I'm trying to prove the following: Let $M$ be a smooth manifold, $\omega\in \Gamma(\bigwedge^n TM^*)$ non-vanishing. Then $M$ is orientable. My approach is the following. I want to define an atlas of charts, s.t. for each chart $\varphi$ on some $U\subset M$ it holds: $\omega(d\varphi(v_1),...,d \varphi(v_n))=\lambda de^1 \wedge... \wedge de^n$ for some $\lambda>0$, where $v_1,..,v_n \in TM$ and $e_i$ are the standard coordinates on $\mathbb{R}^n$. Now to show that this works I need to
1) guarantee that I can cover $M$ with such charts
2) Show that this imples orientability in the sense that the jacobian of the transition maps are positive.
For 2), my idea is to write for two such charts $det(d (\varphi \circ \psi))=det(d \varphi \cdot d \psi)=w(v_1,...,v_n)^2 >0$ for suitably chosen $(v_1,...,v_n)$.
Let $\varphi=x_1,...,x_n)$. Now: $w(\dfrac{\partial}{\partial x_1},...,\dfrac{\partial}{\partial x_n})= \\ \lambda de^1 \wedge...\wedge de^n(d \varphi (\dfrac{\partial}{\partial x_1}),..., d \varphi ( \dfrac{\partial}{\partial x_n}) = \\ \lambda \begin{pmatrix} de^1(d \varphi (\dfrac{\partial}{\partial x_1})) & ... & de^1(d \varphi (\dfrac{\partial}{\partial x_n}))\\ ... & & ... \\ de^n(d \varphi (\dfrac{\partial}{\partial x_1})) &...& de^n(d \varphi (\dfrac{\partial}{\partial x_n})) \end{pmatrix}$
correct? Can I somehow transform this matrix into $d \varphi$? And is there a way to write $w(\dfrac{\partial}{\partial x_1},...,\dfrac{\partial}{\partial x_n})$ as $d \psi^{-1}$? Because then the proof would be finished, right?
And how to show 1)? Or is maybe my whole approach wrong?
Thanks in advance!