"Electromagnetic tensor" -- u for up index, d for down index gdd = ((1,0,0,0),(0,-1,0,0),(0,0,-1,0),(0,0,0,-1)) guu = inv(gdd) A = (Ax(),Ay(),Az()) Au = (phi(),Ax(),Ay(),Az()) Ad = dot(gdd,Au) B = curl(A) E = -d(phi(),(x,y,z)) - d(A,t) Bx = B[1] By = B[2] Bz = B[3] Ex = E[1] Ey = E[2] Ez = E[3] X = (t,x,y,z) Add = d(Ad,X) Fdd = transpose(Add) - Add "Verify equation (1)" check(Fdd == ((0,Ex,Ey,Ez),(-Ex,0,-Bz,By),(-Ey,Bz,0,-Bx),(-Ez,-By,Bx,0))) "ok" "Verify equation (2)" Fuu = dot(guu,Fdd,guu) check(contract(dot(transpose(Fdd),Fuu)) == 2 dot(B,B) - 2 dot(E,E)) "ok" "Verify equation (3)" check(det(Fdd) == dot(B,E)^2) check(det(Fuu) == dot(B,E)^2) "ok" Ju = contract(d(Fuu,X),1,3) "Verify equation (4)" check(contract(d(Ju,X)) == 0) "ok" "Verify equation (5)" Jx = Ju[2] Jy = Ju[3] Jz = Ju[4] check((Jx,Jy,Jz) == curl(B) - d(E,t)) "ok" Run