Curl

Curl is a term used in various contexts in mathematics and physics. In vector calculus, it is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. In physics, it is used to define the rotation of a field, such as the magnetic or velocity fields.

Definition

In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. At every point in the field, the curl of that point is represented by a vector. The attributes of this vector (length and direction) characterize the rotation at that point.

The curl of a vector field F, denoted by curl F, or ∇ × F, is defined as the vector field having magnitude equal to the maximum "circulation" at each point and to be oriented perpendicularly to this plane of circulation for each point.

Mathematical Formulation

The curl of a vector field F, denoted by curl F or ∇ × F, is defined using the del operator (∇) and the cross product (×). The curl of F is given by:

∇ × F = ( ∂Fz/∂y - ∂Fy/∂z ) i + ( ∂Fx/∂z - ∂Fz/∂x ) j + ( ∂Fy/∂x - ∂Fx/∂y ) k

where i, j, and k are the unit vectors for the x, y, and z directions, respectively.

Physical Interpretation

In physics, the curl of a field is used to determine the rotation of that field. For example, the curl of a velocity field gives the vorticity, or the local spinning motion of a fluid near some point, as opposed to the motion of the fluid at that point.

See Also

• Vector calculus
• Vector field
• Del operator
• Cross product
• Vorticity

References

• MathWorld - Curl
• PhysicsWorld - Curl

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