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Questions tagged [smooth-functions]

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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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I'm looking for a formula that would work to elevate my students' grades. What I'm trying to say is when the minimum score gotten by my student is $0$ and the maximum is $42$, I want to convert them ...
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Suppose $f$ is a differentiable real-valued function of a real variable. By linearisation, we can write $$f(x)=f(0)+xf'(0)+xh(x)$$ where $\lim_{x\to 0} h(x)=0$. If $f$ is twice-differentiable then we ...
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I've got a question in the context of smooth and convex optimization: Let $f_i\in C(\mathbb{R}^d)$ for all $i\in[N]$ be convex and Lipschitz-smooth. That means $||\nabla f_i(x)-\nabla f_i(y)\leq L||x-...
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I'm learning differential geometry properly for the first time and I'm having a hard time understanding how the notion of a tangent vector or a derivative in the context of smooth manifolds squares ...
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For didactical / illustrative pourposes I’m searching a real valued function with the following properties: $\mathcal{C}^\infty$ over all $\mathbb{R}$ or better analytic over the entire complex plane....
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This is a past analysis exam problem: Let $f \in C^{\infty}(\mathbb{R})$ be an infinitely differentiable real-valued function on $\mathbb{R}$ so that $f(x)=1$ for all $x \in[-1,1]$ and $f(x)=0$ for ...
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Let $f,g\in C^{\infty}(\mathbb{R})$ such that \begin{equation*} f(1/n)=1, g(1/n)=\frac{n}{1-n^2} \end{equation*} for $n=2,3,4,\dots$ What are the possibile values of $f(\pi)$ ? Do you have any ...
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I just start reading Introduction to Manifolds by Loring W.Tu, and on page 6 it states Lemma 1.4 (Taylor's theorem with remainder). Let $f$ be a $C^\infty$ function on an open subset U of $\mathbb{R}^...
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I am reading "An Introduction to Manifolds Second Edition" by Loring W. Tu. Problem 1.3.(b) Let $a,b$ be real numbers with $a < b$. Find a linear function $h: \mathopen]a,b\mathclose[ \...
tchappy ha's user avatar
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Let $K$ be a constant and $x$ be a variable. What is a smooth, monotonic function that is as close to $\min(K,x)$ as possible, but never exceed $\min(K,x)$? Also f(x)>=0 for x>=0 and f(0)=0 ...
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I’ve recently been introduced to sheafs and it made me wonder weather the following statement is true: $f:\mathbb R^n \rightarrow \mathbb R^m$ is $C^k$ iff it maps $C^k$ curves to curves $C^k$. One ...
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I'm looking for a map from the square (identifying the 2-Torus) onto the closed unit disc, especially regarding: Surjectivity: I want a map from the square onto the closed unit disc, Smoothness: ...
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Let $\Omega \subseteq \mathbb{R}^n$ be neighborhood of $0$ and let $f : \Omega \to R$ be a smooth function, when does there exist a diffeomorphism $c : U \to \Omega$ with $c(0) = 0$ such that $f \circ ...
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I recently trying to solve the questions from John Lee's Smooth Manifold. And for the question 2.5, I gave this proof. But I felt something is not right in my proof, is there any modification that I ...
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I am reading "Introduction to Analysis 1" (in Japanese) by Sin Hitotumatu. Definition 7.6 A function $f(t)$ defined on $a \leq t \leq b$ is said to be piecewise smooth if it is ...
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Consider $F:\mathbb{R}^2\to \mathbb{R}$, $F(x,y)=x^3-6xy+y^2$. I am trying to find all $t\in \mathbb{R}$ whose level set $F^{-1}(t)$ is a submanifold of $\mathbb{R}^2$. Since the critical point of $F$ ...
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Let $s\in \left(0, 1\right)$, and consider the task of finding the 'nicest-possible' function $f$ such that $f(0) = 0, f^\prime (0) = s$, and $f(1) = 1, f^\prime (1) = 0$, where 'niceness' is framed ...
πr8's user avatar
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Assume we have $f\in C^\infty(\mathbb{R})$ with $f(x)=0$ for any $x\leq 0$ and $f(x)=1$ for any $x\geq 1$. What's the best possible bound for $|f''|$? I know from the lecture notes that we can choose $...
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In reading this blog post by Terence Tao, and in Remark 2, he mentions that Cesàro summation can be viewed as a form of smooth summation. Remark 2 The most famous instance of smoothed summation is ...
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Context: Let $\Omega$ be an open set in $\mathbb C$ and $K$ be a compact subset of $\Omega$. Question: find a $\alpha$ $\in$ $C^\infty_0(\Omega)$ such that it is 1 on $K$. So far: I found that by ...
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I am Reading "Differentiable manifolds, a first course" by Lawrence Conlon and I can not understand the following part: "Recall that, if $X \subset R^n$ is an arbitrary subset, a ...
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My task is: Find an injective, smooth immersion $f:]0,1[ \times ]0,1[ \rightarrow \mathbb{R}^3$ such, that its image $f(]0,1[ \times ]0,1[)$ is compact. I know, that it is possible to find the ...
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I'm looking at the proof of the Lemma 3.11, P.58 from John Lee's "Introduction to Smooth Manifolds". And I'm confused towards the end. Here's the screenshot of the whole proof: I'm ...
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Wikipedia defines the line integral of a scalar field $\int_C f({\bf s})\, ds$ for a "piecewise smooth curve $C$". Unfortunately, there does not seem to be a consistent definition across the ...
tparker's user avatar
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Consider the bump function $f \colon \mathbb{R}^n \to \mathbb{R}$ given by $f(x) = \exp( 1/(|x|^2 - 1))$ for $|x| < 1$ and $f(x) = 0$ for $|x| \geq 1$. I aim to show that $f$ is $C^\infty$, i.e., ...
