Induced orientation without tangent space and outward pointing vector I know that there is a definition of orientability of a manifold (M) that only relies on the atlas (A) of the manifold. If the Jacobian of any transition function of any two overlapping coordinate charts is everywhere positive on the domain of the transition function, then (M, A) is called oriented. If there is such an atlas for M, then M is called orientable. This definition does not need the concept of "tangent space".
In contrast, the usual definition of "induced orientation" for the boundary of a manifold with boundary relies on the concepts of "tangent space" and an "outward pointing vector". I just wonder whether there is an alternate definition for induced orientation that only relies on A without the concept of tangent space and outward pointing vector.
 A: After reading Munkres' Analysis on Manifolds, I have found an answer to my question. First of all, under Munkres' theory, the coordinate chart of a subset of a $k$ dimensional manifold $M$ (where $k > 1$) containing a boundary point should be one whose domain is a subset of $H^k$ instead of $\mathbb{R}^k$ (Munkres defines a coordinate chart as a mapping from an open subset of the Euclidean space to the manifold rather than the other way around). Here $H^k$ is the subspace of $\mathbb{R}^k$ with $x_k \geq 0$.
Munkres also defines an operation called "restriction". If $\phi: O \rightarrow U$ is a coordinate chart where $O$ is a subset of $H^k$ and $U$ is a subset of $M$ containing part of $\partial M$, the boundary of $M$, then one can get a coordinate chart of $U \cap \partial M$ by letting $x_k = 0$ in $\phi$.
Then Munkres defines the induced orientation of $\partial M$ (where dim $M = k > 1$) as follows (see P.288 of the book). If $k$ is even, it is the orientation obtained by simply restricting coordinate charts belonging to the orientation of $M$. If $k$ is odd, it is the opposite of the orientation of $\partial M$ obtained in this way.
For $k = 1$, the situation is a bit different. According to Munkres, the coordinate charts of subsets of $1$-dimensional manifolds containing $\partial M$ can be either $H^1$, which is in fact the right half real line; and $L^1$, which is the left half real line.
Then Munkres defines the induced orientation of $p \in \partial M$ (where dim $M = k = 1$) as follows (see P.307 of the book). If the domain of the coordinate chart of the set around $p$ is a subset of $H^1$, the induced orientation of $p$ is $-1$. If the domain of the coordinate chart of the set around $p$ is a subset of $L^1$, the induced orientation of $p$ is $+1$.
