0
votes
1answer
14 views

Reference for transformation of integrals over Lipschitz boundaries

Let $D\subseteq\mathbb{R}^d$, $d\ge 2$, be a bounded Lipschitz domain. Then according to page 314 of [Function spaces, Alois Kufner, 1977] one can define a surface integral of a real valued function ...
0
votes
1answer
37 views

A notational confusion on gradient

Given a parametrized function $f_{w}: \Bbb R ^{m} \to \Bbb R ^{k}, w \in \Bbb R^d$, I see in a book the following notation $\bigtriangledown ^ {w} f_{w}(.)$ denote its gradient w.r.t. $w$. What is the ...
3
votes
0answers
31 views

A singular $n-$cube and a circumference defined the border than 2-cube

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...
0
votes
1answer
27 views

one problem on multivariable claculus

Suppose $\phi(\bar{x}(t))$ be a function which takes vectors (parameterized by $t$) as argument. Now take $c$ be a minimum point of the function $\phi$. consider a curve $\gamma(t)$ which passes ...
1
vote
1answer
39 views

For which values of $a$ is this set a manifold?

Let $f:\mathbb{R}^3\to\mathbb{R}, f(x,y,z)=(x-y+z-1)^2$. For which values of $a$ is $\{(x,y,z)\in\mathbb{R}^3:f(x,y,z)=a\}$ a 2-manifold? Instead of $(x-y+z-1)^2=a$ seems a better idea to write ...
1
vote
1answer
34 views

Prove that inverse of $f$ defines a manifold

Let $f:\mathbb{R}^3\to\mathbb{R}$ be given by $f(x,y,z)=z^2$. Prove that $0$ is not a regular point but $f^{-1}(\{0\})$ is a manifold. I divided this in two parts: $(1)\; 0$ is not a regular ...
0
votes
1answer
38 views

k+1 Differential form

Consider the k-form given by, $ w = \sum_{i_{1}<i_{2}<...<i_{k}} w_{i_{1}i_{2}...i_{k}} dx^{i_{1}}\wedge dx^{i_{2}}\wedge...\wedge dx^{i_{k}}$ Define $k+1$ form $dw$ , the differential of ...
2
votes
1answer
52 views

Proving a set of $2\times 3$ matrices is a manifold?

The way I have always been told to check if something is a manifold (I haven't had a whole lot of experience with them), is to check if the derivative of the function representing the loci of the ...
5
votes
1answer
110 views

Directional, differential and lie derivatives on manifolds intuition?

Trying to translate elementary multivariable calculus into the language of manifolds: Is the directional derivative on a manifold just a way of finding the rate of change of a vector in a single ...
0
votes
0answers
39 views

finite-dimensional continuous vector bundle

Let $M$ a compact metric space and $\pi: F \rightarrow M$ a finite-dimensional continuos vector bundle over $M$, endowed with a continuous Riemannian metric. I was wondering if it will be true that: ...
0
votes
0answers
23 views

Derivative of function on parametrized manifold

Given $U\subset\mathbb{R}^d$. Let $\alpha:U\mapsto \mathbb{R}^k$ be an injective function such that $\alpha(U)$ is $d-$dimensional parametrized manifold. Now define $\beta:\alpha(U)\mapsto U$ by ...
0
votes
1answer
24 views

Derivative on parametrized manifold

Let $U\subset \mathbb{R}^m$ is open and $\alpha: U\mapsto\mathbb{R}^n$ is $m-$dimensional parametrized manifold with $m\leq n$. I have two following questions: If we are given $v\in \alpha(U)$, ...
0
votes
2answers
79 views

Supremum Definition in context of a Rectifiable/Differentiable Curve

I am trying to prove that if I have 2 paramterizations of the same curve $\gamma$ and $\sigma$ (i.e. there is continuous bijective map $\phi$ such that $\sigma = \gamma \circ \phi$) then if the curve ...
6
votes
1answer
93 views

How to obtain a diffeomorphism between $\mathrm{SL}(2,\mathbb{R})$ and $ (\mathbb{C}\!\smallsetminus\!\{0\})\times\mathbb R$?

