Length of Curves

Path, Curve

A path in \( \mathbb{R}^{n} \) is a continuous function \( \gamma:I\mapsto \mathbb{R}^{n} \) where \( I \) is an interval in \( \mathbb{R} \). The image set \( \gamma(I) \) is called a curve.

Velocity vector, Speed

Let \( \gamma:I \mapsto \mathbb{R}^{n} \) be a diffetiable path. Then \( \mathbf{v}(t) = \gamma'(t) \) is called velocity vector of the path, and its length \( \| \mathbf{v}(t) \| \) is called the speed of the path.

“Proposition”. Let \( \gamma: [a,b] \mapsto \mathbb{R}^{n} \) be a diffetiable path of class \( C^{1} \). Then the length of \( \gamma \) is given by

\[ \int_{a}^{b} \| \gamma'(t) \|\, dt \]