As we know that the Curl of linear velocity is equal to twice the rotational velocity (Mathematically). In practical life if it is possible to construct a Curl operator which transform the linear velocity into rotational velocity, then it will be very helpful for our society. Please share your opinion/s....
What do you mean for "practical life"? From a mathematical point of view curl v is a new vector that requires v to be differentiable and results in a new direction.
In fluid mechanics this operation produces the vorticity vector field and we have several pratical use. I think also about the traffic flow dynamics in our society.
Thank you Professor Filippo for giving comments.
My thinking about "practical life" means a package (that gives direct results, likes matlab, mapple, etc) that converts linear velocity into rotational velocity. For example, When we are driving a car, wheels of the car are rotating. If possible, we will supply linear velocity and package (curl operator) will transform it into rotational (circular) velocity.
It sounds like you are describing a rotation matrix or possibly the special case of infinitesimal rotations. In the case of a car with wheels changing direction, if I remember correctly you would need to use the function of the angle, f(θ), of the wheels in the rotation matrix. The partial derivatives of f(θ) will become the entries in the rotation matrix. We did these in linear algebra, and they are closely related to to the curl.
Here is a wikipedia article about rotation matrices: https://en.wikipedia.org/wiki/Rotation_matrix and the special infinitesimal case: https://en.wikipedia.org/wiki/Rotation_group_SO(3)#Infinitesimal_rotations. This seems to jive with a geometric interpretation of curl: https://en.wikipedia.org/wiki/Curl_(mathematics)#Curl_geometrically.
Also, a Wolfram article about the same topic displayed with a touch more rigor: http://mathworld.wolfram.com/InfinitesimalRotation.html
Additionally, here is a derivation of the infinitesimal rotation matrix: https://arxiv.org/ftp/arxiv/papers/1311/1311.6010.pdf
It is quite simple to construct a 3x3 matrix for the curl operator, for example in the Cartesian system as
i j k
d/dx d/dy d/dz
u(x) v(x) w(x)
You can also use the Maple software for symbolic calculus
https://www.maplesoft.com/support/help/maple/view.aspx?path=VectorCalculus%2FCurl
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