Lemma. Expression of $[X,Y]$ on a local chart.
Let $x:U \subset M \rightarrow \mathbb{R}^{n}$ be a local chart. Denote by $\dfrac{\partial}{\partial x_{i}}$ the vector field on $U$ defined by $\dfrac{\partial}{\partial x_{i}}(q) = (dx(q))^{-1}\ . \ e_{i}$, where $\lbrace e_{1},\ldots, e_{n}\rbrace$ is the canonical basis of $\mathbb{R}^{n}$. If $X$ and $Y$ are vector fields on $M$, we can write
$ X = \sum_{i=1}^{n}a_{i}\dfrac{\partial}{\partial x_{i}} \quad , \quad Y = \sum_{j=1}^{n}b_{j}\dfrac{\partial}{\partial x_{j}}. $
Then
$ [X,Y] = \sum_{i=1}^{n}c_{i}\dfrac{\partial}{\partial x_{i}}; \quad \textrm {where} $
$ c_{i} = \sum_{j=1}^{n}(a_{j}\dfrac{\partial b_{i}}{\partial x_{j}} - b_{j}\dfrac{\partial a_{i}}{\partial x_{j}}) $
My progress on the proof of the lemma is as follows:
$[X,Y] \ = \ \sum_{i,j=1}^{n} \left[ a_{i} \dfrac{\partial}{\partial x_{i}} \ , \ b_{j} \dfrac{\partial}{\partial x_{j}}\right] \ = \ \sum_{i,j=1}^{n} \left(\left(a_{i} \dfrac{\partial}{\partial x_{i}}\right) \left(b_{j} \dfrac{\partial}{\partial x_{j}}\right) \ - \ \left(b_{j} \dfrac{\partial}{\partial x_{j}}\right) \left(a_{i} \dfrac{\partial}{\partial x_{i}}\right)\right)$
Applying the product rule
$\sum_{i,j=1}^{n} \left(a_{i}\left( \dfrac{\partial}{\partial x_{i}} b_{j}\right)\dfrac{\partial}{\partial x_{j}} \ + \ a_{i}b_{j} \left( \dfrac{\partial}{\partial x_{i}} \dfrac{\partial}{\partial x_{j}}\right) - b_{j}\left(\dfrac{\partial}{\partial x_{j}} a_{i}\right)\dfrac{\partial}{\partial x_{i}} - b_{j} a_{i}\left( \dfrac{\partial}{\partial x_{j}} \dfrac{\partial}{\partial x_{i}}\right) \right)$
Schwarz's Theorem
$\sum_{i,j=1}^{n} \left( a_{i}\left( \dfrac{\partial}{\partial x_{i}} b_{j}\right)\dfrac{\partial}{\partial x_{j}} \ - \ b_{j}\left(\dfrac{\partial}{\partial x_{j}} a_{i}\right)\dfrac{\partial}{\partial x_{i}}\right)$
$[X,Y] = \sum_{i,j=1}^{n} \left( a_{j}\dfrac{\partial}{\partial x_{j}} b_{j} \ - \ b_{j}\dfrac{\partial}{\partial x_{j}} a_{i} \right) \dfrac{\partial}{\partial x_{i}} = \sum_{i=1}^{n} c_{i} \dfrac{\partial}{\partial x_{i}} $
But I want to prove the following:
If $X$ and $Y$ are two vector fields defined on an open set $U\subset \mathbb{R}^{n}$, then $[X,Y]\ (q)\ = \ DY(q)\ .\ X(q)\ - \ DX(q)\ .\ Y(q) \ ; \ q\in V$
I would appreciate any ideas on how to approach the proof.
