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Questions tagged [smooth-functions]

For questions about infinitely or arbitrarily differentiable (smooth) functions of one or several variables. To be used especially for real-valued functions; for complex-valued functions, the tag holomorphic-functions is more appropriate.

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1answer
34 views

Smooth, approximately space-filling curves in high dimensions

I'm looking for smooth (infinitely differentiable everywhere) functions (curves) $\mathbb{R}\rightarrow\mathbb{R}^d$ that are approximately space-filling, i.e. scaling allows the curve to eventually ...
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0answers
53 views

Exercise verification in John Lee's Introduction to Smooth Manifolds, Part 2

Let $\pi: E\to M$ be a smooth vector bundle, and $V$ an open subset of $E$ with the property that $V \cap E_p$ is non empty and convex for all $p\in M$. By "a section of $V$" we mean a (local or ...
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60 views

Exercise verification in John Lee's Introduction to Smooth Manifolds, Part 1

Let $\pi: E\to M$ be a smooth vector bundle, and $V$ an open subset of $E$ with the property that $V \cap E_p$ is non empty and convex for all $p\in M$. By "a section of $V$" we mean a (local or ...
1
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1answer
14 views

If $\Gamma(f):=\frac a2|f'|^2$, are there $0\le\eta_k\in C_c^\infty$ with $\eta_k\uparrow1$ and $\Gamma(\eta_k)\le1k$?

Let $$\Gamma(f):=\frac a2|f'|^2\;\;\;\text{for }f\in C_c^\infty(\mathbb R)$$ for some $a>0$. How can we show that there is a $(\eta_k)_{k\in\mathbb N}\subseteq C_c^\infty(\mathbb R)$ with $$0\le\...
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0answers
9 views

Real advantage in considering germs of smooth functions

I went back to read some manifold theory recently and I realized that I can't justify to myself the reason to consider germs of smooth functions over simply smooth functions other than formalism, ...
1
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1answer
12 views

Not Simultaneously Zero

I am reviewing Chapter 4, Section 6.1, Theorem 6 of D. Widder's Advanced Calculus. It states $1. f(x,y,\alpha),g(\alpha),h(\alpha) \in C^1$ $2. f_1^2+f_2^2 \ne 0$ $3. (g')^2+(h')^2 \ne 0$ $4. f(g(\...
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26 views

$M$ compact, $0$ is a regular value of $f:M\to\mathbb R,$ show that $f^{-1}(0)$ is diffeomorphic to $f^{-1}(\varepsilon)$ for small $\varepsilon.$

The exact statement of the problem is: If $M$ is compact and $0$ is a regular value of $f:M\to\mathbb R,$ then there is a neighborhood $U$ of $0\in\mathbb R$ such that $f^{-1}(U)$ is diffeomorphic ...
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23 views

constructing a special function using mollifiers.

I am reading a book where a lot of functional analysis is required and I am struggling to prove a statement. In the book it is said that, for any constant $c> \pi + \epsilon$, where $\epsilon >...
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22 views

How can prove or disprove that bounded smooth functions have a bounded derivative? [duplicate]

How can prove or disprove that bounded smooth functions have a bounded derivative ? For example cosinx is bounded and smooth and has bounded derivative . I can’t think of a proof to show that the ...
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1answer
61 views

Derivative of a exponentially decaying smooth function

Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function such that $|f(x)|=O(e^{-c|x|})$ for some $c>0$. Is it possible to infer something about the decay of its derivative?
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2answers
40 views

Mapping $\mathbb{R}^n$ “close” to integer points

For $\epsilon>0$ and a given $n$, define $$S_\epsilon=\bigcup_{x\in\mathbb{Z}^n}B_\epsilon(x)\subset\mathbb{R}^n$$ where $B_\epsilon(x)$ is the open ball of radius $\epsilon$ around the point $x$ ...
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11 views

Convolution of two exponentials e^(-1/(x-c)) and e^(-1/(d-x))

Let \begin{equation} f_c(x)=\left\{\begin{aligned} &e^{-\frac{1}{x-c}}\phantom{.......}x>c\\ &0\phantom{...........}x\le c\\\end{aligned}\right.\phantom{........}\text{ e }\phantom{..........
2
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1answer
37 views

Smooth curve segments and smooth charts

Let $M$ be a smooth manifold with or without boundary, and let $\gamma:[a,b]\to M$ be a smooth curve. I want to show that there exist a finite partition $a=a_0<a_1<\dots<a_k=b$ such that $\...
1
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1answer
16 views

Restricting a smooth function to a smaller domain is not a surjective map.

