Curl:
There are two operations on vector fields that are fundamental to the study of fluid
flow, and electricity and magnetism. Each operation resembles a differentiation, but
one produces a vector field, and the other produces a scalar field.
If we write:
∇= ⟨ ∂ ∂ ∂
, ,
∂x ∂ y ∂z ⟩
so that
∇f= ⟨ ∂ ∂ ∂
, ,
∂x ∂ y ∂z
f ⟩
then we can view ∇ as a vector operation that resembles differentiation.
If F is a vector field on R3 , the curl of F is given by:
curl ( F )=∇ × F
If F=⟨ P , Q , R ⟩ ,
( ∂∂ Ry − ∂∂Qz ) i,( ∂∂ Pz − ∂∂ Rx ) j ,( ∂Q
curl ( F )=
∂x ∂y)
−
∂P
k
If f is a function of three variables, with continuous second-order partial derivatives,
then curl ( ∇ f ) =0
If a vector field, F , is conservative, then:
curl ( F )=0
Conversely, if a vector field, F , is not conservative, then:
curl ( F ) ≠ 0
Similarly, if F is a vector field whose domain is R3, whose component functions have
continuous partial derivatives, and if curl ( F )=0 , then F is a conservative vector field.
This operation is the curl because it is associated with rotations. For instance,
suppose F represents velocity in a fluid flow, then particles near (x , y , z) tend to
rotate or curl around vector curl (F ( x , y , z ) ). The magnitude of the vector indicates
how quickly the particles do so.
If curl ( F )=0 at a point P , then P is free from rotation at P , and F is called irrotational
at P . In other words, there is no ‘whirlpool’ or ‘eddy’ at P .
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