Tagged Questions
5
votes
0answers
62 views
Invariant submanifolds
Let $M$ be a smooth manifold, and let $N$ be a submanifold. Let $V$ be a smooth vector field on $M$ which generates a flow $\Phi_t$ on $M$. My intuition tells me (perhaps modulo some technical ...
0
votes
0answers
41 views
Geodesic equation for a 2D manifold
I am having trouble understanding how the following statement (taken from some old notes) is true:
For a 2D manifold such that $$ds^2=\frac{1}{u^2}(-du^2+dv^2)$$
If we assume that $$\dot x^a\dot ...
3
votes
1answer
39 views
Smoothness in Banach space
I need a reference about a definition. Let $n$ be an integer and $G$ be a group of $H^n$(Sobolev) automorphisms of a vector bundle $E$ on some manifold $M$ and $C$ be the space of connections of class ...
2
votes
0answers
36 views
Orientation-preserving diffeomorphism [duplicate]
Can you help for solving this please. Although I study this subject I could not solve this question please help me ı am willing to learn this question.
3
votes
0answers
60 views
Kähler Geodesics
Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric
$$\begin{pmatrix} ...
-5
votes
0answers
41 views
Find approximate value using diffrentials [closed]
Find approximate value of sin 45° using differentials.
Be detailed about the answer because I don't have time to read the chapter and have a test in 30 minutes.
3
votes
1answer
118 views
Lemme 2.4 in Morse theory by Milnor
This is lemma 2.4 from "Morse theory" by Milnor ,with the prove
I have some questions about this prove :
1) why $\displaystyle\frac{dc}{dt}(f)=\lim_{h\rightarrow 0} \frac{fc(t+h)-fc(t)}{h}$ and ...
1
vote
0answers
40 views
Cohomologies of $\mathbb R^n$ with rational differential forms
We can consider de Rham complex $0 \to \Omega^1 \to \Omega ^ 2 \to...$ on $\mathbb R^n$, where $\Omega ^r$ are $r$-forms on $\mathbb R^n$ with rational coefficients. What are homologies of this ...
0
votes
0answers
42 views
Drone targeting [closed]
Imagine a drone and a target point on a 2d plane. There are eight parameters:
...
0
votes
0answers
41 views
solve this equation $bu'^2+\cos(u)(a+b\cos(u)) v'^2=0$ [closed]
hi every can any one please solve this equation
$$bu'^2+\cos(u)(a+b\cos(u)) v'^2=0$$
where $u'= \frac{du}{dt}$ and $v'= \frac{dv}{dt}$?
I try to solve but i did not get any idea to solve
I hope ...
1
vote
0answers
23 views
Unique continuation for elliptic operators
Consider the following system of linear elliptic equations:
$ \Delta s_i = \sum_{j=1}^{d} l_{ij} s_j $ for $ i=1,\ldots d $ where $ l_{ij} = l_{ji} $.
It should be true that the following unique ...
4
votes
0answers
77 views
Clarification in a paper
This is regarding a clarification in page 384 of a paper published in Annals of Statistics by Amari.
In page no. 384, he defines $$R_i(t)=\frac{\partial}{\partial \theta_i} ...
2
votes
1answer
55 views
Does there exist a vector field tangent to a given curve?
Let $\gamma : \mathbb{R} \to \mathbb{R}^2$ be an injective $C^2$ curve. Does there exist a $C^1$ vector field $X : \mathbb{R}^2 \to \mathbb{R}^2$ tangent to $\gamma$, ie. such that $\gamma'(t)= ...
11
votes
1answer
205 views
When does a vector field admit orthogonal fields?
My question is:
Let $\,X$ be a nonvanishing smooth vector field over an open subset $U \subset \mathbb{R}^3$. Which conditions on $X$ guarantee the existence of a smooth nonvanishing vector field ...
1
vote
0answers
22 views
Boundaries- regularity and local parametrization
Suppose we have a bounded domain $\Omega \subset \mathbb{R}^3$ with $C^2$ boundary.Let $x_0 \in \partial \Omega$. We choose a $X_1,x_2,x_3$- coordinate system such that the $x_1,x_2$-plane is ...
