All Questions
Tagged with differential-calculus dg.differential-geometry
26 questions
9
votes
3
answers
696
views
I want to find a smooth section of the map from the Stiefel manifold to the Grassmanian manifold
The following question is related to research I am doing on reinforcement learning on manifolds.
I have a set of basis vectors $\boldsymbol{B} = \{\boldsymbol{b}_1,\dots,\boldsymbol{b}_k\}$ that span ...
- 155
2
votes
1
answer
168
views
Validity of formula $u(x)=\frac{1}{4\pi}\int_G \nabla_y \frac{1}{\lvert x-y \rvert} \times \omega(y) \, d^3y +A(x)$ for periodic boundary case
I think it is better to provide context in which the previous question Any formula or estimates the Green function for the Laplacian in $3D$ periodic box? has been raised.
The motivation is the ...
- 3,477
0
votes
1
answer
154
views
Conditions for surface area of surface of revolution to be product of arclengths
Given a circle $C$ in the xz-plane which does not intersect the $z$-axis, we can build a smooth 2-torus with surface area $(2\pi a)(2\pi b)$ where $a$ is the radius of the circle $C$ and $b$ is the ...
- 511
3
votes
1
answer
175
views
Example of homeomorphism that lifts to real blow up but not C^1?
Given smooth manifold $M$, let $Bl_\Delta(M\times M)$ be the (say oriented; you can ask this question for the unoriented case too) real blow up of $M\times M$ along the diagonal and let $\pi:Bl_\Delta(...
- 41
4
votes
1
answer
1k
views
Laplace-Beltrami of the mean curvature
For a surface $S$ defined in 3D space, denote its mean curvature as $H$, and the Laplace-Beltrami operator as $\Delta_S$. I know that there is a result for the Laplace-Beltrami of coordinate functions:...
- 141
3
votes
0
answers
164
views
Extension of normal vector field to a domain
Let $\Omega \subset \mathbb R^3$ be a bounded regular simply connected domain contained in a ball $S$. Assume also that $\Omega$ is simply connected by surfaces (i.e. every regular closed surface ...
- 31
4
votes
0
answers
112
views
Properness of real analytic maps?
Fix a polynomial mapping $\mathbb R^n\overset{f}{\to} \mathbb R$. This answer shows that if the top degree homogeneous component of $f$ is zero only at the origin, then $f$ is proper. Intuitively, ...
- 10.5k
5
votes
1
answer
670
views
Signed distance function and level set
For $\phi\in C^1(\mathbb{R}^N)$ with $$\omega_{\phi}=\{x\in\mathbb{R}^N\ |\ \phi(x)>0\}$$ being a bounded set with $\nabla\phi (x)\neq 0,\ \forall\ x\in\phi^{-1}(0)=\partial\omega_{\phi}\neq \...
- 1,759
0
votes
1
answer
210
views
The derivation of thin plate spline interpolation energy function？ [closed]
I am trying to derive the "thin plate energy functional". Given a thin plate $z = z(x,y)$, how does one derive easily the energy functional
$$\iint_{\mathbb{R}^2} \,\left[\left(\frac{\...
- 103
58
votes
22
answers
12k
views
Which high-degree derivatives play an essential role?
Q. Which high-degree derivatives play an essential role
in applications, or in theorems?
Of course the 1st derivative of distance w.r.t. time (velocity), the 2nd derivative (acceleration),
and the ...
3
votes
2
answers
483
views
Curl as a divergence... Is it possible? [closed]
I want to know if it is possible to express the operation
$$
\nabla \phi \times (\nabla \times \mathbf A)
$$
as the divergence of second order tensor field $T$. Here $ \phi$ is a scalar field and $\...
- 153
0
votes
2
answers
238
views
exactness of a 1-form [closed]
This may be a trivial question. If so, apologies in advance.
Let a $1$-form $\omega$ be given. Suppose that for any line segments $C_1=[i,j]$, $C_2=[j,k]$, $C_3=[i,k]$, we have that
$\int_{C_1} \...
- 393
1
vote
2
answers
280
views
Uniqueness of tangent space given local injectivity of orthogonal projection onto it
Definition. Let $X\subset \mathbb R^n$ be a locally Euclidean subset. Say it has a tangent space at $p\in X$ if there exists a linear subspace $V\leq \mathbb R^n$ satisfying the following conditions.
