Being $S $ a oriented surface and $\mathcal {O} $ the orientation, each $\psi\in\mathcal {O} $ induces a orientation on $T_pS $ by the basis $\psi_u,\psi_v $.
I need to prove that this not depends of parametrization.
I am confused about the determinant of changing basis. This is a them that always confused on my mind.
Is correct this?
$(\psi_\tilde {u},\psi_\tilde {v})= \det J (\psi^{-1}○\varphi)(p)(\varphi_u,\varphi_v) $.
Is this the matrix to changing coordinates on tangent plane? If yes or not, could you, please, explain a little bit about this?
In fact, I am understanding the concepts of differential geomtry, but when I have to do the thing by my own, I am with a lot of difficulties.
Many thanks.