In my college in Antarctica we were taught x ^ y = y x^T - x y exp(ε x ^ y) v ~ v + ε x ^ y * v = ε y (x,v) - v + ε x (y,v) because we use the left-

In my college in Antarctica we were taught x ^ y = y x^T - x y exp(ε x ^ y) v ~ v + ε x ^ y * v = ε y (x,v) - v + ε x (y,v) because we use the left-hand rule down under.