TeXipedia

circlearrowleft

Represents a counterclockwise circular arrow, commonly used to denote rotation, cyclic processes, or circulation in mathematical and scientific contexts.

Overview

Essential in various mathematical and scientific notations where counterclockwise motion or cyclic behavior needs to be indicated.

  • Frequently used in physics to represent angular momentum, rotational motion, or magnetic field circulation
  • Common in group theory and abstract algebra to denote cyclic groups or permutations
  • Appears in thermodynamics and engineering diagrams to indicate counterclockwise processes or cycles
  • Often paired with \circlearrowright for showing bidirectional or reversible processes

Examples

Representing a counterclockwise rotation in a physical system.

ω=2π↺\omega = 2\pi \circlearrowleft 
\omega = 2\pi \circlearrowleft

Indicating cyclic order of elements in group theory.

(a↺b↺c)(a \circlearrowleft b \circlearrowleft c) 
(a \circlearrowleft b \circlearrowleft c)

Denoting counterclockwise line integral in complex analysis.

∮γf(z) dz=2πi↺\oint_{\gamma} f(z)\,dz = 2\pi i \circlearrowleft 
\oint_{\gamma} f(z)\,dz = 2\pi i \circlearrowleft