Tagged Questions
2
votes
2answers
46 views
Smooth maps on a manifold lie group
$$
\operatorname{GL}_n(\mathbb R) = \{ A \in M_{n\times n} | \det A \ne 0 \} \\
\begin{align}
&n = 1, \operatorname{GL}_n(\mathbb R) = \mathbb R - \{0\} \\
&n = 2, \operatorname{GL}_n(\mathbb ...
-1
votes
2answers
96 views
The euclidean space $\Bbb R^n$ is orientable as a manifold.
I know that
The euclidean space $\Bbb R^n$ is orientable as a manifold.
I think that it is orientable because it has a nowhere vanishing $n$-form.
But I am not sure.
Please can you explain ...
2
votes
0answers
33 views
Stoke's theorem application to curl theorem. I did. Please can you check it?
Now, I need to apply stoke's theorem to curl theorem.
My teacher gave a hint.
Accourding to the hint, I accept $w=Pdy∧dz +Q dz∧dx + R dx∧dy$ $\in Ω^2(M)$
$dim(M)=2$
M is the subset of $\Bbb ...
2
votes
0answers
50 views
Real projective space is Hausdorff
I could not understand the proof of thıs proposition can you help me and give clear explanation.Just can you say how we have (n+1)x2 matrix??
This prove is correct or I need to add something ?? ...
1
vote
1answer
53 views
Real Projective Space
How I can prove this corollary 7.15.I consider that ı can apply the corollary 7.10.But I could not can you help me.
1
vote
1answer
31 views
Locally finite or not
I am tryıng to learn locally finite and can you give an explanation for my green writing please thank you
3
votes
1answer
72 views
What is overlop
I want to ask something that what is overlop.My teacher said that For Ex1, Everything is overlop hence it is not locally finite.For example 2,it doesnt overlop.Actually ı could not understand.please ...
1
vote
2answers
54 views
An open cover that is not locally finite
I could not understand that why is not locally finite for example 13.4 can you give me explanation please.
2
votes
0answers
74 views
Show that the projection map is Orientation preserving iff n is even
My question is that
Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball. Let $U$ be the upper hemisphere
$U =${$x∈S^n |x^{n+1} >0$}.
It is a coordinate chart on ...
2
votes
1answer
52 views
Manifolds with boundary and definition
Can you help for understanding this definitions in a good way.What is the my problem is that I can not image this definition and proposition in my mind and also ı dont understand the reason that ı ...
2
votes
1answer
45 views
Boundary orientation for a cylinder
Please help me.I am think that I can use stokes theorem but ı could not apply.This question is very benefical for me to learn the subject please help me :(
1
vote
1answer
33 views
Now I am asking that the topological and manifold boudary for real line I am grateful to explain me more clearly and instructively.
Let M be the subset $[0,1[$ $∪ $ {$2$} of the real line. Find its topological boundary $bd(M)$ and its manifold boundary $∂ M$.
I know that while I find the topological boundary, I need to show ...
4
votes
1answer
68 views
The open Möbius Band is not orientable
Can you explain my green underlying please.I have confused and ı dont understand why by the continuity of the orientation at the points $(0,0)$ and $(1,0)$ are also $e_{1},e_{2}$
2
votes
1answer
81 views
Why is the cylinder surface on $\Bbb R^3$ orientable?
Why is the cylinder surface on $\Bbb R^3$ orientable? Please can someone explain me clearly?
1
vote
1answer
58 views
Diffeomorphism of open intervals in $\mathbb{R}$ with specified values
I know two open intervals on $\mathbb{R}$ are diffeomorphic to each other. My question is if I have a intervals $(a-\varepsilon,b+\varepsilon)$ and $(c-\delta,d+\delta)$, is there a diffeomorphism ...
2
votes
1answer
55 views
$F(x,y) = (x^2 +y^2,xy)$. compute $F^{∗}(u \, du+v \, dv)$
Let $F : \Bbb R^2 → \Bbb R^2$ be given by
If $u$,$v$ are the standard coordinates on the target $\Bbb R^2$,
compute $F^{∗}(u \, du+v \, dv)$.
$$F(x,y) = (x^2 +y^2,xy).$$
I am confused so much. I ...
0
votes
1answer
49 views
Partitions of unity and bump function
I can not image this guestion in my mind.can you give me graph and help how ı can prove this question please.
0
votes
0answers
56 views
Prove a torus is a 2-dimension C∞-Manifold and find its tangent space
(i) A torus is a doughnut-shaped surface in $\mathbb{R}^3$ that can be constructed as follows. Let $a > b > 0$ and consider the circle $C$ of radius $a$ in the $xy$-plane. By definition, the ...
0
votes
1answer
59 views
Find a $1$-form $ω$ on $\mathbb R^2 −\{(0,0)\}$ such that $ω(X) = 1$ and $ω(Y) = 0$.
Please ı dont know what I need to do. thus, help me to solve.
1
vote
1answer
48 views
Show that this is a diffeomorphism
I have a function $F:(0,2\pi) \times \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}^2$
with $(\phi,r)\mapsto(r(\phi-\sin(\phi)),r(1-cos(\phi)))$ and want to show that this is a smooth(meaning ...
