# Tagged Questions

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### 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 $E$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $|f|<1$, ...
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### 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 ...
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### 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 ...
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### 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 $\mathbb S^1\times \mathbb S^1$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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 $\phi : U ... 1answer 413 views ### Prove that the tangent space has the same dimension as the manifold I asked this question a couple of days ago. And I thought that I totally understood the question. However it turned out that I didn't, since the argument I constructed was proved to be wrong just now: ... 1answer 150 views ### Prove that the dimension of the tangent space$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$. ... 1answer 59 views ### normal form of an n-form It is known, that one can convert any function$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$df\neq 0$. ... 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$by \begin{equation*} g(x) = \begin{cases} f(x) & x \in E \\ 0 & x ... 2answers 146 views ### The most general notion of a directional derivative Questions: I know you can define a directional derivative on some subset of$\mathbb R^n$, but what can be said about an arbitrary set of points,$S$? What are the most general criteria$S$must ... 1answer 156 views ### Equivalent definition of Tangent Spaces There are about 4 definitions of tangent spaces 1) using velocities of curve 2) via derivations 3)via cotangent spaces 4) as directional derivatives. I am not getting the intuition about what tangent ... 1answer 104 views ### Diffeomorphism between a triangle and a square? Is there always a diffeomorphism between$(0,1)^2$and any given (not degenerate) triangle? 0answers 37 views ### Existence of a smooth function with a given kernel Let$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 $$(1)\quad f\ge 0,$$ $$(2) \quad f^{-1}(0)=K.$$ 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 ... 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$\mu(0)\!>\!\varepsilon$, ... 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 ... 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 ...
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 ...