How do I show that the projection map $π : U → R^n , π (x_1 , . . . , x_n , x_{n+1} ) = (x_1 , . . . , x_n )$, is orientation-preserving if and only if $n$ is even
My idea is to calculate the Jacobian of the map and find out if it is greater than zero. Is there a way of doing this without calculating Jacobian?