4
votes
0answers
51 views
Heat Kernel Asymptotics on Manifold with Boundary
On a closed Riemannian manifold $M$, the heat kernel $k_t(x, y)$ of the Laplace-Beltrami operator (or more general of any generalized symmetric Laplace-type operator acting on sections of a vector ...
1
vote
1answer
62 views
Flow trajectories of a vector field with singular point
Definition: a flow line (of 2D vector field) is a curve that vector field is tangent everywhere to it. If the vector field is zero at some point (singular point) the definition is ambiguous. Does this ...
3
votes
0answers
58 views
Existence Theorem for Geodesics
The text I am reading now defined geodesics to be those curves that satisfy the following differential equation:
$\ddot{\gamma}^k(t)+\dot{\gamma}^i(t)\dot{\gamma}^j(t)\Gamma^k_{ij}(\gamma(t)) = 0$
...
0
votes
0answers
29 views
Three-dimensional images of the geodesics
On an n-dimensional Riemannian manifold , the geodesic equation written in a coordinate chart with coordinates is:
$$\frac {\mathcal D^2x^a}{ds^2}=\frac{d^2x^a}{ds^2}+\Gamma^a_{bc}\frac ...
0
votes
0answers
40 views
Which types of differential equations in mathematics are able to broken into the extrinsic normal component?
Which types of differential equations in mathematics are able to broken into the extrinsic normal component like covariant differential
$${\mathcal D^2x^a}=d^2x^a+\Gamma^a_{bc}dx^bdx^c$$
where the ...
3
votes
1answer
102 views
Outward vectors to an Ellipsoid and Euclidean metrics
I'm reading Arnold's proof of the topologically equivalence of the equations $\dot{x}=Ax$ and $\dot{x}=x$ when all the eigenvalues of the $n \times n$-matrix $A$ have positive real part. The proof is ...
0
votes
0answers
19 views
UCP for Dirac operators
Let $M$ be a closed manifold and $D$ dirac operator on it.
Unique continuations principle for Dirac operator says that if $D(f) = 0$ and $f = 0$ on an open set $U$ then $f = 0$ everywhere.
Why this ...
0
votes
1answer
48 views
Problem with one step in a proof of fundamental theorem of curves
Let $k$, $l$ be smooth functions from an interval $I$ into $\mathbb R$ and $k>0$.
Let's consider system of differential equations
$$
t'=k n,
$$
$$
n'=-k t-l b,
$$
$$
b'=ln
$$
with unknow ...
0
votes
0answers
46 views
A doubt about fuchsian functions in physics?
I'm not sure if this is the right place (or physics.stackexchange?) to ask the next
What is the difference between fuchsian, theta-fuchsian, and kleinian functions?
Please, suggest me an ...
0
votes
0answers
24 views
Divergent on $M^ n$-Submanifolds of $R^{n+p}$
I was reading a proof (I won't tell by who 'cause I don't if it is true) the author come up with
$\int_M \exp (-|x|^2) Div_M(\nabla _V V)^T=\int_M \exp (-|x|^2) <\nabla _V V,x^T>$
where ...
1
vote
2answers
75 views
Any idea on how to solve this system of coupled ODEs?
I'm trying to find solutions for the system of ODEs
$$ y_1'(t) = y_1(t)y_2(t) \\
y_2'(t) = 2y_2(t)^2 - y_1(t)^6 $$
And I'm assuming $ y_1(t), y_2(t) > 0 $. This comes from trying to find the ...
2
votes
0answers
52 views
Reconstructing paths on the sphere from the ratio of acceleration to velocity
Given a path $\gamma:[0,1]\to \mathbb C$, we can determine $\gamma$ from information about its derivatives. For example, $\gamma$ is determined by $\gamma(0), \gamma'(0)$, and $\gamma''(t)$. This ...
1
vote
1answer
56 views
non-linear partial differential operators
I am looking for some literature on non-linear partial differential operators used in geometry or analysis. Can you give me some reference. Thanks in advance.
eric
0
votes
1answer
583 views
How to find Singular solutions of differential equation based physical model?
To get singular solutions, do we always need a guess or experiment?
