I'm trying to show that the manifold of a tangent bundle is orientable. However, I evidently made a mistake in my proof, and was wondering if someone could help me point out where I'm going wrong. I'm using notation from Lee's "Introduction to Smooth Manifolds."
Suppose $M$ is an $n$-manifold and we're given two charts $(U,\phi),(V,\psi)$ on $M$ with non-trivial intersection. Write $\phi(p) = (x_1(p),\ldots,x_n(p))$, $\psi(p)=(y_1(p),\ldots,y_n(p))$, and for charts on the trivializations, also write
$$ \bar \phi\left(p, \sum_i v_i {\partial\over \partial x_i}|_p\right) = (x_1(p),\ldots,x_n(p),v_1,\ldots,v_n), \\ \bar \psi\left(p, \sum_i v_i {\partial\over \partial y_i}|_p\right) = (y_1(p),\ldots,y_n(p),v_1,\ldots,v_n). $$
We want to show that the Jacobian of $\bar \psi \circ \bar \phi^{-1}$ has positive determinant.
To compute this, I'm getting:
$$\begin{eqnarray} \bar \psi \circ \bar\phi^{-1}(x_1,\ldots,x_n,v_1,\ldots,v_n) &=& \bar \psi (p, \sum_i v_i {\partial \over \partial x_i}|_p) \\ &=& \bar \psi(p, v_1(\sum_i \frac{\partial y_i}{\partial x_1} {\partial \over \partial y_i}|_p) + \ldots + v_n(\sum_i \frac{\partial y_i}{\partial x_n} {\partial \over \partial y_i}|_p)) \\ &=& \bar \psi(p, (\sum_i v_i {\partial y_1 \over \partial x_i}|_p) {\partial \over \partial y_1}|_p + \ldots + (\sum_i v_i {\partial y_n \over \partial x_i}|_p) {\partial \over \partial y_n}|_p) \\ &=& (y_1(p), \ldots, y_n(p), \sum_i v_i {\partial y_1 \over \partial x_i}|_p, \ldots, \sum_i v_i {\partial y_n \over \partial x_i}|_p). \end{eqnarray}$$
Taking the Jacobian of this last expression, I'm getting a matrix of the form
$$\begin{bmatrix} J & 0 \\ 0 & J \end{bmatrix}$$
where $J$ is the Jacobian matrix of $\psi \circ \phi^{-1}$, i.e., the matrix $\left( \frac{\partial y_i}{\partial x_j}|_p \right)_{ij}$.
From here, I was going to argue that the determinant of this block matrix above is positive. However, my professor indicated that that block matrix representation of the Jacobian is wrong. Could anyone tell me why?