What is curl in mathematical terms?

Answer 1

Curl represents the force of rotation in a 3-D vector field. Generally, the curl vector at a given point is the answer to the question, "What would happen if I stuck something there that could spin but couldn't move?" Unless the curl is zero, it would spin perpendicularly to the curl vector (according to the right-hand rule), and the longer the vector is, the faster.

Curl is mathematically defined in a given direction as the limit of "circulation over area", i.e. the line integral of a circle around the point, divided by the area of the circle, with the circle shrinking towards the point. More practically, the actual vector can found by taking the cross product of the gradient operator with the function that defines the field:

curl_x = ∂F/∂y - ∂F/∂z

curl_y = ∂F/∂z - ∂F/∂x

curl_z = ∂F/∂x - ∂F/∂y