curl

Intuitively, the curl measures how much is a vector field rotating or spinning around a point. It is often written as or .

Demonstrations

The following situation also has a non-zero curl:

Curl in 2D

Given , and and both exist, When the rotation is counter-clockwise, we have positive curl. And when the rotation is clockwise, we have negative curl.

Intuition

Consider the following counter-clockwise rotation configuration.

• If we want to be larger as becomes larger, we want .
• Similarly, if we wish to be smaller as becomes larger, then we want .

This is the intuition why , as we try to make the curl positive when we have counter-clockwise rotation.

Curl in 3D

Given ,

Note that this is the reason we often write curl as cross product , where is the Del operator (). Then we can use the following determinant mnemonic of cross product:

3-dimensional curl can be understood as rotation around a specific axis (where the direction of the curl is the rotation axis, following the right-hand rule). In the same sense, the 2D curl can be interpreted as a 3D curl rotation around the axis: .

Relationship with Gradient Fields and Conservative Vector Fields

See: conservative vector fields and curl

If has continuous second order partial derivatives then . As a direct consequence, if is a conservative vector field, then .

On the other direction, if is defined on every point, all components have continuous first-order partial derivatives, and , then is a conservative vector field.

See Also

• divergence
• Green’s theorem

References

• Divergence and curl: The language of Maxwell’s equations, fluid flow, and more - YouTube