Let V $\subset$ U open sets of $\mathbb{R}^d$, then $\phi: C^{\infty}(U) \to C^{\infty}(V)$ such that $\phi(f) = f\mid_{V}$ is not a surjective function. I've been trying to prove this but I don't ...
2
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1answer
65 views

$[a,b]$ as a smooth manifold with boundary has global coordinates?

Consider a compact interval $[a,b]$. If $[a,b]$ had global coordinates, then there would be an homeomorphism $f:[a,b]\to U$ where $U$ is an open subset of $\mathbb{R}$ or an open subset of $[0,\infty)$...
4
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1answer
247 views

Discrepancy between calculus methodologies - Is it significant?

Two of the ways of doing calculus with algebra are non-standard analysis NSA and smooth infinitesimal analysis SIA. NSA has a technique called 'taking the standard part' which neglects incremental (or ...
2
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0answers
51 views

About change of coordinates of vector fields in smooth manifold theory

Background Suppose $M$ is a smooth $n$-manifold and $(U,\varphi=(x^i))$ a smooth chart on $M$. For each point $p\in U$ we know that $(\frac{\partial}{\partial x^i}|_p)_{i=1}^n$ is a basis for the ...
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35 views

Show that embeddings, diffeomorphisms, etc. are stable classes of maps

This is part of Problem 16 in Chapter 6 of Lee's Smooth Manifolds. Let $N,M,S$ be smooth manifolds. A smooth family of maps is a collection $\{F_s:N\to M \;|\; s\in S\}$ such that $F_s(x)=F(x,s)$ ...
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31 views

When the slope is flipping

My question is about function derivatives. There are cases when the slope line changes its orientation (flipping). For example, for the sigmoid function $\sigma(x)=\frac{1}{1+e^{-x}}$ at $x=0$ the ...
3
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2answers
56 views

Intuition of a Smooth Curve

I'm currently taking an intro to Differential Geometry course, and am having trouble with the definition of a smooth curve. If you consider the curve $\lambda(t) = \left (\cos^3(t),\sin^3(t)\right )$ ...
3
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1answer
35 views

Is the integral of a smooth function on a closed smooth real manifold finite?

Let $M$ be a closed smooth real manifold and $f$ a $C^\infty(M)$ function. I need to prove whether $$\int_M f(p)\mathbb dp\in\Bbb R$$ I think yes, because for all $p\in M$, $f(p)\leq k$ for some $k\...
3
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2answers
108 views

Questions on the real bump function and conjuction of smooth functions

Before I ask you my question which I will mark in bold I will tell you what I already gathered so far. In a previous result I have showed that the bumpfunction is smooth. The bumpfunction is defined ...
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1answer
11 views

If $f:M\times M\to\Bbb R$ is $C^\infty$ and $M$ a compact smooth manifold, is $\bar f(p)=\int_M f(p,q)dq$ smooth?

Let $M$ be a compact smooth manifold subset of some $\Bbb R^n$ and $f:M\times M\to\Bbb R$ a $C^\infty$ function. I need to check wether $\bar f(p)=\int_M f(p,q)dq$ is smooth, but I have no clue where ...
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0answers
17 views

Smooth functions and equalizers

Preliminaries: Let $(\mathcal{G}_0,\mathcal{G}_1)$ be a Lie groupoid. In particular, $\mathcal{G}_0,\mathcal{G}_1$ are smooth manifolds and we have smooth maps $s,t:\mathcal{G}_1\to \mathcal{G}_0$ ...
0
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1answer
33 views

Homotopy between a map and it's antipodal map

My question is similar to another on this site: Two maps are homotopic if never antipodal Let $f:S^n\rightarrow S^n$ be a continuous map such that $f(x)\neq -f(-x)$ for all $x$, I want to define a ...
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0answers
17 views

