# Definition:Smooth Path/Real Cartesian Space

## Definition

Let $\R^n$ be a real cartesian space of $n$ dimensions.

Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval.

Let $\rho: \left[{a \,.\,.\, b}\right] \to \R^n$ be a path in $\R^n$.

That is, let $\rho$ be a continuous real-valued function from $\left[{a \,.\,.\, b}\right]$ to $\R^n$.

For each $k \in \left\{ {1, 2, \ldots, n}\right\}$, define the real function $\rho_k: \left[{a \,.\,.\, b}\right] \to \R$ by:

- $\forall t \in \left[{a \,.\,.\, b}\right]: \rho_k \left({t}\right) = \pr_k \left({\rho \left({t}\right)}\right)$

where $\pr_k$ denotes the $k$th projection from the image $\operatorname{Im} \left({\rho}\right)$ of $\rho$ to $\R$.

Then $\rho$ is a **smooth path (in $\R^n$)** if and only if:

- $(1): \quad$ all of $\pr_k$ are continuously differentiable

- $(2): \quad$ for all $t \in \left[{a \,.\,.\, b}\right]$, at least one $\rho_k' \left({t}\right) \ne 0$, where $\rho_k'$ denotes the derivative of $\rho_k$ with respect to $t$.

### Closed Smooth Path

Let $\rho: \left[{a \,.\,.\, b}\right] \to \R^n$ be a **smooth path in $\R^n$**.

$\rho$ is a **closed smooth path** if and only if $\rho$ is a closed path.

That is, if and only if $\rho \left({a}\right) = \rho \left({b}\right)$.

### Simple Smooth Path

Let $\rho: \left[{a \,.\,.\, b}\right] \to \R^n$ be a **smooth path in $\R^n$**.

$\rho$ is a **simple smooth path (in $\R^n$)** if and only if:

- $(1): \quad \rho$ is injective on the half-open interval $\left[{a \,.\,.\, b}\right)$

- $(2): \quad \forall t \in \left({a \,.\,.\, b}\right): \rho \left({t}\right) \ne \rho \left({b}\right)$

That is, if and only if $t_1, t_2 \in \left({a \,.\,.\, b}\right)$ with $t_1 \ne t_2$, then $\gamma \left({a}\right) \ne \gamma \left({t_1}\right) \ne \gamma \left({t_2}\right) \ne \gamma \left({b}\right)$.

## Also see

## Sources

There are no source works cited for this page.I have come up with this as an intuitive idea of what I believe it ought to be, based on extrapolating from the complex case. The same applies to the pages dependent on this. This is so I can get the basics of vector analysis down as it has been presented in Spiegel's Mathematical Handbook. As soon as I find some actual proper sources I will work on them properly.Source citations are highly desirable, and mandatory for all definition pages.Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page. |