In Spivak's Calculus on Manifolds after showing that partitions of unity exists, Spivak defines integrals of functions over open sets as follows. He says:

"An open cover $\mathcal{O}$ of an open set $A\subset \mathbb{R}^n$ is called admissible if each $U\in \mathcal{O}$ is contained in $A$. If $\Phi$ is a partition of unity subordinate to $\mathcal{O}$, $f : A\to \mathbb{R}$ is bounded in some open set around each point of $A$, and the set of discontinuities of $f$ has measure zero, then for each $\varphi \in \Phi$, $\int_A \varphi |f|$ exists. We define $f$ to be integrable if $\sum_{\varphi \in \Phi}\int_A \varphi |f|$ converges. This implies the convergence of $\sum_{\varphi \in \Phi}|\int_A \varphi f|$ and hence absolute convergence of $\sum_{\varphi \in \Phi}\int_A \varphi f$ which we define to be $\int_A f$."

Now, this indeed shows to be a great theoretical tool, because it made the proof of many theorems really nice, like the change of variables theorem. And this makes sense pretty fine intuitively. However, in this way we are defining the integral as the sum of a series. Proving convergence is something we do all the time in analysis, but finding the sum can be tricky as I know. The methods I know for finding for example that $\sum_{n=1}^\infty 1/n^2=\pi^2/6$ involves techniques of integration by residues on $\mathbb{C}$ and so on.

In that case, is it feasible to compute integrals with this definition? Or this definition just serves for theory and there are other ways then to compute this kind of integral over open sets?

Thanks very much in advance!

  • $\begingroup$ It is generally not feasible to compute integrals using this definition. Theorems like Stokes' Theorem are used instead. $\endgroup$ – Potato Oct 31 '13 at 19:14

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