# 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.

125 questions
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### 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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### 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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### 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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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 \}... 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 ... 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 (... 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 ... 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 ... 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 dimM=n. Let (U,\phi) be a smooth chart for M. Then \phi^{-1}:\phi(U)\to U is a diffeomorphism and for each x\... 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 ... 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 ... 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{...
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 ...
### 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 $\mathbb{R}^n$ ...