Definition:Smooth Path/Real Cartesian Space

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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


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In particular: 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.
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