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.
871 questions
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Is there a differentiable function with a "slow" linearisation?
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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How does the differential geometry notion of a differential align with the standard notion of a derivative?
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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If $f\in C^\infty(\mathbb R)$ and $C> 0$ then find $n \in \mathbb N$ and $\xi \in \mathbb R$ such that $|f^{(n)}(\xi)| > C$
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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A Question Regarding Taylor's Remainder Theorem
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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Is there a smooth function approximating the minimum of a constant and a variable?
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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$C^\infty $ map from the square onto the closed unit disc
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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Induced Smooth Map in R
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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Determining smoothness of a level set of a critical value
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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Bounds for derivatives of smooth function
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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Find a smooth function with compact support
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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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.
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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What are the smoothness requirements for the curve on which a line integral is defined?
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 ...
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Can we make $C_0^\infty(\mathbb{R})$ into a complete locally convex topological vector space?
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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Request: Image of a nowhere analytic, smooth everywhere, and flat function
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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Question about the definition of smooth vector bundle at page 250 in "Introduction to smooth manifolds" by John M Lee
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 ...
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Smooth numerical AND function with N > 2 parameters for a C++ optimization engine
(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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a continuously differentiable function on the whole space is necessarily Lipschitz-gradient on any compact set
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. ...
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A help on a proof and/or to understand why it is false
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 ...
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Does a collection of smooth map determines unique differentiable structure of a manifold?
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 ...
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Regularity of weak solutions to the heat equation
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\...
- 701
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Curves in the plane, up to diffeomorphism
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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When is a multivariate vector function smooth and when is it conformal
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 $F:S\rightarrow π,[x,y,z]\rightarrow\ [f(z)x,f(z)y,0 ]$ ...
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Could a $C_c^{\infty}$ function be homogeneous of degress $-n$?
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 ...
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Can we define differentiable structures on topological spaces which are not manifolds?
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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A smooth map of a sphere to itself is homotopic to a map with isolated fixed points
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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