---
tag: C57T562NKB
kind: concept
name: Differentiable curve
msc: [53A04]
specializes: [0G7WM8DX4P]
sources:
  - url: https://en.wikipedia.org/wiki/Differentiable_curve
---

## Summary

A differentiable curve is a [[parametrised curve|0G7WM8DX4P]] whose coordinate functions have continuous derivatives up to some order $k \ge 1$.
It is the basic object of the differential geometry of curves.
Tangent vectors, arc length, curvature and torsion are all defined for differentiable curves of suitable order.

## Explanations

### L1 human

Think of a point that moves smoothly through space, with a velocity at each moment.
A differentiable curve records the position of that point as a function of time.
Being differentiable means that the velocity exists and changes continuously.
For example, $t \mapsto (\cos t, \sin t, t)$ is a differentiable curve, the helix, which winds around a cylinder.