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I am so confused at the definition for integrate differential forms over manifold.

I am looking at the definition introduced in N. Hitchin's note. You can see it at http://people.maths.ox.ac.uk/hitchin/hitchinnotes/manifolds2012.pdf. The definition is on the lower half of page 61.

He says:

If $\omega$ is a differential form, $M$ is an oriented manifold, and $\{(U_\alpha, \psi_\alpha)\}$ is the atlas on it, such that $\{U_\alpha\}$ covers $M$, and that the determinant of the Jacobian of the change of coordinates is positive, then we choose a partition of unity $\{\varphi_\alpha\}$ subordinate to the covering $\{U_\alpha\}$. Then we look at $\omega|_{U_\alpha}$. We can write it in local coordinates as $$ \omega|_{U_\alpha}=f_\alpha(x_1,\ldots,x_n)dx_1\wedge\cdots\wedge dx_n $$ Then, $$ \varphi_i\omega|_{U_\alpha}=g_i(x_1,\ldots,x_n)dx_1\wedge\cdots\wedge dx_n $$ Now, here's a part that I have trouble with. He claims that this $g_i$ is a compactly differentiable function defined on all $\mathbb{R}^n$. I can see how $g_i$ can be a differentialable function defined on $\psi_\alpha(U_\alpha)\subset \mathbb{R}^n$. But how does this extends to a globally defined differentiable function?? Indeed, you only can write $\varphi_\alpha$ in local coordinates defined on a subset of $\mathbb{R}^n$, since it's a function on the manifold.

I appreciate any help! Thanks!

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up vote 4 down vote accepted

The coordinate chart $\psi_\alpha$ embeds $U_\alpha$ into $\mathbb{R}^n$. Since $\phi_\alpha$ has support on $U_\alpha$, we can define $g$ as an "extension" of $\phi_\alpha$ to $\mathbb{R}^n$ by $$g(x) = \begin{cases}\phi_\alpha(\psi_\alpha^{-1}(x)), & x\in \psi_\alpha (\operatorname{supp}\phi_\alpha), \\ 0, & \mbox{else.} \end{cases}$$ As long as our partition of unity has compact support (which we may do by refining the open cover), then $g$ has compact support. All functions involved are smooth, so $g$ is smooth.

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