All Questions
Tagged with integration dg.differential-geometry
43 questions
43
votes
6
answers
10k
views
Why do I need densities in order to integrate on a non-orientable manifold?
Integration on an orientable differentiable n-manifold is defined using a partition of unity and a global nowhere vanishing n-form called volume form. If the manifold is not orientable, no such form ...
- 843
31
votes
4
answers
4k
views
The "ds" which appears in an integral with respect to arclength is not a 1-form. What is it?
The only reasonable way to interpret "$ds$" as a functional on tangent vectors has to be that it takes a tangent vector and spits out its length, but this is not linear. So $ds$ is not a 1-form. It ...
- 12.1k
22
votes
1
answer
2k
views
A difficult integral for the Chern number
Cross post from Maths stack exchange
The integral
$$
I(m)=\frac{1}{4\pi}\int_{-\pi}^{\pi}\mathrm{d}x\int_{-\pi}^\pi\mathrm{d}y\phantom{,} \frac{m\cos(x)\cos(y)-\cos x-\cos y}{\left( \sin^2x+\sin^2y +...
- 333
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 ...
- 37.8k
14
votes
1
answer
2k
views
The perturbation of non-Hamiltonian algebraic vector fields
In this question, we are interested in the number of limit cycles which appears in the following perturbational system:
\begin{equation}\cases{
x'=y -x^{2}+\epsilon P(x,y) \\
y'=-x+\epsilon Q(x,y) }
\...
- 356
13
votes
2
answers
2k
views
Stokes theorem for manifolds without orientation?
In textbooks Stokes' theorem is usually formulated for orientable manifolds (at least I couldn't find any version not using orientability). Is Stokes theorem: $\int\limits_{M}d\omega=\int\limits_{\...
- 675
7
votes
3
answers
431
views
Identity involving an improper integral (with geometric application)
Is it (for some reason) true that
$\lim_{c\to 0^+}\int_c^{\pi/2}\frac{c}{t}\sqrt\frac{1+t^2}{t^2-c^2}dt=\frac{\pi}{2}$?
Numerical evidence (from Mathematica):
when $c=1/5$, the integral is $\...
- 3,212
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 $...
- 5,407
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 $...
- 622
6
votes
1
answer
400
views
Densities, pseudoforms, absolute differential forms and measures, differential forms, etc
Apologies if this question is too basic, but I figured I first heard of most of these concepts on MO, so perhaps I can ask here.
Gelfand’s definition, copied from AlvarezPaiva [My edit, could be ...
- 831
5
votes
3
answers
1k
views
Area of metric spheres in Riemannian manifolds
I am trying to estimate the integral $\int \mathbb{e} ^{-d(x_0,x)^2} \mathbb{d}x$ on a Riemann manifold $(M,g)$, for some arbitrary fixed $x_0 \in M$ and $d$ the usual distance. The only thing that I ...
- 5,407
5
votes
1
answer
319
views
Spherical average of $\frac{1}{x}$
Let $X_1,...,X_n$ be points on $\mathbb S^1.$
We then define the expectation value $E(X)=\frac{1}{n}\sum_{i=1}^n X_i.$
Let $\frac{dS(X_1)}{2\pi}$ be the normalized surface measure of $\mathbb S^1,$ i....
- 124
5
votes
1
answer
752
views
Gaussian integral over a ball
How to compute the following integral?
$$\int_{\|x\|^2\leq R} \exp(-x^\ast G x+2\mathcal{Re}(x^\ast a)) \,dx,$$
where $x$ is an $M \times 1$ vector ($M\gg 1$), $G$ is a positive definite matrix, and $...
- 129
4
votes
0
answers
71
views
Integration of volume forms over manifolds with corners
Suppose that $M$ is a (compact, oriented, smooth) manifold with corners.
Let $M_{\leq 1}$ the manifold with boundary of all points of index $\leq 1$ (following the notations here). This manifold is an ...