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Let $$ C_0^\infty(\mathbb{R}) = \{f \in C^\infty(\mathbb{R}) \mid \forall n \in \mathbb{N}: f^{(n)} \text{ vanishes at infinity}\} $$ Does there exist a reasonable family of seminorms on $C_0^\infty(\...
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Let $S^n$ be the $n$-sphere. It is well-known that $S^n$ is the one-point compactification of $\mathbb{R}^n$ via the stereogarphic projection. Moreover, the Schwartz space on $\mathbb{R}^n$ may be ...
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I’d like an image and/or a series for a real, nowhere analytic, smooth everywhere function $f(x)$ with a Maclaurin series of $0$ i.e. $f^{(n)}(0)=0$ for $n\in\mathbb{N}$. The easiest way to generate ...
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we are looking at a function of the form $F(w)=\frac{1}{n}\sum_{i=1}^n f_i(w)$ with $f_i\in C^1$, $n\in\mathbb{N}$, $w\in\mathbb{R}^d$. Now my paper says that if $F$ is average L-smooth, that is \...
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At page 250, in the middle of the page, the definition of smooth vector bundle is given. It is said that if M and E are smooth manifolds with or without boundary, π is a smooth map, and the local ...
Jeongyeon Park's user avatar
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So I have the following exercise to solve in a geometry class about manifolds. Let $U \subset \mathbb{R}^n$ be an open set with its standard smooth structure and let $f : U \to \mathbb{R}^k$ be a map. ...
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(This is a repost from StackOverflow.) Consider an optimization engine that uses target functions and constraints that requires smooth (with at least first-order continuous derivative) functions to ...
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Let $E\subset\mathbb{R}^N$ and $0<T<\infty$. Are weak solutions $u\in L^2(0,T;W^{1,2}(E))$ to the Heat equation $$\partial_t u - \Delta u =0 \qquad \text{in}\,E\times(0,T)$$ actually (locally) ...
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Question: Function f is continuously differentiable on the whole space, it is necessarily also Lipschitz-gradient on compact set $K$. Is the following proof true: We use the fact that $K$ is compact. ...
AVA's user avatar
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Let $\mathcal{D} \,' = \mathcal{D} \,'(\mathbb{R})$ be the space of distributions (continuous linear functionals on $C_c^\infty (\mathbb{R})$). Given $f \in C^\infty(\mathbb{R})$, do we have an ...
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his is my second post on the maths stackExchange Forum, and I'm still on a subject that I've asked in 2024, my question has been answered clearly, but I've recently found this paper that contradict ...
Economos's user avatar
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I just learned about Green's theorem and learned that it applies only to positively oriented, piecewise smooth and simple closed curves. However, I don't understand what are piecewise smooth curves ...
Rupa Gaming's user avatar
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I am a begginer in Differential geometry and i only know about smooth manifold and smooth functions and this question is coming in my mind. Given a topological manifold, and given a collection of ...
Ricci Ten's user avatar
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I'm asking these three questions because I'm still missing a point in the proof of the fact that the solid pyramid $P$ in $\mathbb{R}^3$ is not a manifold with corner, as the top vertex is not a ...
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I am looking for some reference where it is explicitly stated that any weak solution to the heat equation $$\partial_t u - \Delta u = 0 \,\,\,\,\text{in}\,E\times(0,T)$$ (for some bounded domain $E\...
HelloEveryone's user avatar
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I'm approximating a smooth function $f: [0, 1] \rightarrow \mathbb{R}$ using, $$ g = \frac{1}{N_{stoc}} \sum_{m = 1}^{N_{stoc}} \langle r_m | f \rangle r_m $$ where $r_m$ is a random array of values ...
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Suppose there is a smooth curve $\gamma \subset \mathbb{R}^2$ that intersects $\mathbb{R}^1\times 0 \subset \mathbb{R}^2$ only at $(0,0)$ and is infinitely tangent to it and is in the upper-half-plane ...
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I am encountering some difficulties in understanding definitions of smoothness. My course is based on John M. Lee Introduction to Smooth Manifolds $\textbf{Definition}$ : Let $F : M \rightarrow N$ be ...
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This is an exercise from my book. We re tasked with choosing a positive, increasing, $C^{\infty}$$f:\mathbb{R}\rightarrow\mathbb{R}$ so that $F:S\rightarrow π,[x,y,z]\rightarrow\ [f(z)x,f(z)y,0 ]$ ...
Antonis Mamouras's user avatar
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14 votes
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I don't know much about the fondations of the theory of manifolds, but the way additional structure if defined on manifolds doesn't feel right to me. For example, to define a smooth manifold, we first ...
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Does there exist a function that belongs to $ C_c^{\infty} $ and is homogeneous of degree $ -n $? If so, can you provide a concrete example? In the weak sense, does there exist a function that ...
xxxg's user avatar
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I was looking at GLSL's smoothstep, aka the Hermite for $f(0)=0, f'(0)=0, f(1)=1, f'(1)=0$. That led to looking at the quintic Hermite as well, to get the ...
me22's user avatar
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I've been thinking about exactly how one determines differentiability of continuous maps between topological spaces. If $f$ is a map between topological vector spaces, the general idea is that $f$ ...
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This is a question my teacher raised in his lecture, but I don't know how to solve it. For a $C^{k}$ differential system, $k\in\mathbb{N}\bigcup\{\infty\}$, $\dfrac{dx}{dt}=f(x)$, $x\in\mathbb{R},f(x)\...
APTEX's user avatar
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Let $v:S^k\to S^k$ be a smooth map of a sphere into itself. Such a map possibly can have nonisolated fixed points (e.g. the identity map of $S^k$). Can we always homotope $v$ to a smooth map $S^k\to S^...
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