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Results tagged with integration
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user 126183
Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...
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Accepted
An abstract characterization of line integrals
Again, this is an assertion that certain known properties of line integration characterise it uniquely; this doesn't provide a "new" construction of line integration. … Theorem:
For any manifold $M$, line integration is the unique function $I\colon\Omega^1(M)\times\mathcal C_M\to\mathbb R$ satisfying the following properties:
(additivity in the path) Suppose that $c_ …
- 1,790
10
votes
An abstract characterization of line integrals
I don't know if it's exactly what you're looking for, but line integration is the unique way to assign a real number $I(\omega,c)\in\mathbb{R}$ to every pair of a smooth $1$-form $\omega$ on a smooth manifold … Remark:
This is the approach that one takes when defining iterated integration of a sequence $\omega_1,\dots,\omega_n$ of $1$-forms along a path $c$. …
- 1,790