Skip to main content

All Questions

Filter by
Sorted by
Tagged with
4 votes
0 answers
70 views

Integration of volume forms over manifolds with corners

Suppose that $M$ is a (compact, oriented, smooth) manifold with corners. Let $M_{\leq 1}$ the manifold with boundary of all points of index $\leq 1$ (following the notations here). This manifold is an ...
Phil-W's user avatar
  • 1,035
6 votes
2 answers
428 views

An abstract characterization of line integrals

Let $M$ be a smooth manifold (endowed with a Riemann structure, if useful). If $\omega \in \Omega^1 (M)$ is a smooth $1$-form and $c : [0,1] \to M$ is a smooth curve, one defines the line integral of $...
Alex M.'s user avatar
  • 5,407
3 votes
0 answers
185 views

References on integration on non-compact manifolds

I am looking for references on integration on non-compact Riemannian manifolds, specially on the change of variables theorem. In particular I have non-compact manifold $M$ and I have an integral (in ...
Fernanda's user avatar
6 votes
1 answer
1k views

Fubini's theorem on arbitrary foliations

In what follows $ \mathbb{R}^{n+m} = \{(x,y): x \in \mathbb{R}^n, \ y \in \mathbb{R}^m \} \ .$ Suppose $G: U \to V $ is a $C^1$-diffeomorphism from an open subset of a manifold to an open subset of $...
Behnam Esmayli's user avatar
2 votes
3 answers
803 views

A Curved/Warped Version of Fubini's Theorem

I will think of $ \mathbb{R}^{n+m}$ as $\mathbb{R}^n \times \mathbb{R}^m$. Let $ V \subset \mathbb{R}^{n+m}$ be open and $g:V \to U \subset \mathbb{R}^{n+m} $ be a $C^1$ diffeomorphism. For a fixed ...
Behnam Esmayli's user avatar
3 votes
0 answers
426 views

Integration over a Surface without using Partition of Unity

Suppose we are given a compact Riemann surface $M$, an open cover $\mathscr{U}=\{U_1,U_2,\dots\}$ of $M$, charts $\{(U_1,\phi_1),(U_2,\phi_2),\dots\}$, holomorphic coordinates, $\phi_m:p\in U_m\mapsto ...
Wakabaloola's user avatar
3 votes
1 answer
155 views

volume of region between two manifolds

The question is motivated by a simple example: the area of a ring is $\pi(R^2-r^2)$, where $R$ and $r$ are the radii of the outer and inner circles respectively. Let $C$ be the 'middle circle' with ...
Bill J's user avatar
  • 65
3 votes
1 answer
263 views

An Stokes type theorem for some operations other than integral

Let $M$ be a compact $m$ dimensional manifold with boundary $\partial M$. Assume that $I_{1}, I_{2}$ are two linear functionals on $\Omega^{m}(M), \Omega^{m-1}(\partial M)$, respectively. Assume ...
Ali Taghavi's user avatar
3 votes
1 answer
938 views

Stokes theorem for manifolds with boundary as disjoint union of submanifolds

Looking at the generalizations of Stokes theorem, I did find a version for manifold with corners, but I was surprised this generalization doesn't contain a simple example such as the cone. So my ...
Jon-S's user avatar
  • 549
14 votes
0 answers
573 views

Reference for a proof of the fiberwise Stokes theorem

The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary, the difference between the fiberwise integral of the differential and the ...
Dmitri Pavlov's user avatar