Could someone please give me a tip on how to show that the map $\mathrm{SL}(2,\mathbb{R}) \to (\mathbb{C}\setminus\{0\})\times\mathbb{R}$ \begin{equation}\begin{pmatrix} a & b\\ c & d ...
7
votes
0answers
213 views

Munkres' Question on Manifolds

In Munkres' 'Analysis on Manifolds' on pg. 208 there's a question which reads: QUESTION: Let $f:\mathbb R^{n+k}\to \mathbb R^n$ be of class $\mathscr C^r$. Let $M$ be the set of all the points ...
3
votes
1answer
94 views

Integral curves of $X = z \dfrac{\partial}{\partial \theta} - \sin \theta \dfrac{\partial}{\partial z}$ on a cylinder

Consider coordinates $(\theta, z)$ on $S^1 \times \mathbb R$, and a vector field $$X = z \dfrac{\partial}{\partial \theta} - \sin \theta \dfrac{\partial}{\partial z}.$$ Show that the integral curve of ...
0
votes
2answers
82 views

Surface of graph of a function.

Let $G:=\{(x,y,z)\in\mathbb{R}^3:|x|<|z|^2,|y|<|z|,0<z<1\}$ and $f:G\to\mathbb{R},f(x,y,z)=2x+2y+z^3$. Calculate the surface of the graph of f. We recently got introduced to Stokes' ...
4
votes
0answers
35 views

Cone is a submanifold iff it is a vector subspace

Could you tell me how to prove the following? Let $\emptyset \neq M \subset \mathbb{R}^n $ be a cone ($\forall x \in M, t \in \mathbb{R} : tx \in M $ ), $0 \le d \le n.$ Prove that $M$ is a $d$ ...
2
votes
1answer
28 views

sufficient condition for being an integral factor

Let $ f: \mathbb {R}^m \rightarrow \mathbb {R}-\{0\} $ function $C^{\infty}$ class and $w$ a one-form $C^{\infty}$ class in $\mathbb {R}^m $. If $\alpha=w-\dfrac{1}{f}dx_{m+1} $ satisfies $\alpha ...
5
votes
1answer
90 views

Volume of neighborhood of the curve

Let $\gamma:[0,1] \to \mathbb R^n$ be a smooth closed curve, such that $\gamma(0)=\gamma(1), \gamma'(0)=\gamma'(1), |\gamma'(a)|=1$ and let $B_r=\{x| \exists t, |\gamma(t)-x|<r\}$. How can i show ...
0
votes
1answer
41 views

Another exercise from Fleming's Functions of Several Variables.

I'm using Flemming's book Function of Several Variables. In it, the author defines Manifolds like this: Let $1\le r\lt n,\ q\ge1$. A nonempty set $M \subset \mathbb{R}^n$ is a manifold of dimension ...
0
votes
1answer
50 views

Intersection of a manifold with open set

I'm using Flemming's book Function of Several Variables. In it, the author defines Manifolds like this: Let $1\le r\lt n,\ q\ge1$. A nonempty set $M \subset \mathbb{R}^n$ is a manifold of dimension ...
0
votes
2answers
70 views

Use implicit function theorem to show $O(n)$ is a manifold

In class today our teacher mentioned that one can use the implicit function theorem to show that $O(n) \subseteq \mathbb{R}^{n^2}$ is a submanifold...that is, map $A \mapsto A^* A$, and set it equal ...
1
vote
1answer
27 views

Tangent vectors as curves equivalence relation

I do not understand the definition of the equivalence relation that is defined on the curves creating a tangent vector space. Let $X$ be any manifold, a point $x \in X$, two curves $\alpha:(-a,a) \to ...
4
votes
1answer
58 views

