Definition:Smooth Path

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 $\closedint a b$ be a closed real interval.

Let $\gamma: \closedint a b \to \C$ be a path in $\C$.

That is, $\gamma$ is a continuous complex-valued function from $\closedint a b$ to $\C$.

Define the real function $x: \closedint a b \to \R$ by:

$\forall t \in \closedint a b: \map x t = \map \Re {\map \gamma t}$

Define the real function $y: \closedint a b \to \R$ by:

$\forall t \in \closedint a b: \map y t = \map \Im {\map \gamma t}$

where:

$\map \Re {\map \gamma t}$ denotes the real part of the complex number $\map \gamma t$

$\map \Im {\map \gamma t}$ denotes the imaginary part of $\map \gamma t$.

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 $t \in \closedint a b$, either $\map {x'} t \ne 0$ or $\map {y'} t \ne 0$.