;}\\bm{A} =',A
print '\\text{8-component real spinor\\;\\;}\\bm{\\psi} =',psi
dirac_eq = (st4d.grad*psi)*I*sig_z-e*A*psi-m*psi*g0
dirac_eq = dirac_eq.simplify()
dirac_eq.Fmt(3,r'%\text{Dirac Equation\;\;}\nabla \bm{\psi} I \sigma_{z}-e\bm{A}\bm{\psi}-m\bm{\psi}\gamma_{t} = 0')
return
