Velocity Vector of a Curve¶

Definition¶

Let $\boldsymbol{\gamma}\colon I\to\mathbb{R}^n$ be a differentiable curve, that is, a path parametrized by an interval $I\subseteq\mathbb{R}$ rather than by $[0,1]$ . If each component $\gamma_i$ is differentiable at $t_0$ , then the velocity vector of $\boldsymbol{\gamma}$ at $t_0$ is

\boldsymbol{\gamma}'(t_0)= \begin{pmatrix} \gamma_1'(t_0)\\ \vdots\\ \gamma_n'(t_0) \end{pmatrix}.

The velocity is the tangent vector to the curve at the point $\boldsymbol{\gamma}(t_0)$ .

Interpretation¶

The velocity vector records the instantaneous direction and rate of motion along a path: each component says how fast one coordinate of the position changes at time $t_0$ .

A curve gamma(t) in the plane with its velocity vector at gamma(t0), tangent to the curve and split into its horizontal and vertical components.

The red arrow $\boldsymbol{\gamma}'(t_0)$ starts at the point $\boldsymbol{\gamma}(t_0)$ and touches the curve there. The dashed legs are its components $\gamma_1'(t_0)$ and $\gamma_2'(t_0)$ , the derivatives of the coordinate functions. Its length $\|\boldsymbol{\gamma}'(t_0)\|$ is the speed. If the velocity is nonzero, the tangent line is the line $\boldsymbol{\gamma}(t_0)+s\boldsymbol{\gamma}'(t_0)$ $s γ^{'} (t_{0})$ , $s\in\mathbb{R}$ $R$ .

The velocity $\boldsymbol{\gamma}'(t)$ is tangent to the curve; its components are the derivatives of the coordinates, and its length is the speed.

A curve here is a map $t\mapsto\boldsymbol{\gamma}(t)$ $γ (t)$ , not only its image. The unit circle traced as $(\cos t,\sin t)$ has speed $1$ , traced as $(\cos 2t,\sin 2t)$ it has speed $2$ : same image, different velocities. The condition $\boldsymbol{\gamma}'(t_0)\neq\boldsymbol{0}$ in the tangent statement is needed: $\boldsymbol{\gamma}(t)=(t^3,t^2)$ has velocity $\boldsymbol{0}$ at $t=0$ $0$ , and its image has a sharp point (a cusp) at the origin, so the velocity there gives no tangent direction.

For $n=1$ the velocity is the ordinary derivative, and in general it is the single column of the Jacobian matrix of $\boldsymbol{\gamma}$ . Combined with a gradient, it gives the rate of change of a function along the curve, the chain rule on a curve. For the straight line $\boldsymbol{\gamma}(t)=\boldsymbol{a}+t\boldsymbol{u}$ with unit $\boldsymbol{u}$ the velocity is $\boldsymbol{u}$ , and that rate at $t=0$ is the directional derivative $D_{\boldsymbol{u}}f(\boldsymbol{a})$ .

Key Results¶

Velocity is computed component by component.
If $\boldsymbol{\gamma}'(t_0)\neq\boldsymbol{0}$ $0$ , then $\boldsymbol{\gamma}'(t_0)$ gives a tangent direction to the image of the curve.
The speed of the curve is $\|\boldsymbol{\gamma}'(t_0)\|$ when a norm is being used.

Examples¶

For $\boldsymbol{\gamma}(t)=(t,t^2)$ $(t, t^{2})$ , the velocity is $\boldsymbol{\gamma}'(t)=(1,2t)$ $(1, 2 t)$ , so at $t=1$ the curve passes through $(1,1)$ with velocity $(1,2)$ .
For $\boldsymbol{\gamma}(t)=(\cos t,\sin t)$ $(cos t, sin t)$ , the velocity is $\boldsymbol{\gamma}'(t)=(-\sin t,\cos t)$ $(- sin t, cos t)$ , which is tangent to the unit circle.

Prerequisites¶

Sources¶

Math IV: Multivariate Calculus · Week 1, Lecture 2

Aliases: velocity of a curve, velocity vector Keywords: derivative of a parametrized curve, tangent vector Symbols: element of, gamma, real numbers, subset

Last updated: 20 October 2026 03:02