Definition:Smooth Curve

Definition

Let $M$ be a smooth manifold.

Let $I$ be a open real interval, considered as a smooth manifold of dimension $1$.

This article, or a section of it, needs explaining. In particular: We have various pages defining "manifolds" of various types (differentiable, smooth, complex), but not one defining a basic "manifold". It can be assumed by inference that a manifold is "a second-countable locally Euclidean space of finite integral dimension" but this needs to be rigorously and unambiguously clarified by including that definition at the top level of the page Definition:Topological Manifold. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Then a smooth mapping $\gamma : I \to M$ is called a smooth curve.

$3$-Dimensional Real Vector Space

Let $\R^3$ be the $3$-dimensional real vector space.

Let $I$ be an bounded or unbounded open interval.

A smooth curve in $\R^3$ is a mapping $\alpha : I \to \R^3$ defined as:

$\map \alpha t := \tuple {\map x t, \map y t, \map z t}$

where $\map x t, \map y t, \map z t$ are smooth real functions.

Also defined as

A smooth curve can also be seen defined as a curve which has a continuous first derivative.

Note that this definition does not correspond with the definition of smooth curve as defined on $\mathsf{Pr} \infty \mathsf{fWiki}$, which requires that all higher derivatives are all also continuous.

Also see

• Results about smooth curves can be found here.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Lengths and Distances