Induced Orientation of $\Delta ^{n}$'s Faces. In Rotman's An Introduction to Algebraic Topology, pp.62~63, Rotman defines some terms involving the word 'orientation'. I'm very confused at his way of defining the terms. He begins by;

Definition. An orientation of $\Delta ^{n}=[e_{0},e_{1},\ldots,e_{n}]$ is a linear ordering of its vertices.
An orientation thus gives a tour of the vertices. For example, the orientation $e_{0}<e_{1}<e_{2}$ of  $\Delta ^{2}$ gives a counterclockwise tour. It is clear that two different orderings can give the same tour; thus $e_{0}<e_{1}<e_{2}$ and $e_{1}<e_{2}<e_{0}$ and $e_{2}<e_{0}<e_{1}$ all give the counterclockwise tour, while the other three orderings (orientations) give a clockwise tour.
If $n=3$, the reader should see that there are essentially only two different tours, corresponding to the left-hand rule and right-hand rule, respectively.
Definition. Two orientations of $\Delta ^{n}$ are the same if, as permutations of $\{e_{0},e_{1},\ldots,e_{n}\}$, they have the same parity(i.e, both are even or both are odd); otherwise the orientations are opposite.
Given an orientation of $\Delta ^{n}$, there is an induced orientation of its faces defined by orienting the $i$th face in the sense $(-1)^{i}[e_{0},\ldots,\hat{e_{i}},\ldots,e_{n}]$, where $-[e_{0},\ldots,\hat{e_{i}},\ldots,e_{n}]$ means the $i$th face (vertex $e_{i}$ deleted) with orientation opposite to the one with the vertices ordered as displayed. ...

My questions are:


*

*When he says the $i$th face in that last paragraph, does he mean the face opposite to $e_{i}$ or the face opposite to the $i$th vertex of the given linear ordering? In more details, for example, when I order the vertices by $e_{3}<e_{2}<e_{1}<e_{0}$ in $\Delta ^{3}$, what induced orientations do I get?

*According to his definition, there are more than one linear orderings of the vertices that give the same orientation on $\Delta ^{n}$. Then is the concept 'induced orientation' well defined? Do the 'same' orientations on $\Delta ^{n}$ give the 'same' induced orientations?

*After those definitions he draws a picture of a triangle as an example and says: 'It is plain that these orientations of the edges are "compatible" with the orientation of $\Delta ^{2}$'. What does he mean by that word "compatible"? I can certainly see something in 2-dimension, but not in higher dimensions...

*What is the motivation behind the definition of an induced orientations? I know that expressions such as $(-1)^{i}[e_{0},\ldots,\hat{e_{i}},\ldots,e_{n}]$ leads to the definition of the boundary operator and the whole theory of singular homology, but I'm not just that motivated enough. The only picture I can draw is just a triangle, and I'm not really sure what have motivated topologists to define things that way. 
I add the picture of page 63 the book. This single pages confuses me a lot that I can't go any further. Any answer will be appreciated. Thanks in advance.

 A: *

*The $i$th face $[e_0,\dots,\hat{e_i},\dots,e_n]$ has all the vertices of $\Delta^n$ except from $e_i$. Thus, it's the face opposite to $e_i$ (this doesn't depend on the ordering). 
For example, if $e_3<e_2<e_1<e_0$, then the 2nd face is $[e_0,e_1,e_3]$ and the induced orientation is $e_3<e_1<e_0$ (since $(-1)^2=1$) and the 3rd face is $[e_0,e_1,e_2]$ and the induced orientation is $e_0<e_1<e_2$ (opposite of $e_2<e_1<e_0$ since $(-1)^3=-1$). 

A: It seems to me that the indexed notation $e_0,e_1,...,e_n$ is used in two different ways in this portion of the book.
First, in the few paragraphs at the top of page 63 up through the paragraph "Definition" (and, I presume, back on page 62), it seems as if the sequence of vertices of $\Delta^n$ has been given a fixed notation $e_0,e_1,...,e_n$. 
On the other hand, starting with the paragraph on the "induced orientation", the notation $e_0,e_1,...,e_n$ is being used instead as the "permuted" notation for a given ordering of the vertices. In other words, whatever notation for the vertices may have been used before, starting in this paragraph there is a given ordering of the vertices and the vertex subscripts have been rewritten so that the given ordering is $e_0 < e_1 < \cdots < e_n$.
Some authors, instead of rewriting vertex subscripts, will instead write different orderings in the form $e_{\sigma(0)} <\cdots < e_{\sigma(n)}$ where $\sigma$ is a permutation of the set $\{0,1,...,n\}$. Other authors  would rather avoid the notational overload of another symbol such as $\sigma$, at the risk of introducing ambiguities which can trip up the reader.