Show that $\dot{n_s}=-\kappa_s t$

I found the question in a differential geometry textbook while studying. This question seems so intesting to me. So please help me solving it. Show that, if $\gamma$ is a unit-speed plane curve, ...
2
votes
0answers
43 views

problem with submersion

Given $\varphi:\mathbb{R}^{m+n}\longrightarrow \mathbb{R}^m$ is $C^{ k}$ class. If there $a\in \mathbb{R}^{m+n}$ with $\varphi^{\prime}(a)$ is surjective. Then there a mergullo $f:V\longrightarrow ...
1
vote
2answers
84 views

What is the geometric meaning of the number of independent derivatives of $\gamma$?

Let $\gamma:I \to \mathbb{R}^n$ be a curve. I want to see, what is a geometric meaning of the number of independent derivatives of $\gamma$. I guessed it is it's dimension but it was not. Can you help ...
3
votes
1answer
159 views

Problem 3-26 in Spivak´s Calculus on Manifolds

Let $ f: [a,b] \to \mathbf{R}$ be integrable and non-negative and let $$ A_f = \{ (x,y) : a \leq x \leq b \mbox{ and } 0 \leq y \leq f(x)\}$$ Show that $A_f$ is Jordanmeasurable and has area $ ...
1
vote
0answers
48 views

Prove equivalence of two conditions to be a smooth $k$-manifold $M^k \subseteq \mathbb{R}^n$

For the first couple classes of differential geometry, we have used the more concrete characterization of a manifold (given in #1 below). I am trying to prove that the following two conditions are ...
6
votes
1answer
131 views

Is there any relationship between Cauchy-Riemann equations and vector fields on manifolds?

Well, suppose we have $f : \mathbb{C} \to \mathbb{C}$ analytic, then if $f = u + iv$ the functions $u,v : \mathbb{C} \to \mathbb{R}$ satisfy the Cauchy-Riemann equations: $D_1u=D_2v$ and $D_2u=-D_1v$. ...
2
votes
0answers
240 views

local parametrization of regular surface

I am doing excercises of Do Carmo's dg of curves and surfaces Chapter 2.2 and need some help with the following excercise: Show that the set $S=\{(x,y,z)\in R^3;z=x^2-y^2\}$ a regular surface and ...
0
votes
1answer
44 views

Basic (multivariable) calculus question

I need some help with basic calculus. I asked a question the other day and got a decent answer but there is one step in the answer I just don't understand. Why is ${\partial y_1 \over \partial x_1} $ ...
3
votes
1answer
79 views

Polar decompostion should be a diffeomorphism, right?

I seem to have gotten stuck in the mud verifying what I thought was going to be a completely straightforward fact. I would appreciate if somebody could help dig me out. Inside the $n \times n$ ...
4
votes
1answer
109 views

Prove that the circle $S^1$ is not the boundary of any compact manifold with boundary in $\mathbb R^2-{(0,0)}$

Suppose it were, then define a 1-form $w:=\frac{1}{x^2+y^2}(-y\,\mathrm dx+x\,\mathrm dy)$. Firstly , I try to evaluate $\int_{S^1}w$ by two ways . Firstly, let $F\colon[0,2 \pi]\to S^1$ defined by ...
2
votes
1answer
110 views

Finding a direct basis for tangent space of piece with boundary of an oriented manifold.

I have the following definition (from Hubbard's vector calculus book) for an oriented boundary of piece with boundary of an oriented manifold: Let $M$ be a $k$ dimensional manifold oriented by ...
2
votes
2answers
96 views

Vector Field Generating Variation Along Curve

I'm learning a proof of the fact that length extremising curves are geodesics of the Levi-Civita connection, and have found something I don't understand. The argument states the following. Suppose ...
2
votes
1answer
220 views

Incomplete vector field

Is there a way I can tell if a vector field on a manifold or just $\mathbb{R}^n$ is incomplete simply by just looking at its formula. For instance on $\mathbb{R}$, $\displaystyle X= (x^2+1) ...
1
vote
1answer
54 views

Can you express nice manifolds as the zero locus of functions with linearly independent derivatives?

