I am an undergraduate student who is studying Spivak's calculus on manifolds. I have several questions in the pages 119 and 120 of the book, which are about the tangent space at a boundary point of a manifold.

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1) I think I understand why there are exactly two unit vectors in $M_x$ which are perpendicular to $(\partial M)_x$. This is because the dimension of the subspace of $M_x$ that is orthogonal to $(\partial M)_x$ is 1, and in a one dimensional space, there are two unit vectors which have the same size but the different directions (Is this correct?). And Spivak says in the picture that "..., then only one of these unit vectors is $f_*(v_0)$ for some $v_0$ with $v^k < 0$. Is it right that "these unit vectors" mean the two unit vectors perpendicular to $(\partial M)_x$ and that it is 'only one' because it cannot have two different directions at the same time?

2) Spivak says it is easy to see that given an orientation of a k dimensional manifold with boundary $M$, the orientations $(\partial \mu)_x$ for $x \in \partial M$ are consistent on $(\partial M)_x$. My reasoning is that following the Spivak's definition of consistent choices of orientations, consider a chart $f: W \subset H^k \rightarrow \mathbb{R}^n$ and $a, b \in W$ ($a$ is an interior point and $b$ is an interior point). Because by assumption $M$ is orientable, if $\mu_{f(a)} = [f_*((e_1)a), . . . , f_*((e_k)a)]$, then $\mu_{f(b)} = [f_*((e_1)b), . . . , f_*((e_{k - 1})b), n(b)]$ where $n(b)$ is an outward unit normal. Then, we can choose $[f_*((e_1)b), . . . , f_*((e_{k - 1})b)]$ as $(\partial \mu)_b$. In this way, by restricting $f$ to the intersection of its image and $\partial M$, we can get the same orientation for all $x \in \partial M$; thus, if $M$ is orientable, so is $\partial M$. Please check if my reasoning is okay.

3) At the bottom of the picture, Spivak says the induced orientation on ... is $(-1)^k$ times the usual orientation. I really cannot understand how this happens.

4) In the next page, he says that the vectors $n(x)$ vary continuously on $M$. Also, conversely, if a continuous family of unit normal vectors $n(x)$ is defined on all of $M$, we can determine an orientation of $M$. Could you explain why this is the case?

Thank you!


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