Degree (with regards to smooth manifolds) of a rational function

I have the following two parts question I would like to get a grasp on: Given two non-zero polynomials $p,q$ over $\mathbb{C}$ we can consider the function: $f(z):=\frac{p(z)}{q(z)}$ (i) Show that $...
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0answers
49 views

Confusion on $\vert\vert \varphi\vert\vert_{p}$ where $p \in [1,\infty[$

Let $\varphi \in C^{\infty}(\mathbb R^n)$ and for $\epsilon > 0$ define $\varphi_{\epsilon}(x):=\epsilon^{-n}\varphi(x/\epsilon)$ such that $\varphi_{\epsilon} \in C^{\infty}(\mathbb R^n)$ with ...
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41 views

Smoothing a function

Given a function $f(x)$ does there exist a sequence of smooth functions that $f_{n}(x) \to f(x)$ as $n \to 0$? I am currently trying to smooth out a kink for the general function $x^{1/p}$. An ...
1
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1answer
101 views

Understanding what smoothness means

Consider $\phi: A\times B \to C$, with all spaces involved topological spaces. $\phi$ is continuous if for any given neighborhood of the image point, $N_{\phi(a,b)}$, there exist neighborhoods $N_a$ ...
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0answers
41 views

Question about the Definition of Smoothness at a Point

Suppose $M$ and $N$ are smooth manifolds. Let $F:M \to N$ be any map. The definition of smoothness of $F$ is the following: Def: We say that $F$ is smooth if for each $p\in M$ there exist $(U,\phi)$...
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55 views

If $C\subset\mathbb{R}^n$ is closed, then $C=f^{-1}(0)$ for some smooth $f:\mathbb{R}^n\to\mathbb{R}$.

Let $C\subset \mathbb{R}^n$ closed. Prove that there is a smooth function $f:\mathbb{R}^n\to\mathbb{R}$ such that $C=f^{-1}(0)$. I've found this solution in the internet: take a cover of balls $\{B_i\...
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19 views

What manifold structure is natural for the codomain of a differential form defined on a manifold?

It is well known that a differential form $\omega$, of degree $r$, defined in a manifold $M^m$ is an application that at each point $p\in M^m $ associates an alternating $r$-linear form $\omega(p)\in ...
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29 views

A question about a claim from “Riemannian Geometry” by Petersen

Petersen says the following on pg 13 of "Riemannian Geometry" by Petersen: If $\frac{\psi^2-1}{t^2}$ is a smooth function of $t$ at $t=0$, then $\psi^{(1)}=\psi^{(3)}=\dots=\psi^{(odd)}=0$ I don't ...
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38 views

If $\beta$ is a continuous function with $\beta f'=0$ for all $f\in C_c^\infty$, are we able to conclude $\beta=0$?

Let $\beta:\mathbb R\to\mathbb R$ be continuous. If $$\beta f'=0\;\;\;\text{for all }f\in C_c^\infty(\mathbb R),\tag1$$ are we able to conclude $\beta=0$? If, given a compact $K\subseteq\mathbb R$, ...
2
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1answer
60 views

Is any $C_c^\infty(\mathbb R)$-function the product of two $C_c^\infty(\mathbb R)$-functions?

Let $f\in C_c^\infty(\mathbb R)$. Are we able to write $$f=gh$$ for some $g,h\in C_c^\infty(\mathbb R)$? Unfortunately, I've no idea how to I could prove or disprove this.
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0answers
34 views

How can i show a curve is smooth

we have a curve:$$x=f_1(t),y=f_2(t)$$$$t\in I[a,b]$$How can i show that this curve is smooth?$$$$ So far what I read is that the curve must have vertical tangent and perpendicular tangent. But I don’...
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0answers
24 views

Complex smooth maps over $(0 , \infty) \times G$.

Suppose that $G$ is a linear, compact and connected Lie group and that we have a map $$ u:(0,\infty) \times G \longrightarrow \mathbb{C}$$ Also we suppose that for all $t_{0}\in(0,\infty)$ the map $...
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0answers
16 views

Smothing series of calendar dates

After asking this question I obtained Meeus's More Mathematical Astronomy Morsels which leads me to believe Meeus used an entirely different approach, so I will not be pursuing this line of inquiry at ...
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0answers
55 views

Is there a missing definition for addition of vector fields?

