In Tom Apostol calculus volume 2 (page 402) I found that the non zero jacobian determinant of a function whose components have continuous partial derivatives on a set, is either positive or negtive in the whole set.
Why does the determinant has same sign in the whole set?
In the text I found that the sign of the jacobian determinant signifies whether the orientation of a curve (clock or counterclockwise) of a curve will be preserved by that function (positive determinant) or not (negative determinant).
How to prove or disprove that the sign of the determinant signifies whether the orientation of the curve is preserved by the function or not?