TeXipedia

curvearrowleft

Displays a curved arrow pointing counterclockwise, commonly used to indicate cyclic processes or rotational transformations.

Overview

Essential in mathematical notation and diagrams where cyclic or rotational relationships need to be depicted visually.

  • Frequently used in group theory to show cyclic permutations
  • Appears in physics diagrams representing rotational motion or angular momentum
  • Common in algorithmic flowcharts to indicate iterative processes
  • Useful in chemistry to show cyclic reactions or electron movement
  • Often paired with \curvearrowright for showing bidirectional relationships or transformations

Examples

Showing a counterclockwise rotation in a geometric transformation.

R↶=(cos⁡θ−sin⁡θsin⁡θcos⁡θ)R_{\curvearrowleft} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} 
R_{\curvearrowleft} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}

Indicating cyclic permutation of elements.

(a,b,c)→↶(c,a,b)(a,b,c) \xrightarrow{\curvearrowleft} (c,a,b) 
(a,b,c) \xrightarrow{\curvearrowleft} (c,a,b)

Denoting counterclockwise circulation in a physical system.

∮CF⃗⋅dr⃗↶\oint_{C} \vec{F} \cdot d\vec{r} \quad \curvearrowleft 
\oint_{C} \vec{F} \cdot d\vec{r} \quad \curvearrowleft