All Questions
Tagged with integration dg.differential-geometry
16 questions with no upvoted or accepted answers
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
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.
$$
Then I claim that the $N$-form $f(x) dx_1\wedge\dots\wedge dx_N$ is closed, i.e. there exists a vector ...
- 16.2k
4
votes
0
answers
192
views
Can this integral be made nonpositive?
Let $M^2 \subset \mathbb{S}^3$ be a closed and orientable embedded (and minimal, if important) surface. Choose a unit normal vector field $\eta: M \to \mathbb{S}^3$ along $M$ and a point $p_0 \in \...
- 2,095
4
votes
0
answers
211
views
Divergence theorem on stratified spaces
It is very common in physics and engineering to apply the divergence theorem
to compact spaces whose boundary is not smooth. For example, in the wikipedia link I just gave, the picture illustrating ...
- 361
3
votes
0
answers
46
views
Evaluate $\int_\phi e^{tr(RM)} dR$ where $\phi$ is a set of all real orthonormal matrices of a certain size
I am trying to evaluate $\int_\phi e^{tr(RM)} dR$ where $\phi$ is a set of all real orthogonal matrices of a certain size. $M$ is an arbitrary real matrix (of a certain size).
This is equivalent to
$$\...
- 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
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
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