# Integration of forms over manifolds

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?