...
- 10.5k
5
votes
1
answer
704
views
An inequality inspired by the isoperimetric inequality
Let us consider the simplest isoperimetric inequality. Consider a smooth simple closed curve given by $r=\rho(\theta)$ in polar coordinates, where $\rho(\theta)>0$ can be regarded as a smooth ...
- 141
2
votes
2
answers
2k
views
Commuting of exterior derivative and contraction (vector-valued forms)
$\newcommand{\sig}{\sigma}$
$\newcommand{\tr}{\operatorname{tr}_{\eta}}$
$\newcommand{\al}{\alpha}$
$\newcommand{\be}{\beta}$
$\newcommand{\til}{\tilde}$
Let $E$ be a smooth vector bundle over a ...
- 6,741
2
votes
0
answers
202
views
Universal chord theorem for curves
Let $\mathrm{\gamma} : [0,1] \to \mathbb{R}^2$ be a piecewise smooth, simple plane curve.
Assume $\gamma(0) = (0,0)$, $\gamma(1) = (1,0)$ and that the slope of the tangent is not $0$ wherever it's ...
- 121
29
votes
1
answer
3k
views
Is there an explicit formula for the hessian of "Determinant"?
Let $f: G= \mbox{GL}(n,\mathbb{R}) \to \mathbb{R}$ be the determinant function.
Then $\mbox{Hess} (f)$ is a two linear map on $M_{n}(\mathbb{R})\simeq T_{e}(G)$ where $e$ is the neutral element of $G$,...
- 356
0
votes
1
answer
162
views
Solutions of this system of PDE's
This question is related to the existence of Einstein metrics on tangent bundles where the metric is induced by the isotropic almost complex structures on the tangent bundle. I'm trying this on the ...
- 601
3
votes
1
answer
268
views
An answer to this system of PDE's
Planning of the question:
Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle
The isotropic almost complex structures $J_{\delta , \sigma}$ were introduced by Aguilar on the ...
- 601
3
votes
4
answers
1k
views
Intrinsic definition of arc length [closed]
Is there an intrinsic way of defining the arc length of a curve in $\mathbb{R}^{3}$, that is without resorting to a parametrization of the curve?
- 6,962
2
votes
0
answers
130
views
Slice a compact C1 surface in R3 by a moving transverse plane. Does the length of the slice depend C1 on the plane?
To be more precise I am interested in questions similar to the one below
(I asked the question below on math.stackexchange last week but got not answer.)
I have a $C^1$ function $f:[0,1]^2 \to \...
- 1,517
1
vote
1
answer
298
views
Question about extending a solution to Monge-Ampere solution
I am interested in solutions to the Monge-Ampere equation for a smooth function
$h(x,y)$ of two variables(though I suppose I could try to make do with $C^2$ solutions). The equation is:
$$\det[\...
- 1,331
5
votes
0
answers
1k
views
Boundary of an open, bounded and convex set in $\mathbb{R} ^n$
Let $U$ be an open, bounded and convex set in $\mathbb{R} ^n$. Since $\partial U$ is a rectifiable set it follows that up to a set of $H^{n-1}$-measure zero $\partial U$ is contained in a countable ...
9
votes
1
answer
2k
views
Differential Calculus and the De Rham Homotopy Operator
Suppose $M$ is a smooth manifold, $\Omega^k(M)$ the vector space (or $C^\infty(M)$-module in case this is better suited) of $k$-differentialforms on $M$ and
$$I: \Omega^k(M\times \mathbb{R}) \to \...
- 2,074
8
votes
0
answers
307
views
Co-filtered and pro-finite manifolds, filtered algebras, and differential calculus on them
I've come across a lot of questions (and nice answers) on MO, concerning infinite-dimensional manifolds and differential calculus over them, but nothing suiting the simpler and special case I have in ...
- 2,527
25
votes
2
answers
1k
views
Is there a convenient differential calculus for cojets?
I understand the basics of exterior differential geometry and how to do calculus with exterior differential forms. I know how to use this to justify the notation dy/dx as a literal ratio of the ...
- 2,754