4
votes
1answer
129 views

Why the whole theory about differentiable manifolds is based on open sets?

I only have studied basic topology, which means i haven't studied any about differentiable manifolds. I just skimmed pages on wikipedia. Here is a simple illustration on a basic situation. Let $E$ ...
3
votes
1answer
47 views

Can a proper Morse function $\mathbb{R}\to\mathbb{R}$ have infinitely many critical points?

Depending on interpretation, there may be an assumption missing from Exercise 6.1.4(a) in Liviu I. Nicolaescu's Invitation to Morse Theory: Suppose $f : \mathbb{R} → \mathbb{R}$ is a proper Morse ...
0
votes
0answers
25 views

Functions are smooth iff their product is smooth?

For $k+l=n$, we want to prove that $h:\Bbb{R}^n\to\Bbb{R}^k$ is $C^1$ if and only if $F:\Bbb{R}^n\to\Bbb{R}^k\times\Bbb{R}^l, (x,y)\to (x,h(x,y))$ is $C^1$. Write down and prove a more general ...
1
vote
1answer
73 views

Is the classical derivative the weak derivative on any domain?

I was looking at the following definition of the weak derivative: Let $\Omega$ be a domain (ie an open connected subset of $\mathbb{R}^n$). Suppose $u,v \in L_{1,loc}(\Omega)$ and \begin{equation} ...
0
votes
3answers
93 views

What dimensions are possible for contours of smooth non-constant $\mathbb R^n\to\mathbb R$ functions?

While for $n=2$ it is pretty clear that the contours of a non-constant $f:\mathbb R^n\to\mathbb R$ are either extrema (and therefore points) or (the union of) 1-dimensional isolines, for $n=3$ I am ...
1
vote
0answers
89 views

Is this function just zero?

I asked a question like this a few minutes, but am about ready to strike the problem out as errata for the book. The problem defines a function $g(x)=f(x-a)g(b-x)$. The function has bound $|f|<1$, ...
1
vote
1answer
77 views

Lifting problems existence

Let $g:\mathbb{R}^m \longrightarrow \mathbb{R}^m,g\in C^1(\mathbb{R}^m)$ such that: $\|g'(x)(v)\|\geq\|v\|,\forall v\in \mathbb{R}^m,\forall x \in \mathbb{R}^m$ show that any rectilinear path ...
1
vote
2answers
142 views

Restriction of smooth functions.

Consider the following question: Suppose that $X$ is a subset of $\mathbb{R}^n$ and $Z$ is a subset of $X$. Show that the restriction to $Z$ of any smooth map on $X$ is a smooth map on $Z$. (Note: A ...
3
votes
1answer
200 views

Prove that a standard torus is diffeomorphic to $ \mathbb S^1\times \mathbb S^1$

I was asked to prove that a standard torus(which means we don't consider those pathological cases where it intersects with itself, e.g horn torus) is diffeomorphic to $ \mathbb S^1\times \mathbb S^1$. ...
4
votes
1answer
125 views

Show that the tangent space of the diagonal is the diagonal of the product of tangent space

I'm stuck on this question for quite a few days and still haven't got a clue what to do. The question is as follows: If $\Delta$ is the diagonal of $X\times X$ where $X$ is a manifold, show that its ...
2
votes
1answer
78 views

Diagonal Inclusion Map of a manifold $X$

This question actually comes from the question I asked before: Derivative map of the diagonal inclusion map on manifolds And I repeat it as follows: Let $f: X\longrightarrow X\times X$ be the ...
6
votes
1answer
201 views

Derivative map of the diagonal inclusion map on manifolds

I was trying to work through a problem(#10 of $\S$1.2) in Guillemin and Pollack's book $\textit{differential topology. }$ The problem is given as follows. Let $f: X\longrightarrow X\times X$ be ...
0
votes
1answer
71 views

Prove that the tangent space of a hyperplane is itself

I know this might sound really stupid: I was trying to show that the tangent space of a hyperplane is itself. I started by parametrising the hyperplane locally at $x$ with a diffeomorphism $\phi : U ...
2
votes
1answer
413 views

Prove that the tangent space has the same dimension as the manifold

I asked this question a couple of days ago. And I thought that I totally understood the question. However it turned out that I didn't, since the argument I constructed was proved to be wrong just now: ...
1
vote
1answer
150 views

Prove that the dimension of the tangent space $T_x(X)$ of a k-dimensional manifold is k

In section 2, page 9 of Guillemin and Pollack's book $\textit{Differential Topology}$, he gave a proof that the dimension of the tangent space $T_x(X)$ is equal to the dimension of the manifold $X$. ...
3
votes
1answer
59 views

normal form of an n-form

It is known, that one can convert any function $f(x_1,\dots,x_n)$, defined near $0$, into the function $(y_1,\dots,y_n)\mapsto a+y_1$, by a suitable local change of coordinates, provided $df\neq 0$. ...
0
votes
3answers
60 views

Lebesgue Measure with given function?

Suppose $E$ subset $R$ ($R $\is real numbers) where $E$ is Lebesgue measurable, and $f:E\to R$ and defined $g: R\to R$ by \begin{equation*} g(x) = \begin{cases} f(x) & x \in E \\ 0 & x ...
2
votes
2answers
146 views

The most general notion of a directional derivative

Questions: I know you can define a directional derivative on some subset of $\mathbb R^n$, but what can be said about an arbitrary set of points, $S$? What are the most general criteria $S$ must ...
2
votes
1answer
156 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
104 views

Diffeomorphism between a triangle and a square?

Is there always a diffeomorphism between $(0,1)^2$ and any given (not degenerate) triangle?
1
vote
0answers
37 views

Existence of a smooth function with a given kernel

Let $K\subset \mathbb{R}^n$ be a closed set, then is there existing a smooth function $f\in C^{\infty}(\mathbb{R}^n,\mathbb{R})$, such that $$ (1)\quad f\ge 0, $$ $$ (2) \quad f^{-1}(0)=K. $$
6
votes
2answers
340 views

Smooth closed real plane curve intersecting itself at infinitely many points

Can a smooth closed real plane curve intersect itself at infinitely many points? It seems intuitively obvious that the answer should be no, yet I have no idea how to prove this or construct a ...
0
votes
1answer
113 views

smooth function $\mu:\mathbb{R}\rightarrow\!\mathbb{R}$ with $\mu(0)>\varepsilon$, $\:\mu_{[2\varepsilon,\infty)}=0$, $\:-1<\mu'\leq 0$

How can I prove (preferrably without the use of any heavy theorems) the existence of a smooth function $\mu\!:\mathbb{R}\rightarrow\!\mathbb{R}$ with properties $\mu(0)\!>\!\varepsilon$, ...
4
votes
2answers
103 views

Openness and differentiation

Given that $A$ is an open set in $\mathbb R^n$ and $f:A \to \mathbb R^n$ is differentiable, and its derivative is non-singular at every point in $A$, prove that $f(A)$ is open in $\mathbb R^n$ Note ...
0
votes
1answer
99 views

equivalence between differentiability definitions

In analysis course we encounter commonly the following definition of differentiable function: $f:U \rightarrow \mathbb{R^m}$, where $U \subset \mathbb{R^n}$ is differentiable when $\exists \ T \in ...
10
votes
2answers
470 views

Gradient nonzero extensions of a vector field on the circle

Let $\mathbf{v}=(a,b)$ be a smooth vector field on the unit circle $\mathbb{S}^{1}$ such that $a^{2}+b^{2}\neq0$ everywhere in $\mathbb{S}^{1}$ with degree $\deg\mathbf{v}=0$. Suppose also that ...