- 1,035
4
votes
0
answers
221
views
Can Differential Geometry aid in comparing the close contour integrals of $f(z)/z$ and $f(z) / \bar{z}$?
Let us consider a function $f(z)$ holomorphic along and inside a contour $\Gamma$ not surrounding the origin. With reference to the following contour integrals:
$$ \oint\limits_{\Gamma} \frac{f(z)}{z}\...
- 362
4
votes
0
answers
194
views
The Poincaré Lemma
Let me consider an $L^1(\mathbb R^N)$ function $f$ such that $$
\int_{\mathbb R^N} f(x) dx =0.
$$
Then I claim that the $N$-form $f(x) dx_1\wedge\dots\wedge dx_N$ is closed, i.e. there exists a vector ...
- 16.2k
4
votes
0
answers
192
views
Can this integral be made nonpositive?
Let $M^2 \subset \mathbb{S}^3$ be a closed and orientable embedded (and minimal, if important) surface. Choose a unit normal vector field $\eta: M \to \mathbb{S}^3$ along $M$ and a point $p_0 \in \...
- 2,095
4
votes
0
answers
211
views
Divergence theorem on stratified spaces
It is very common in physics and engineering to apply the divergence theorem
to compact spaces whose boundary is not smooth. For example, in the wikipedia link I just gave, the picture illustrating ...
- 361
3
votes
1
answer
985
views
Closed Poincaré dual, why $\int_M \omega \wedge \eta_S$ and not $\int_M \eta_S \wedge \omega $?
My book is Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott of which An Introduction to Manifolds by Loring W. Tu is a prequel.
The characterization of the closed Poincaré dual ...
- 315
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 ...
- 356
3
votes
1
answer
198
views
Volume of 3-dimensional region
Let $G$ be bounded finitely connected domain in $\mathbb{R}^3$ with 2-smooth boundary $\partial G$ each connected component of which has positive Gaussian curvature.
Each sufficiently small open ...
- 197
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 ...
- 65
3
votes
1
answer
330
views
acoustic dipole volume integral w/ dirac delta?
I have an acoustic research problem that leads to the following integral formulation:
\begin{align}
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(\mathbf{y},\tau)\frac{\partial}{\partial y_i}\left(...
3
votes
1
answer
327
views
Let $\gamma(s)$ be a unit speed planar curve with $\kappa(s)$ as its curvature. Now what is $\int \frac{1}{\kappa}ds$?
Let $\gamma(s)$ be a unit speed planar curve with $\kappa(s)$ as its curvature. Now what is $\int \frac{1}{\kappa}ds$ and geometrically which things it represents?
- 930
3
votes
1
answer
848
views
Integration by parts on manifold with corners
Suppose that $M$ is a compact manifold with corners, where each boundary hypersurface is an embedded submanifold. Then, do we have an integration by parts identity? i.e.
\begin{align*}
\int_M g(\nabla ...
- 156
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 ...
- 549
3
votes
0
answers
46
views
Evaluate $\int_\phi e^{tr(RM)} dR$ where $\phi$ is a set of all real orthonormal matrices of a certain size
I am trying to evaluate $\int_\phi e^{tr(RM)} dR$ where $\phi$ is a set of all real orthogonal matrices of a certain size. $M$ is an arbitrary real matrix (of a certain size).
This is equivalent to
$$\...
- 433
3
votes
0
answers
396
views
Differential of exponential map with respect to the base point
Let $(M,g)$ be a smooth Riemannian manifold embedded in $\mathbb{R}^m$. I would like to understand the transformation formula which will allow me to pass from the integral $\int_M \dots dV_g(x)$ to $\...
- 319
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 ...
- 51
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 ...
- 277
2
votes
3
answers
804
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 ...
- 622
2
votes
1
answer
353
views
Volume of submanifold as integral of delta-function
Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations:
\begin{equation}
f_1(\vec x)=0, \\
\vdots \\
f_m(\vec x)=0,
\end{equation}
(where $\vec x$ are ...
