curl
The curl (also known as rotor) is a first order linear
differential operator which acts on vector fields in .
Intuitively, the curl of a vector field measures the extent to which a
vector field differs from being the gradient of a scalar field. The
name ”curl” comes from the fact that vector fields at a point with a
non-zero curl can be seen as somehow ”swirling around” said point. A
mathematically precise formulation of this notion can be obtained in
the form of the definition of curl as limit of an integral
about a
closed circuit.
Let be a vector field in .
Pick an orthonormal basis and write . Then the curl of , notated or or , is given as follows:
By applying the chain rule, one can verify that one obtains the same
answer irregardless of choice of basis, hence curl is well-defined as
a function
of vector fields. Another way of coming to the same
conclusion is to exhibit an expression for the curl of a vector field
which does not require the choice of a basis. One such expression is
as follows: Let be the volume of a closed surface enclosing
the point . Then one has
Where is the outward unit normal vector to .
Curl is easily computed in an
arbitrary orthogonal coordinate system by using the appropriate
scale factors. That is
for the arbitrary orthogonal curvilinear coordinate system
having scale factors .
Note the scale factors are given by
Non-orthogonal systems are more easily handled with tensor analysis or exterior calculus.
Title | curl |
Canonical name | Curl |
Date of creation | 2013-03-22 12:47:39 |
Last modified on | 2013-03-22 12:47:39 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 17 |
Author | rspuzio (6075) |
Entry type | Definition |
Classification | msc 53-01 |
Synonym | rotor |
Related topic | IrrotationalField |
Related topic | FirstOrderOperatorsInRiemannianGeometry |
Related topic | AlternateCharacterizationOfCurl |
Related topic | ExampleOfLaminarField |
Defines | curl of a vector field |