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Curl
Curl In vector calculus, the curl (or cross-product ) operation is a linear transformation that transforms a vector field into another vector field....
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Curl In vector calculus, the curl (or cross-product ) operation is a linear transformation that transforms a vector field into another vector field....
Curl
In vector calculus, the curl (or cross-product) operation is a linear transformation that transforms a vector field into another vector field. It measures the "rotation" or "curvature" of a vector field at a point.
Intuitively:
The curl operation takes a vector field and "pulls" it in a certain direction, creating a new vector field.
The direction of the pull depends on the order of the cross product (right-hand or left-hand).
The magnitude of the curl represents the magnitude of the "rotation" at that point.
Formally:
where:
{\bf v} is the vector field
{\bf i}, {\bf j}, {\bf k} are the standard unit vectors
{\partial/\partial x}, {\partial/\partial y}, {\partial/\partial z} are the partial derivatives
Examples:
curl {\bf i} = 0
curl {\bf j} = 0
curl {\bf k} = 1
curl {\bf i} + {\bf j} = 0
In higher dimensions, the curl operation can be expressed in terms of determinants of matrices