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')
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