0
votes
1answer
33 views
prove that $supp(π^{∗} f) = (supp f)×N.$ Please can you check my answer? Also more explanation please.
My question is that
Let $f \colon M \to R$ be a $C^{\infty}$ function on a manifold $M$.
If $N$ is another manifold and $π \colon M \times N \to M$ is the projection onto the first factor, prove ...
0
votes
2answers
81 views
Lie bracket in local coordinates. Find the formula $c^{k}$ in terms of $a^{i}$ and $b^{j}$
This is from T.U Loring's manifold book. I tried. But I didnt do the question. Please show me how to solve instructively and explicitly. I want to learn this topic. Thank you for help.
4
votes
1answer
64 views
How are the isometries $h:(\mathbb{R}^n,||\cdot||_p)\longrightarrow(\mathbb{R}^n,||\cdot||_p)\;$?
An isometry of $\mathbb{R}^n$ is a function $h:\mathbb{R}^n\longrightarrow\mathbb{R}^n$ that preserves the distance between vectors:
$$||h(x)-h(y)||_p=||x-y||_p\;\;, \;\;p\ge1$$
for all $x$ and $y$ ...
1
vote
1answer
54 views
If $\alpha:[a,b]\longrightarrow \mathbb{R}^n$ is a continuous function and injective $\Longrightarrow $ $\text{int}(\alpha([a,b]))=\emptyset\;$?
If $\alpha:[a,b]\longrightarrow \mathbb{R}^n$ a continuous function and injective, $n>1$
We can say that $\text{int}(\alpha([a,b]))=\emptyset$ ?
Any hints would be appreciated.
0
votes
1answer
50 views
$ l(\alpha)=\int_a^b\|\alpha$ '$(t)\| \;\mbox{dt}$ , $\alpha$ a differentiable function whose derivative is integrable.
Let $\alpha:[a,b]\longrightarrow \mathbb{R}^n$ a differentiable function whose derivative is integrable.
We can say that $\displaystyle l(\alpha)=\int_a^b\|\alpha $'$(t)\| \;\mbox{d}t\;$? , $\alpha ...
2
votes
2answers
92 views
Implicit function theorem - how to approach?
I have a question that I have been working on for a while. I was wondering how I should approach the following question:
Are there any points on the graph of the equation
$$x^3+3xy^2+2xy^3=1$$
...
3
votes
0answers
57 views
State of the art of the Implicit Function Theorem
What is the most general form of the Implicit Function Theorem? Quite a general form of this theorem was given by Kumagai (1980): An implicit function theorem. So I am wondering what are the weakest ...
6
votes
4answers
171 views
why $\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$ means surface area?
Why the following integral means the area of surface $f(x,y)=z$?
$$\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$$
3
votes
1answer
48 views
Every closed $C^1$ curve in $\mathbb R^3 \setminus \{ 0 \}$ is the boundary of some $C^1$ 2-surface $\Sigma \subset \mathbb R^3 \setminus \{ 0 \}$
How can I prove it?
This problem looks similar to Plateau's problem - but it is much more specific. I believe there exists some elementary proof.
(Proving this will help me apply Stokes' theorem to ...
0
votes
0answers
116 views
Baby Rudin, Chapter 10, Problem 23 (d) - Differential forms.
In problems 21 and 22, Rudin defines the differential forms $\eta=\dfrac{xdy-ydx}{x^2+y^2}$ and $\zeta=\dfrac{x dy \wedge dz+ydz \wedge dx+z dx \wedge dy}{r^3}$ and the reader is asked to prove ...
2
votes
2answers
80 views
Is $\omega_n$ exact in $\mathbb R^n -\{0 \}$?
For $n \ge 2$ consider the differential form $\omega_n=r^{-n} \sum_{i=1}^n(-1)^{i-1}x_idx_1 \wedge \ldots \wedge dx_{i-1} \wedge dx_{i+1} \wedge \ldots \wedge dx_n$, defined on $\mathbb R^n \setminus ...
1
vote
1answer
58 views
Is $\zeta=\frac{x dy \wedge dz+y dz \wedge dx+z dx \wedge dy}{r^3}$ exact in the complement of every line through the origin?
$r=\sqrt{x^2+y^2+z^2}$ of course.
If the line is the $z$ axis, it is given in the book (Rudin) that $\zeta=d \left( -\dfrac{z}{r} \dfrac{xdy-ydx}{x^2+y^2} \right)$
I've managed to figure out 2 ...
1
vote
0answers
48 views
Difference between distance between two points and metric
if i have a line element given e.g. $ds^2=\frac{dx^2+dy^2}{2y} $ is it then always possible to derive a distance between two points in this metric? and how would one determine the length of a curve if ...
1
vote
1answer
133 views
shortest distance between two points on $S^2$
Length of Curve in $2D$ is $l_{\gamma}(\mathbb{R}^2)=\int_{0}^{1}\sqrt{(dr/dt)^2+r^2(d\theta/dt)^2}$
Length of a curve in $3D$ is ...
