Spivak start the section 'Stokes theorem on manifolds' of his book 'Calculus on manifolds' with the following definition:

If $\omega$ is a $p$-form on a $k$-dimensional manifold with boundary $M$ and $c$ is a singular $p$-cube in $M$, we define $$ \int_{c}\omega=\int_{[0,1]^{p}}c^{*}\omega $$ precisely as before; integrals over $p$-chains are also defined as before. In the case p=k it may happend that there is an open set $W\supset[0,1]^{k}$ and a coordinate system $f:W\to\mathbb{R}^{n}$ such that $c(x)=f(x)$ for $x\in[0,1]^{k}$; a $k$-cube in $M$ will always be understood to be of this type. If $M$ is oriented, the singular $k$-cube $c$ is called orientation-preserving if $f$ is.

He then defines:

Let $\omega$ be a $k$-form on an oriented $k$-dimensional manifold $M$. If there is an orientation preserving singular $k$-cube $c$ in $M$ such that $\omega=0$ outside of $c([0,1]^{k})$, we define $$ \int_{M}\omega=\int_{c}\omega $$

And then extends the definition:

Suppose now that $\omega$ is an arbitrary $k$-form on $M$. There is an open cover $\mathcal{O}$ of $M$ such that for each $U\in\mathcal{O}$ there is an orientation preserving singular $k-$cube $c$ with $U\subset c([0,1]^{k})$. Let $\Phi$ be a partition of unity subordinate to this cover. We define $$ \int_{M}\omega=\sum_{\varphi\in\Phi}\int_{M}\varphi\cdot\omega $$

The implication I don't understand is the following:

All our definitions could have been given for a $k$-dimensional manifold with boundary $M$ with orientation $\mu$. Let $\partial M$ have the induced orientation $\partial\mu$. Let $c$ be an orientation preserving k-cube in $M$ such that $c_{(k,0)} $ lies in $\partial M$ and is the only face that has any interior points in $\partial M$. As the remarks after the definition of $\partial\mu$ show, $c_{(k,0)}$ is orientation preserving if k is even, but not if k is odd. Thus if $\omega$ is a $(k-1)$-form on $M$ wich is 0 outside of $c([0,1]^{k})$, we have $$ (1)\hspace{0.5 cm}\int_{c_{(k,0)}}\omega=(-1)^k\int_{\partial M}\omega $$ On the other hand, $c_{(k,0)}$ appears with coefficient $(-1)^{k}$ in $\partial c$. Therefore $$ (2)\hspace{0.5 cm}\int_{\partial c}\omega=\int_{(-1)^{k}c_{(k,0)}}\omega=(-1)^{k}\int_{c_{(k,0)}}\omega =\int_{\partial M}\omega $$

For the notation $c_{(i,\alpha)}=c\circ(I^n_{(i,\alpha)})$ and $I^n_{(i,\alpha)}=(x^1,\cdots,x^{i-1},\alpha,x^i,\cdots,x^{n-1})$.

My questions are as follows

  • Why it follows from the remarks that $c_{(k,0)}$ is orientation preserving if k is even but not if k is odd?
  • In (1) where does the coefficient $(-1)^{k}$ comes from?
  • Why the first equality in (2) holds?

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