Hi I have some questions regarding integrals on manifolds.
1) Let $M_n$ be differentiable orientable manifold. The integral of a differential $n$-form $w$ with compact support is:
Let $(\Omega_i, \varphi_i)$ be an atlas compatible with the orientation chosen, and $\{\alpha_i\}$ be a partition of unity subordinate to $\{\Omega_i\}$. On $\Omega_i$, $w = f_i(x)dx_i^1 \wedge … \wedge dx_i^n$. The integral is
$$\int_M w = \sum_i \int_{\varphi_i(\Omega_i)}[\alpha_i(x)f_i(x)]\circ \varphi_i^{-1}dx^1 \wedge … \wedge dx^n $$
This definition confuses me. Why do we need the partition of unity and the inverse of the chart map? Since $x^j$ presumeably denotes the coordinates, can't we just do a normal integral $\int f dx^1 … dx^n$?
2) For a manifold $M$, let $i:\partial M \to M$ be the inclusion map. In Stokes' formula, it is customary to write $\int_{\partial M} w$ to mean $\int_{\partial M} i^*w$.
Can someone explain to me the meaning of this second integral? We are integrating over the boundary of the manifold the integrand which is the pullback of a differential form on $M$. I can't see the intuition at all.
Best Answer
Zhen Lin answered your question (2) in his comment and most of question (1).
To answer the last part of (1) quickly:
Your confusion is coming because you aren't keeping track of where the various functions are defined. Each $\Omega_i \subset M$. The functions $\alpha_i(x) f(x)$ are defined on $\Omega_i$. The term in the integral: \[ \int_{\phi_i(\Omega_i)} [ \alpha_i(x) f_i(x) ] \circ \phi_i^{-1} dx^1 \wedge \dots dx^n \] is integrating in $\mathbb{R}^n$ (over the open set $\phi_i(\Omega_i)$). The composition with $\phi_i^{-1}$ is just "moving" the function $\alpha_i(x) f(x)$ to $\phi_i(\Omega_i)$.