Can we get it from a relation of family of curves of general solution? For example, $(y')^2-xy'+y=0$ has the general solution ...
0
votes
0answers
42 views
When to make a substitution in ODE
The setting is on evolving hypersurfaces. So for each time $t$, $\Gamma(t)$ is a hypersurface given by the zero level set of the function $\phi(x,t)$. Consider a ball, then the hypersurface has ...
2
votes
2answers
575 views
A simple explanation of differential calculus and its link to geometry?
The wikipedia articles on differential calculus and differential geometry are quite long and not so straightforward for a layman like me. Is there a master of math vulgarization out there that could ...
1
vote
1answer
194 views
How to show that the geodesics of a metric are the solutions to a second-order differential equation?
On $\mathbb R^n$, let $\rho: \mathbb R^n\to\mathbb R$ be a smooth function, and $g$ be the metric given by scaling the usual flat metric by $e^{2\rho}$.
I want to know how to show that the geodesics ...
1
vote
1answer
44 views
Is it possible to achieve the following form?
Is it at all possible to, by a change of variables, transform the metric $dx^2+dy^2\over g(r)^2$ where $g$ is a function and $r=\sqrt{x^2+y^2}$ to something of the form $du^2+f(u,v) dv^2$? Thank you.
1
vote
1answer
133 views
Suggestions for a Global Analysis book
can somebody tell me some good books or lecture notes in "global analysis" ?
I am a newcomer in this subject.
thanks in advance.
greetings
trito
3
votes
2answers
186 views
Asymptotic Expansion for heat operator $e^{-t\triangle}$
I'm afraid the question below might turn out to be very stupid -
I just don't know how to make sense of two asymptotic expansions,
given the heat operator $e^{-t\triangle}$ with $\triangle$ a ...
0
votes
1answer
79 views
On Tangent vectors as jets & submanifolds
Here is my second question on understanding jets better:
For a smooth manifold $M$ the set of jets $J^1_0(\mathbb{R},M)$ is the same as the
Tangent bundle $TM$. This implies that any equivalence ...
1
vote
2answers
138 views
On the definition of jets
I have some problems with the definition of jets and it would be
great if someone could help me here:
In many books it is written, that the $r-th$ order jet $j^r_xf$ of
a smooth function $f:M ...
4
votes
2answers
140 views
Changing the manifold, preserving the discrete spectrum
On a Riemannian manifold $M$, the Laplace operator $L$ is uniquely defined.
If $M$ is not compact, then $L$ admits a continuous spectrum.
Is there a way of "changing" $M$ and/or $L$ in say $M'$ ...
1
vote
1answer
96 views
System of inequalities
All functions are smooth and continuous.
I use ' to signfy $d/dt$.
Given:
$q_i'(0)=0$, for i=1,2,3,4.
$q_2(t)q_3(t)>q_1(t)q_4(t)$ for $0<=t<T$. T is finite. At t=T the inequality breaks.
...
15
votes
1answer
534 views
Stochastic interpretation of Einstein Equations
Einsteins theory of gravitation, general relativity, is a purely geometric theory.
In a recent question I wanted to know what the relation of Brownian Motion to the Helmholtz equation is and got a ...
5
votes
2answers
405 views
Manifold interpretation of Navier-Stokes equations
I am wondering about particle trajectories for solutions of the Navier-Stokes equation. Is it possible that there is a Manifold $M$ for which fluid particles move along geodesic's or "straight lines ...
3
votes
2answers
157 views
Harmonic functions on $S^2$
Consider the sphere $S^2 = \lbrace (x,y,z) :\ x^2 + y^2 + z^2 = 1 \rbrace$. This is a smooth manifold in $\mathbb{R}^3$, and for a given point $s \in S^2$, one can consider its coordinate ...
7
votes
1answer
284 views
In which commutative algebras does any derivation possess a flow?
Suppose $A$ is a commutative algebra over $\mathbb{R}$ with unity. $\mathbb{R}$-linear map $\xi\colon A\to A$ is a derivation of $A$ iff $\xi(ab)=a\xi(b)+\xi(a)b$ for any $a,b\in A$.
If $\gamma\colon ...