In proving $\mathfrak X(U)$ is a $C^{\infty}U$-module, for an open subset $U$ of $\mathbb R^n$ my book defines scalar multiplication of smooth vector fields $U$ by smooth functions on $U$ as $$[fX]...
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0answers
31 views

Equivalence of definitions for ring of germs $C_p^{\infty}(\mathbb R^n)$

I want to show the characterizations of $C_p^{\infty}(\mathbb R^n)$ are equivalent: $$A= \{[f] | \text{smooth} \ f: \mathbb R^n \to \mathbb R \}$$ $$B= \{[f] | \text{smooth} \ f: U_p \to \mathbb R \}...
1
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1answer
68 views

Which is the definition of the set of germs $C_p^{\infty}(\mathbb R^n)$? Does $C^{\infty}(U)$ consist of germs or functions?

Which one is the set known as $C_p^{\infty}(\mathbb R^n)$? The set of germs of smooth real-valued functions defined on $\mathbb R^n$ The set of germs of smooth real-valued functions defined on a ...
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1answer
47 views

Can a smooth map between two embedded submanifolds be (locally) smoothly extended?

Consider $\cal M$ and $\cal M'$, smooth embedded submanifolds of two linear manifolds $\cal E$ and $\cal E'$ (respectively). Let $F \colon \cal M \to \cal M'$ be a smooth map. From Lee's textbook (...
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0answers
17 views

Smoothness is local

Let us consider a map $f:M\to R$ where $M$ is a smooth manifold. If every point $p\in M$ has a neighborhood $U$ such that $f|_U$ is smooth, prove that $f$ is a smooth function. My idea is to prove ...
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1answer
127 views

Completion of local frames for the tangent bundle of a smooth manifold

In John M. Lee book Introductions to smooth manifolds Proposition 8.11 is left as exercise. Can anyone give me hints and suggestions to prove it? In particular, i want to show: Let $M$ be a smooth ...
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0answers
60 views

Why is this function well defined and $C^\infty$? [Part 2]

Let $M\subseteq\mathbb{R}^k$ be an embedded submanifold of $\mathbb{R}^k$, with dim$M=n$. Let $(U,\phi)$ be a smooth chart for $M$. Then $\phi^{-1}:\phi(U)\to U$ is a diffeomorphism and for each $x\...
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0answers
21 views

Is it implicitly assumed that a product of manifolds have the product smooth manifold structure?

If $M$ and $N$ are smooth manifolds and $(U,\varphi)$ is a smooth chart of $M \times N$, is it standard for a text to assume that $M \times N$ has the product smooth manifold structure without ...
2
votes
2answers
78 views

“Standard reference” for $C_c^\infty(\mathbb R)$ is dense in $C_c(\mathbb R)$

$C_c^\infty(\mathbb R)$ is dense in $C_c(\mathbb R)$. This can be shown by mollification. This is a well-known, widely used fact. However, I wasn't able to find any book which I could point in a ...
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0answers
25 views

Smooth endpoint map

Consider a control system $$ \dot x(t) =f(x(t),u(t)) \qquad (\star) $$ where $f$ is a smooth vector field and $x\in \mathbb{R}^n$. The endpoint mapping is defined by $$ E:\mathbb{R}^n\times \mathbb{...
1
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1answer
53 views

Smooth map with null differential at each point are constant on the connected component of the domain

Let $F:M\to N$ be a smooth map between smooth manifolds $M$ and $N$ (with or without boundary). I want to show that $dF_p:T_pM\to T_{F(p)}N$ is the zero map for each $p\in M$ if and only if $F$ is ...
0
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1answer
60 views

Definition of smooth functions on arbitrary subsets of $\mathbb{R}^n$ and partial derivatives

Let $A$ be an arbitrary subset of $\mathbb{R}^n$, and let $f:A\to \mathbb{R}$ be a function. We say that $f$ is smooth if for each point $p$ in $A$ there exists an open subset $U$ of $\mathbb{R}^n$ ...