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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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Coordinate-free definition of $C^k$

There is the concept of smoothness classes $C^k$ for a function $f:\mathbb{R}^n \to \mathbb{R}$. Are there a coordinate-free version of it for a function $f:\mathbb{E}^n\to\mathbb{R}$ from the $n$-...
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2answers
37 views

If $f, g$ are both smooth functions of $t$, why does $f \circ g$ is a smooth function of $t$?

Let $f,g: \mathbb{R} \to \mathbb{R}$. I think it is clear that $f \circ g$ has derivatives of all orders since $\dfrac{d^n}{dt^n} (f\circ g)$ only depends of $f, g$ and it's derivatives. But I'm ...
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1answer
53 views

Chain rule and smooth functions

I have read often the following. Let $U\subset \mathbb R^n$ and $V \subset \mathbb R^m$. If $f: U \rightarrow V$ and $g: V \rightarrow \mathbb R^p$ are smooth functions then $g \circ f: U \...
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2answers
28 views

Examples of smooth functions 1 [closed]

I am trying to understand smooth functions. My question is what is an example of a smooth function $f\colon\mathbb{R}\to\mathbb{R}$ such that $f(x)=0$ for all $x\leqslant-1$ and $x\geqslant1$, but $f(...
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1answer
35 views

How can I create a cubic spline to connect these two line segments?

I am trying to create an audio envelope smoothing curve to smooth a steady signal (y=1) as it goes into exponential decay (y=c^-x). I have expressed the graphs of the two line segments I am trying to ...
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1answer
88 views

Smooth images of manifolds are immersed?

in various papers in symplectic geometry, I have encountered the following argument. Statement: Suppose $f: M \rightarrow N$ is a smooth map of constant rank. Then its image $f(M)$ can be equipped ...
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67 views

The identity theorem at the boundary (complex analysis)

Let $\mathbb{D}^2$ be the closed unit disk, and let $f:\mathbb{D}^2 \to \mathbb{C}$ be a smooth map, which is holomorphic on the open unit disk $\text{int}(\mathbb{D}^2)$. Suppose that there exists ...
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28 views

Prove the set of linear continuous inversible operators is open

Let E be a Banach space. Let $L(E)$ be the set of linear continuous operators from E to E. Let A be the set of inversible elements of $L(E)$. Prove that A is open in $L(E)$ and that $T→ T^{−1}$ is ...
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26 views

Flexible grid update algorithm

I have a 2D square grid where the edges are line segments and the vertices have moved from their initial positions. Based on external constraints, the vertices are submitted to further motion, under ...
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2answers
66 views

Smooth approximation of $f(x)=\begin{cases}0&\text{if}\;x<0\\x&\text{if}\;x \geq 0 \end{cases}$ [closed]

I'd like to find a smooth function to approximate $$f(x)=\begin{cases}0&\text{if}\;x<0\\x&\text{if}\;x \geq 0 \end{cases}$$ This function should be differentiable everywhere. Thanks.
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1answer
19 views

Determining the linear independence of tangent vectors at a point on the manifold

We define the tangent space at a point, say $x_0$, on the manifold $M$ as the set of all derivations, i.e maps which maps smooth maps from a neighbourhood of $x_0$ to real numbers to real numbers. ...
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1answer
36 views

Differential and manifold, concrete example of a calculation

I having some problem for computing the differential of the function $f : \mathbb{T}^2 \rightarrow \mathbb{S}^2$ defined as the quotient of the function $F$ from $\mathbb{R}^2$ to $\mathbb{S}^2$ by : ...
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1answer
26 views

If a real multi-variable function is analytic along all analytic curves passing through $0$, is it real-analytic?

Given $f(x):\mathbb{R}^n\rightarrow\mathbb{R}$, if for any curve $\gamma:[-\epsilon,\epsilon]\rightarrow\mathbb{R}^n:t\mapsto(\gamma_1(t),\ldots,\gamma_n(t))$ such that $\gamma(0)=0$ and each $\...
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1answer
22 views

Is the Cartesian product of $C^\infty$ functions a $C^\infty$ function?

Define the Cartesian product of two functions $f:\mathbb{R}^a\to\mathbb{R}^b$ and $g:\mathbb{R}^c\to\mathbb{R}^d$ as $$(f\times g)(x,y)=(f(x),g(y)).$$ If the function $f$ and $g$ are $C^\infty$, is ...
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34 views

Transformation between two measures

If $\mu$ and $\nu$ are two measures, both absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^d$ with smooth densities $p_\mu(\mathbb{x})$ and $p_\nu(\mathbf{x})$, does it always ...
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0answers
24 views

Perimeter of level sets of a smooth function

I've a simple question concerning the perimeter of level sets of a smooth function. Let $f:\Omega \to \mathbb{R}$ be a smooth function defined on a bounded domain of $\mathbb{R}^n$. We set $A_s:=\{f&...
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1answer
32 views

Smooth functions with equal derivatiesof all orders in one point

Let's assume that $f,g:\mathbb{R} \rightarrow \mathbb{R}$ are smooth functions and $\forall n\in \mathbb{N}, \forall x\in \mathbb{R}:\, |f^{(n)}(x)|, |g^{(n)}(x)| < 1 $ prove that if $\forall n ...
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1answer
40 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
58 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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73 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 ...
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1answer
20 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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10 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, ...
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1answer
19 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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31 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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0answers
25 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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23 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
63 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
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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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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{..........
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1answer
41 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 $\...
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1answer
17 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 ...
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1answer
73 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 $[0,\infty)$...
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1answer
259 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 ...
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56 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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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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32 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 ...
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2answers
75 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 $\lambda(t) = \left (\cos^3(t),\sin^3(t)\right )$ ...
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1answer
37 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\...
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2answers
115 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
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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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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 $s,t:\mathcal{G}_1\to \mathcal{G}_0$ ...
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1answer
41 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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19 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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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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45 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 ...
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1answer
106 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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44 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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0answers
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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0answers
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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43 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 $K\subseteq\mathbb R$, ...