Definition:Smooth Path

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Definition

Real Cartesian Space

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 $t$.


Complex Analysis

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

Let $\gamma: \left[{a \,.\,.\, b}\right] \to \C$ be a path in $\C$.

That is, $\gamma$ is a continuous complex-valued function from $\left[{a \,.\,.\, b}\right]$ to $\C$.


Define the real function $x : \left[{a \,.\,.\, b}\right] \to \R$ by:

$\forall t \in \left[{a \,.\,.\, b}\right]: x \left({t}\right) = \operatorname{Re} \left({\gamma \left({t}\right)}\right)$

Define the real function $y: \left[{a \,.\,.\, b}\right] \to \R$ by:

$\forall t \in \left[{a \,.\,.\, b}\right]: y \left({t}\right) = \operatorname{Im} \left({\gamma \left({t}\right)}\right)$

where:

$\operatorname{Re} \left({\gamma \left({t}\right)}\right)$ denotes the real part of the complex number $\gamma \left({t}\right)$
$\operatorname{Im} \left({\gamma \left({t}\right)}\right)$ denotes the imaginary part of $\gamma \left({t}\right)$.


Then $\gamma$ is a smooth path (in $\C$) if and only if:

$(1): \quad$ Both $x$ and $y$ are continuously differentiable
$(2): \quad$ For all $t \in \left[{a \,.\,.\, b}\right]$, either $x' \left({t}\right) \ne 0$ or $y' \left({t}\right) \ne 0$.