Closed curve

Definition

Let $a < b$, and let $\gamma \colon [a, b] \to \mathbb{R}^n$ be a parametrised curve. The curve $\gamma$ is closed if $\gamma(a) = \gamma(b)$. A closed curve is simple if, in addition, $\gamma(s) \ne \gamma(t)$ for all $s < t$ in $[a, b]$ other than the pair $s = a$, $t = b$.

Remarks

The unit circle $\gamma(t) = (\cos t, \sin t)$, $t \in [0, 2\pi]$, is a simple closed curve. The curve $t \mapsto (\sin 2t, \sin t)$, $t \in [0, 2\pi]$, is closed but not simple: it passes through the origin at $t = 0$, $t = \pi$ and $t = 2\pi$. The line integral of a vector field around a closed curve is often written $\oint_\gamma F \cdot d\mathbf{r}$. It is the circulation of $F$ around $\gamma$. A simple closed curve in the plane bounds a region. This is the setting of Green's theorem.

Explanations

L1 human

A closed curve is a path that comes back to where it started, like a lap around a track. It is simple if it never crosses itself on the way.