- 521
2
votes
1
answer
935
views
Exterior derivative independence from coordinate systems
In the book Mathematical Methods of Classical Mechanics by V.I. Arnold, the author introduces (p.189) the concept of exterior derivative as "the principal linear part of the increment" of the function ...
- 141
2
votes
0
answers
168
views
Geometric sets determined by chains (for integration and Stokes' theorem)
I have asked a similar question on mathSE more than a year ago, which received no answers, only a few comments which did not really help me. I am now re-asking this question here but reformulated ...
- 2,792
1
vote
1
answer
269
views
Example of a smooth function in a manifold whose integration vanishes [closed]
Let $M$ be a complete Riemannian manifold. Now for a fixed $p\in M$, is there any non-constant smooth function $u:M\rightarrow\mathbb{R}$ such that
$$\int_{B_r}udV=0\ \forall 0\leq r<\infty,$$
...
- 930
1
vote
1
answer
245
views
evaluating an integral related to the volume of Hessenberg orthogonal matrices
Consider the following integral,
$$
{1 \over 4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}
\sqrt{\, 9 -\sin^{2}\left(\theta_{1} \over 2\right)
\sin^{2}\left(\theta_{2} \over 2\right)\,}
\,{\rm d}...
- 4,466
1
vote
1
answer
241
views
Integrals of the type $\delta(g^{n})$ on $\mathrm{SU}(2)$
I posted this question previously to MathSE. However, I have still not solved it, so lets try to ask it here. When doing some calculations with spin-foam models for 3d quantum gravity for some ...
- 1,171
1
vote
1
answer
151
views
A marginal space splitting $\{ \psi \}^{\perp}$
Let $\psi \in L^2(\mathbb R^2,\mathbb C)$. Is there a continuous projection from $\{ \psi \}^{\perp}$ onto
$$
\left\{ \varphi \in L^2(\mathbb R^2) \:\:\Big| \int \overline{\psi}(x,y) \varphi(x,y)\...
- 31
1
vote
0
answers
88
views
Existence of $H^{1/2}(\partial\Omega)$-regular unit tangent field on smooth surface
Suppose that $\Omega$ is a bounded, smooth, simply connected domain in $\mathbb{R}^3$. My goal is to show that there is a $p(x) \in H^1(\Omega,\mathbb{S}^2)$ such that $p(x)$ lies on the tangent plane ...
- 54
1
vote
0
answers
123
views
Is this integral zero?
I'd like to know if one integral expression I have can be shown to be zero for all possible cases. Let me introduce some notation.
Consider $\mathfrak{g}=C^{\infty}(M)$ and the dual $\mathfrak{g}^*=\...
- 327
1
vote
0
answers
263
views
Does a growing manifold fixed at a point converge to its tangent plane?
Let $M$ be a smooth compact $(n-1)$ dimensional submanifold in $\mathbb{R}^n$. Let $H$ be the $(n-1)$ dimensional Hausdorff measure. Let $f(x,y,t)$ is a function for $x\in\mathbb{R}^n$, $y\in\mathbb{R}...
- 65
1
vote
2
answers
353
views
How to show this integral on boundary of Lipschitz domain is finite?
Sorry for asking a basic question but this did not get answered on M.SE.
Let $\Omega \subset \mathbb{R}^n$ be a Lipschitz domain. How do I show rigorously that
$$\int_{\partial\Omega} \frac{1}{|y|^{...
-1
votes
1
answer
578
views
A non-trivial upper bound on the integral of Lipschitz functions over a bounded support
Let $x \in \mathcal{X} = [0,1]^n$, and $f(x)$ be an $L$-Lipschitz function. Let $f(0)=0$. What is (the exact or a non-trivial upper bound on) $\int_{x\in\mathcal{X}} |f(x)|\,\mathrm{d}x$?What about $\...
- 482