equivalent definitions of orientation I know two definitions of an orientation of a smooth n-manifold $M$:
1) A continuous pointwise orientation for $M$. 
2) A continuous choice of generators for the groups $H_n(M,M-\{x\})=\mathbb{Z}$.
Why are these two definitions equivalent? In other words, why is a choice of basis of $\mathbb{R}^n$ equivalent to a choice of generator of $H_n(\mathbb{R}^n,\mathbb{R}^n-\{0\})=\mathbb{Z}$?
See comments for precise definitions.
Thanks!

Recall that an element of $H_n(M,M-\{x\})$ is an equivalence class of singular $n$-chains, where the boundary of any chain in the class lies entirely in $M-\{x\}$.  In particular, any generator of $H_n(M,M-\{x\})$ has a representative consisting of a single singular $n$-simplex $\sigma\colon \Delta^n\to M$, whose boundary lies in $M-\{x\}$. Moreover, the map $\sigma$ can be chosen to be a differentiable embedding.  (Think of $\sigma$ as an oriented simplex in $M$ that contains $x$.)
Now, the domain $\Delta^n$ of $\sigma$ is the standard $n$-simplex, which has a canonical orientation as a subspace of $\mathbb{R^n}$.  Since $\sigma$ is differentiable, we can push this orientation forward via the derivative of $\sigma$ onto the image of $\sigma$ in $M$.  This gives a pointwise orientation on a neighborhood of $x$.
