print 'K|X =',KX
print 'F =',F
(EBkst.grad*F).Fmt(3,'grad*F = 0')
gradF_reduced = (EBkst.grad*F)/(I*exp(I*KX))
gradF_reduced.Fmt(3,r'%\lp\bm{\nabla}F\rp/\lp ie^{iK\cdot X} \rp = 0')
print r'%\mbox{Previous equation requires that: }e_{E}\cdot e_{B} = 0'+\
r'\mbox{ if }B\ne 0\mbox{ and }k\ne 0'
gradF_reduced = gradF_reduced.subs({EBkst.g[0,1]:0})
gradF_reduced.Fmt(3,r'%\lp\bm{\nabla}F\rp/\lp ie^{iK\cdot X} \rp = 0')
(coefs,bases) = linear_expand(gradF_reduced.obj)
eq1 = coefs[0]
eq2 = coefs[1]
B1 = solve(eq1,B)[0]
B2 = solve(eq2,B)[0]
print r'\mbox{eq1: }B =',B1
print r'\mbox{eq2: }B =',B2
eq3 = B1-B2
print r'\mbox{eq3 = eq1-eq2: }0 =',eq3
eq3 = simplify(eq3 / E)
print r'\mbox{eq3 = (eq1-eq2)/E: }0 =',eq3
def EM_Waves_in_Geom_Calculus():
#Print_Function()
X = (t,x,y,z) = 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')
E = st4d.mv('E','vector')
B.set_coef(1,0,0)
E.set_coef(1,0,0)
B *= g0
E *= g0
F = E+i*B
kx, ky, kz, w = symbols('k_x k_y k_z omega',real=True)
kv = kx*g1+ky*g2+kz*g3
xv = x*g1+y*g2+z*g3
KX = ((w*g0+kv)|(t*g0+xv)).scalar()
Ixyz = g1*g2*g3
F = F*exp(I*KX)
print r'\text{Pseudo Scalar\;\;}I =',i
print r'%I_{xyz} =',Ixyz
F.Fmt(3,'\\text{Electromagnetic Field Bi-Vector\\;\\;} F')
gradF = st4d.grad*F
print '#Geom Derivative of Electomagnetic Field Bi-Vector'
gradF.Fmt(3,'grad*F = 0')
gradF = gradF / (I * exp(I*KX))
gradF.Fmt(3,r'%\lp\bm{\nabla}F\rp /\lp i e^{iK\cdot X}\rp = 0')
g = '1 # 0 0,# 1 0 0,0 0 1 0,0 0 0 -1'
X = (xE,xB,xk,t) = symbols('x_E x_B x_k t',real=True)
(EBkst,eE,eB,ek,et) = Ga.build('e_E e_B e_k t',g=g,coords=X)
i = EBkst.i
E,B,k,w = symbols('E B k omega',real=True)
F = E*eE*et+i*B*eB*et
kv = k*ek+w*et
xv = xE*eE+xB*eB+xk*ek+t*et
KX = (kv|xv).scalar()
F = F*exp(I*KX)
print r'%\mbox{set } e_{E}\cdot e_{k} = e_{B}\cdot e_{k} = 0'+\
r'\mbox{ and } e_{E}\cdot e_{E} = e_{B}\cdot e_{B} = e_{k}\cdot e_{k} = -e_{t}\cdot e_{t} = 1'
print 'g =', EBkst.g
print 'K|X =',KX
print 'F =',F
(EBkst.grad*F).Fmt(3,'grad*F = 0')
gradF_reduced = (EBkst.grad*F)/(I*exp(I*KX))
gradF_reduced.Fmt(3,r'%\lp\bm{\nabla}F\rp/\lp ie^{iK\cdot X} \rp = 0')
print r'%\mbox{Previous equation requires that: }e_{E}\cdot e_{B} = 0\mbox{ if }B\ne 0\mbox{ and }k\ne 0'
gradF_reduced = gradF_reduced.subs({EBkst.g[0,1]:0})
gradF_reduced.Fmt(3,r'%\lp\bm{\nabla}F\rp/\lp ie^{iK\cdot X} \rp = 0')
(coefs,bases) = linear_expand(gradF_reduced.obj)
eq1 = coefs[0]
eq2 = coefs[1]
B1 = solve(eq1,B)[0]
B2 = solve(eq2,B)[0]
print r'\mbox{eq1: }B =',B1
print r'\mbox{eq2: }B =',B2
eq3 = B1-B2
print r'\mbox{eq3 = eq1-eq2: }0 =',eq3
eq3 = simplify(eq3 / E)
print r'\mbox{eq3 = (eq1-eq2)/E: }0 =',eq3
print 'k =',Matrix(solve(eq3,k))
print 'B =',Matrix([B1.subs(w,k),B1.subs(-w,k)])
return
