:()()!Let:is a vector field in the space (3-D Velocity field):

For x-y plane, 3-D vector field reduced to 2-D vector field ()

To analysis swirl about z-axis, use polar coordinateat point:

Where and are unit vectors.Angular velocity about the z-axis is:

1 2 (reduce by Mean value Theorem)1:

2:

=>1:

=>2:

!,(w=v/dr) =>!

Since the integral , over one period , of ,,are zeroAnd since, this average is seen to be

About y-z plane and x-z plane

;

We want the curl of a vector field to express its tendency to swirl; so,Dropping factorfor convenience, we formulate the following definition:The curl of a vector field is the vector field:

Rather than memorize, it is advised to write the curl in the form of a symbolic determinant:The curl of a vector field :

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