Calculus of Vector-Valued Functions and Parametrization

Vocabulary and key concepts regarding the calculus of vector-valued functions, arc length, motion vectors, and curve parametrization.

Definite integral of a vector-valued function

A vector, rather than a real number.

Arc length parametrization ($r(s)$)

A parametrization where the tangent vector $r'(s)$ is always a unit vector for any $s$.

Arc length formula (from $t = a$ to $t = b$)

$s = \int_{a}^{b} |r'(t)|\,dt$.

Arc length parameter

The arc length function defined as $s = s(t) = \int_{a}^{t} |r'(u)|\,du$.

$$\frac{ds}{dt}$$

The derivative of the arc length function with respect to time, which equals speed ($|r'(t)|$).

Velocity vector ($r'(t)$)

A vector tangent to the path that gives the instantaneous direction of motion.

Unit tangent vector ($T(t)$)

A vector that points in the direction of motion, defined as $T(t) = \frac{r'(t)}{|r'(t)|}$.

Acceleration vector

The derivative of the velocity vector $r'(t)$. It is a vector and not the derivative of the speed.

Speed

The magnitude of the velocity vector ($|r'(t)|$), which is a numerical value.

Parametrization Orientation

The direction of a curve determined by the increasing values of the parameter.

Non-uniqueness of parametrizations

The principle that every curve $C$ can be parametrized in infinitely many ways, with different speeds or directions.

Unit circle parametrization

A trigonometric representation of a circle, for example, $x = \sin(t)$ and $y = \cos(t)$.