Tagged Questions
2
votes
1answer
86 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
78 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 ...
2
votes
0answers
36 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
44 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
97 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
86 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 $\vec x\in U$, ...
10
votes
1answer
320 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
121 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
222 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 Fubini´s theorem reads:
Use Fubini´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
39 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 ...
0
votes
2answers
193 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
192 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 ...
