# Inertial Steady 2D Vector Field Topology # Maple sheet to show Eq. (5) restart; with(LinearAlgebra): # 2D vector field U(x,y) in a first-order Taylor approximation uf := ufc + ux*x + uy*y; vf := vfc + vx*x + vy*y; # 4D vector field UTilde(x,y,u,v) in which inertial trajectories are tangent curves ut1 := u; ut2 := v; ut3 := (uf-u)/r + ug; ut4 := (vf-v)/r + vg; # Jabian matrix of U J := Matrix([ [diff(uf,x), diff(uf,y)], [diff(vf,x), diff(vf,y)]]); # Jabian matrix of UTilde # this also proves Eq. (2) J4 := Matrix([ [diff(ut1,x), diff(ut1,y), diff(ut1,u), diff(ut1,v)], [diff(ut2,x), diff(ut2,y), diff(ut2,u), diff(ut2,v)], [diff(ut3,x), diff(ut3,y), diff(ut3,u), diff(ut3,v)], [diff(ut4,x), diff(ut4,y), diff(ut4,u), diff(ut4,v)]]); # e1, e2: eigenvalues of J ev2 := Eigenvalues(J): e1 := ev2[1]; e2 := ev2[2]; # eigenvalues of J4 ev4 := Eigenvalues(J4); # eigenvales of J4 following Eq. (5) f11 := (-1 - sqrt(1+4*r*e1))/(2*r); f12 := (-1 + sqrt(1+4*r*e1))/(2*r); f21 := (-1 - sqrt(1+4*r*e2))/(2*r); f22 := (-1 + sqrt(1+4*r*e2))/(2*r); # if Eq. (5) holds, the following 4 expressions must be zero: simplify(f11 - ev4[2]); simplify(f12 - ev4[1]); simplify(f21 - ev4[4]); simplify(f22 - ev4[3]);