---- > [!definition] > > A [[parameterized curve]] in a [[smooth manifold]] $M$ is a map $\gamma:I \to M$. The **velocity of $\gamma$ at $t_{0} \in I$** is defined as the [[differential of a smooth map between smooth manifolds|pushforward]] $\dot{\gamma}(t_{0}):= \gamma_{*}\left ( \frac{d}{dt} |_{t_{0}} \right) =d \gamma (\frac{d}{dt} |_{t_{0}}) \in T_{\gamma(t_{0})}M$ > where $\frac{d}{dt} |_{t_{0}}$ is the standard [[tangent space at a point of a smooth manifold|coordinate basis]] vector in $T_{t_{0}}\mathbb{R}$. The notation $\dot{\gamma}(t_{0})= \frac{d \gamma}{dt} |_{t=t_{0}}$ is also used. This [[tangent vector to a smooth manifold|tangent vector]] acts on functions as $\dot{\gamma}(t_{0})\, (f: M \to \mathbb{R})= \frac{d}{dt} |_{t_{0}} \big( f \circ \gamma \big)=(f \circ \gamma)'(t_{0}),$ > where the latter is an [[derivative|ordinary derivative]] of a function $\mathbb{R} \to \mathbb{R}$. > > > [[differential of a smooth map between smooth manifolds|In coordinates]] $(U, (x^{j}))$, writing $x^{j}(\gamma(t))=\gamma^{j}(t)$, > $\dot{\gamma}(t_{0})= \frac{d \gamma^{j}}{dt} |_{t_{0}} \, \textcolor{Thistle}{\frac{ \partial }{ \partial x^{j} } |_{\gamma(t_{0})}}$ > which is essentially the same formula as in Euclidean space: it is the [[tangent vector to a smooth manifold|tangent vector]] whose components in a [[tangent space at a point of a smooth manifold|coordinate basis]] are the derivatives of the component functions of $\gamma$. > [!definition] Definition. ([[velocity vector of a parameterized curve]]) > Let $\alpha$ be a [[derivative|differentiable]] [[parameterized curve]] in $\mathbb{R}^{n}$ on an [[open interval]] $I$. Given $t \in I$, $\alpha'(t)$ is called the **velocity vector of $\alpha$ at $t$**. <iframe src="https://www.geogebra.org/calculator/rebrrchh?embed" width="800" height="600" allowfullscreen style="border: 1px solid #e4e4e4;border-radius: 4px;" frameborder="0"></iframe> ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```