Let ${\bf f}:U\to \mathbb R^{n-k}$ be a continuously differentiable function. Then ${\bf f}^{-1}(0)$ is a manifold if $[{\bf D}{\bf f}(x)]$ is surjective at all $x$. This is equivalent to the ...
4
votes
2answers
383 views

Pushforward of Lie Bracket

I am trying to figure out why the following equality is true : $$f_*[X,Y]=[f_*X,f_*Y]$$ where $f:M\rightarrow N$ is a diffeomorphism, $M$, $N$ are smooth manifolds, $X$, $Y$ are smooth vector fields ...
1
vote
1answer
229 views

show that $[fX,gY]= fg[X,Y]+f(Xg)Y−g(Yf)X$.

If $f$ and $g$ are $C^{∞}$ functions and $X$ and $Y$ are $C^{∞}$ vector fields on a manifold $M$, show that $[fX,gY]= fg[X,Y]+f(Xg)Y−g(Yf)X$ This is a proposition in a book. But I cannot prove this:( ...
0
votes
1answer
41 views

I have done the second direction of the proof. Hopefully, it is true. Please show my mistakes?

Show that two $C^{∞}$ vector fields $X$ and $Y$ on a manifold $M$ are equal if and only if for every $C^{∞}$ function $f$ on $M$,we have $Xf =Yf$. I have sone one direction of the proof. let $p ∈ ...
0
votes
1answer
122 views

Problem about differential of a linear map

Please can you tell how to solve this problem clearly? Please solve this explanatorily. Thank you
1
vote
2answers
65 views

Compute the velocity vector.

Can you solve explicitly? please. I don't know how to solve. Thank you for help.
-1
votes
1answer
62 views

Is $S$ a regular submanifold?

$M=M_{n\times n}(\Bbb R)$ $S=\operatorname{SL}(n, \Bbb R) = \left \{ A \in M \mid \det(A)=1 \right \}$ $M$ is an $n^{2}$ dimensional $C^{\infty}$ manifold. Is $S$ a regular ...
1
vote
1answer
166 views

How general formulation of Stoke's theorem relate to Kelvin-Stokes theorem

I asked a similar question, but I realized the question is too vague and it's better to start a new one: We know that there are two usually used formulations of Stoke's theorem. One is vector ...
3
votes
2answers
102 views

Is the image of a parametrization a manifold?

Consider this definition of the parametrization of a manifold, found in Hubbard & Hubbard: A parametrization of a $k$-dimensional manifold $M\subset\mathbb{R}^n$ is a mapping $\gamma:U\subset ...
0
votes
1answer
134 views

A “Manifold with Boundary” Question

I'm given the following scenario: Letting $U$ be an open subset of $\mathbb{R}^n$ and $f,g:U\rightarrow \mathbb{R}$, two smooth functions such that $f(\vec x)\lt g(\vec x)$ for all $\vec x\in U$, ...
11
votes
1answer
514 views

Problem 3-38 in Spivak´s Calculus on Manifolds

This is not homework. Problem 3-38 reads: Let $A_{n}$ be a closed set contained in $(n,n+1)$. Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for ...
3
votes
2answers
484 views

Showing something isn't a manifold

So I'm following some notes that are introducing manifolds with pretty minimal prerequisites. What I want to do is show where the image of $\phi: \mathbb{R}\rightarrow \mathbb{R^2}$ $t\mapsto ...
2
votes
0answers
76 views

laplacian for functions problem with the integral on manifolds

I'm following the proof of the local expression for the Laplacian on a compact manifold and I'm having problems understanding how the integral on a manifold translates into an integral in $R^n$, in ...