-1
votes
1answer
91 views
Length of a curve on $S^2$
$1.$ Could any one tell me what is the shortest distance between $2$ points on $S^2$?
$2.$ Could any one tell me how to measure explicitly a length of a curve on the $S^2$ using polar co-ordinates?
...
1
vote
0answers
73 views
Prove $\left.{\partial X\over \partial Y}\right|_Z=-\frac{\left.{\partial Z\over \partial Y}\right|_X}{ \left.{\partial Z\over \partial X}\right|_Y}$
$$\left.{\partial X\over \partial Y}\right|_Z=-\frac{\left.{\partial Z\over \partial Y}\right|_X}{ \left.{\partial Z\over \partial X}\right|_Y}$$
The above is an identity frequently used in ...
2
votes
1answer
60 views
Rademacher theorem for Riemannian manifold
Let $M$ be an open set of $\mathbb R^n $ and let $ ds^2 $ be some metric on $M$. Let $ d $ be the distance induced by $ ds^2 $ on $M$. If $ f $ is a Lipschitz function with respect to $ d $, is it ...
4
votes
2answers
74 views
Composite of an immersion with the inverse map of another immersion is a diffeomorphism
Let $U\subset \mathbb{R}^k$ be an open set, $n>k$ and $\varphi_1,\varphi_2 : U\to \mathbb{R}^n$ be immersions, meaning continuously differentiable such that the differential taken in any point of ...
2
votes
1answer
87 views
Implicit function theorem-show that in a neighbourhood of the point -can be described by a pair of functions
Let $g_1(x,y_1,y_2)$= $x^2(y_1^2+y_2^2)$-5 and $g_2(x,y_1,y_2)$=$(x-y_2)^2$+$y_1^2$-2. Use implicit function theorem to show that in a neighbourhood of the point x=1, $y_1$=-1, $y_2$=2, the curve of ...
1
vote
1answer
120 views
any two simply connected open set in the plane R^2 are diffeomorphic
Prove that any two simply connected open set in the plane R^2 are diffeomorphic. I know that in the complex plane any simply connected open set is diffeomorphic to either complex plane or open unit ...
1
vote
1answer
70 views
Equivalent definition of Tangent Spaces
There are about 4 definitions of tangent spaces 1) using velocities of curve 2) via derivations 3)via cotangent spaces 4) as directional derivatives. I am not getting the intuition about what tangent ...
1
vote
1answer
35 views
Find tangential plane of a surface given as an equation on a given point
I have a surface given by the equation $f(x,y,z)=0$ and need to find the Tangential plane on a given point $p$.
5
votes
2answers
195 views
Why does the condition of a function being differentiable always require an open domain?
Going through Spivak's Calculus on Manifolds and in his definition of a differentiable function from a subset $A$ of $\mathbb{R}^n$ to $\mathbb{R}^m$, $f$ is said to be differentiable if it can be ...
1
vote
0answers
54 views
Computing evolute o f a curve by finding the surronding of the normal rects.
So, I have to compute the evolute of the curve:
$y^{2} = 2px$
But I have to do that by computing the surronding of the normal rect family. So I start taking the positive side of the function and ...
2
votes
0answers
44 views
what are conormal distributions?
According to the first answer in this post, a conormal distribution $u$ on a manifold $X$ relative to a (closed, embedded) submanifold $Y$ is an element of a Banach (or Hilbert) space $H$ such that ...
0
votes
1answer
39 views
Hessian equivalence
Let $F: R^n \longrightarrow R$ be twice differentiable and $x,y \in R^n$ with $F(x)=F(y)$. Further let $\phi [0,1] \rightarrow R^n$ be a nice curve with $\phi(0)=x$ and $\phi(1)=y$.
If we know that ...
2
votes
1answer
121 views
Uniform convergence in $\mathbb{R}^2$
$a$, $b$ are $2$ points in $\mathbb{R}^{2},\rho_{n}(t)\,:\,[0,1]\to\mathbb{R}^{2}$ is a sequence of continuously differentiable constant speed curves with $\|\rho_n(t)\|=L_n$ for all $t$ from $0$ to ...
3
votes
3answers
78 views
understanding change of variable
the following is drawn from a rather rough set of lecture notes and I am not sure I understand it.
the goal is to determine for which values of $p$ we have
$$
\int_{|x|\leq 1} \frac{1}{|x|^p} \,dx ...
1
vote
1answer
40 views
Rank of the differential of a composition
Let $U$ be an open subset of $\mathbb{R}^2$ and $f:U\to \mathbb{R}$ a differentiable function. Let $S=\{(x,y,f(x,y)): x\in U\}$ be the graph of $f$. Let $V$ be an open subset of $\mathbb{R}^2$ and ...
1
vote
1answer
33 views
Differentiable surface question
Let $U$ be an open set in $\mathbb{R}^2$ and $F:U\to \mathbb{R}^3$ a one-to-one differentiable function such that its inverse from $F(U)$ to $\mathbb{R}^2$ is also continuous. Is it possible that ...
