Arc Length

Tangent Line

Let $C (t)$ be a path that is differentiable so that

C^{'} (t) = [\begin{matrix} x_{1}^{'} (t) \\ ⋮ \\ x_{n}^{'} (t) \end{matrix}]

The tangent line to $C (t)$ at $t_{0}$ is

ℓ (t) = C (t_{0}) + C^{'} (t_{0}) (t - t_{0})

Assume $C : [t_{0}, t_{1}] \to R^{n}$
Arc length is:

L (C) = \int_{t_{0}}^{t_{1}} | | c^{'} (t) | | d t

Which can be thought of the sum of infinitesimal slices of the tangent lines along $[t_{0}, t_{1}]$