-2
votes
0answers
18 views

what does diffeomorphism mean for differential map?

Does isomorphic differential map mean $det(DF)\neq 0$ ? Is $de(DF)\neq 0$ in the following example? For y regular value of $F:M^{n+1}\to N$, we have $dim(F^{-1}(y))=1$ i.e. circles and arcs ...
0
votes
0answers
10 views

My mistake on proving “$deg(f,y)=0$ if f can be extented”.

Statement: Show $deg(f,y)=0$, when $f:\partial M^{n}\to N ^{n}$, y is a regular value, $\exists$ extension $F:M^{n+1}\to N$ and M and N are compact smooth mflds. The outline: 1) $F^{-1}(y)$ is a ...
0
votes
0answers
32 views

How to compute saddle point index using sourcing flow lines?

Prove $Index_{p}(\bigtriangledown f)$= "dimension of sourcing flow lines from p" ,where p is a critical point. Attempt Near $ p \in Cr(f) $ in some coord. $ f(x) - f(p) = $ $\sum_j x_j^2 - \sum_k ...
1
vote
0answers
40 views

Proving the existence of a neighborhood and left inverse using calculus in $\mathbb{R}^n$

Suppose that $m < n$, that $U$ is an open set in $\mathbb{R}^m$ and that $f:U \to \mathbb{R}^n $ is a $C^1$ function that has maximal rank (rank $m$) everywhere in $U$. Show that, for each $x \in ...
3
votes
1answer
61 views

An immersive map is locally left invertible

Question: Suppose that $m < n$, that $U$ is an open set in $\mathbb R^m$ and that $f : U \rightarrow \mathbb R^n$ is a $C^1$ function that has rank $m$ everywhere in $U$. Show that for every ...
0
votes
0answers
14 views

Equivalent vector field with finitely many nondegenerate zeroes

This is an attempt to understand Milnor's proof that it's always possible to compute a vector field's index using a non-degenerate field. The strategy is simple: take $z$ to be an isolated zero of ...
1
vote
1answer
38 views

Proving Borsuk-Ulam with Stokes

What is the easiest way to deduce the Borsuk-Ulam theorem in the case $n=2$ by using integration on manifolds and Stokes theorem? So I want to prove the following: Given a map $f\colon S^2\rightarrow ...
1
vote
0answers
32 views

Constructing a non-degenerate vector-field from old one

This is an attempt to understand Milnor's proof that it's always possible to compute a vector field's index using a non-degenerate field. The strategy is simple: take $z$ to be an isolated zero of the ...
3
votes
2answers
211 views

What is the difference between differential topology and calculus on manifolds?

I'm trying to teach myself one and bought a book on the other. It seems to me that they both cover about the same material. This leads to the question: What is the difference between differential ...
2
votes
1answer
89 views

Integration formula with wedge product.

On $\mathbb{R}^3$, consider a compactly supported $2$-form $$\omega = f_1 \, dx_2 \wedge dx_3 + f_2 \, dx_3 \wedge dx_1 + f_3 \, dx_1 \wedge dx_2.$$ Choose for $S$ the parametrization $h: ...
6
votes
0answers
121 views

1-form with positive integral over a path

Given any smooth path $\gamma$ on a manifold $M$, when can we construct a 1-form $\omega \in \Omega^1(M)$ so that $\int_{\gamma} \omega > 0$? It is easy to see that such $\omega$ exists if the path ...
5
votes
2answers
67 views

$T: \mathbb{R}^n\to \mathbb{R}^m$, $n>m$, $\operatorname{rank}(dT)=m$, show $T$ maps open sets to open sets.

Suppose $T: \mathbb{R}^n\to \mathbb{R}^m$, $n>m$, with $dT$ having rank $m$ at all points in an open set $D \subset \mathbb{R}^n$. What is a proof that $T$ maps $D$ into an open set in ...
0
votes
1answer
65 views

Proof that $\operatorname{rank}(dT)=1$ implies the image is a curve

I have a question about the proof that if the differential $dT$ of a transformation has rank 1 (2) at each point in a domain, then the image will be a curve (surface). Stated more precisely (in ...
2
votes
1answer
187 views

Prove not a violation of Stokes theorem

The question is as follows: Define the vector field ${\bf F}$ on the complement of the $z$-axis by $${\bf F}(x,y,z)= \frac{-y{\bf i} +x{\bf j}}{x^{2}+y^{2}}.$$ i) Show that ...
1
vote
1answer
277 views

Proof that cone not diffeomorphic to plane

What is the simplest way to show that the cone $\{(x,y,z)\in\mathbb R^3\,|\,z=\sqrt{x^2 + y^2}, z\geq 0\}$ is not diffeomorphic to $\mathbb R^2$? After some comments, I realize that this question The ...
3
votes
0answers
87 views

Extension of Brouwer's degree to continuous functions.

I am studying the first chapter of this book: Topological Degree Theory and Applications At page 13 of this document, Definition 1.2.5, it is essentially said that to define the degree of a ...
0
votes
1answer
87 views

Explicitly Proving a parametrization for $x^2 + y^2 - z^2 = a$ for $a < 0$ is a Diffeomorphism.

Problem: I'd like to parametrize the manifold given by $\{(x,y,z)\in{\mathbb R}^{3}\,|\, x^2 + y^2 - z^2 = a\}$ for $a < 0$. The two mappings we'd use are $f(x,y) = (x,y,\sqrt{x^2 + y^2 - a})$ and ...
1
vote
3answers
118 views

A manifold being orientable vs oriented.

I read that an oriented manifold is a manifold with a choice of orientations for each tangent space so that for $p \in M$, there is an open set $U$ and a collection of vector fields $X_1,...,X_n$ so ...
0
votes
1answer
118 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
436 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 ...
1
vote
2answers
174 views

Do we need additional assumptions for problem 3-37 (b) in Spivak´s calculus on Manifolds?

Problem 3-37 (b) reads: Let $A_{n}=[1-1/2^{n},1-1/2^{n+1}]$. Suppose that $f:(0,1)\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for any $x\notin$ any $A_{n}$. Show that ...
3
votes
2answers
316 views

Solving problem 3-29 in Spivak´s Calculus on Manifolds without using change of variables

Problem 3-29 (p. 61) in the section treating FubiniĀ“s theorem reads: Use FubiniĀ“s theorem to derive an expression for the volume of a set of $\mathbb{R}^{3}$ obtained by revolving a Jordan-measurable ...
2
votes
1answer
73 views

Need help understanding proof about critical values of the determinant map.

In , problem 5 the author shows that the differential of the determinant map $d(\det)_A$ for an invertible matrix $A$ is nonsingular by only showing that $d(\det)_A(A) \ne 0$. I don't really see why ...
1
vote
2answers
281 views

Explain the Stokes -theorem from differential from into Integral form

I want to understand the Stokes -theorems deeper. I am trying to understand the operation from $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$$ to ...
5
votes
1answer
313 views

Stokes for integration along the fiber

I want to use a version of Stokes theorem for integration along the fiber and I need some help in proving a general statement. Let $F$ be a $k$-manifold with boundary and let $E \to M$ be a smooth ...