In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a three-dimensional vector field. At a given point in the field, the curl is represented by a vector. The length and direction of this vector characterize the magnitude and axis of the maximum rotation or circulation.

Intuitive Understanding

Imagine a vector field representing the velocity of water in a flowing river. If you place a tiny paddle wheel in the water, the curl measures the tendency of the paddle wheel to rotate:

  • Magnitude of Curl: A high curl magnitude means the paddle wheel would spin rapidly.
  • Direction of Curl: The direction of the curl vector aligns with the axis around which the paddle wheel would rotate, using the right-hand rule.

Formal Definition

The curl of a vector field F, denoted by curl(F) or ∇ × F, is defined as:

\(curl(F) = ∇ × F =    det (      \begin{matrix}       \hat{\imath} & \hat{\jmath} & \hat{k} \\        \frac{\partial}{\partial x} & \frac{\partial}{\partial y} &  \frac{\partial}{\partial z} \\       F_x & F_y & F_z     \end{matrix}    )\)


  • is the del operator (represents partial differentiation)
  • F = (\(F_x, F_y, F_z\)) is the vector field
  • î, ĵ, and are unit vectors in the x, y, and z directions.


  • Irrotational Fields: A vector field with zero curl everywhere is called irrotational. These fields are conservative, meaning their line integrals are path-independent.
  • Relation to Circulation: The circulation of a vector field around a closed loop is equal to the line integral of the curl of the field over the enclosed surface (this is Stokes' Theorem).


  • Fluid Dynamics: The curl helps analyze fluid rotation and vorticity.
  • Electromagnetism: The curl of the electric field is related to the time-varying magnetic field, and the curl of the magnetic field is related to the current density (Maxwell's Equations).
  • Physics and Engineering: The curl is broadly used in physics and engineering whenever phenomena involving rotation or circulation are studied.