curl

curl

The

curl

(also known as

rotor

) is a first order linear

differential operator

which acts on

vector fields

in

ℝ

3

.

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

F

be a vector field in

ℝ

3

.

Pick an

orthonormal basis

{

e

1

→

,

e

2

→

,

e

3

→

}

and write

F

→

=

F

1

⁢

e

1

→

+

F

2

⁢

e

2

→

+

F

3

⁢

e

3

→

. Then
the curl of

F

, notated

curl

⁡

F

→

or

rot

⁡

F

→

or

∇

→

×

F

→

, is given
as follows:

curl

⁡

F

→

=

[

∂

⁡

F

3

∂

⁡

q

2

-

∂

⁡

F

2

∂

⁡

q

3

]

⁢

e

1

→

+

[

∂

⁡

F

1

∂

⁡

q

3

-

∂

⁡

F

3

∂

⁡

q

1

]

⁢

e

2

→

+

[

∂

⁡

F

2

∂

⁡

q

1

-

∂

⁡

F

1

∂

⁡

q

2

]

⁢

e

3

→

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

V

be the volume of a closed surface

S

enclosing
the point

p

. Then one has

curl

⁡

F

→

⁢

(

p

)

=

lim

V

→

0

⁡

1

V

⁢

∫

∫

S

n

→

×

F

→

⁢

𝑑

S

Where

n

is the outward

unit normal

vector to

S

.

Curl is easily computed in an
arbitrary

orthogonal coordinate system

by using the appropriate

scale factors

. That is

curl

⁡

F

→

=

1

h

3

⁢

h

2

⁢

[

∂

∂

⁡

q

2

⁢

(

h

3

⁢

F

3

)

-

∂

∂

⁡

q

3

⁢

(

h

2

⁢

F

2

)

]

⁢

e

1

→

+

1

h

3

⁢

h

1

⁢

[

∂

∂

⁡

q

3

⁢

(

h

1

⁢

F

1

)

-

∂

∂

⁡

q

1

⁢

(

h

3

⁢

F

3

)

]

⁢

e

2

→

+

1

h

1

⁢

h

2

⁢

[

∂

∂

⁡

q

1

⁢

(

h

2

⁢

F

2

)

-

∂

∂

⁡

q

2

⁢

(

h

1

⁢

F

1

)

]

⁢

e

3

→

for the arbitrary

orthogonal

curvilinear coordinate

system

(

q

1

,

q

2

,

q

3

)

having scale factors

(

h

1

,

h

2

,

h

3

)

.
Note the scale factors are given by

h

i

=

(

d

d

⁢

x

i

)

⁢

(

d

d

⁢

x

i

)

∋

i

∈

{

1

,

2

,

3

}

.

Non-orthogonal systems are more easily handled with

tensor analysis

or exterior

calculus

.

(

curl

⁡

F

→

)

i

=

ϵ

i

⁢

j

⁢

k

⁢

∇

j

⁡

F

k

curl

F

→

=

*

d

(

F

1

d

x

1

+

F

2

d

x

2

+

F

3

d

x

3

)

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

Generated on Fri Feb 9 22:18:35 2018 by

LaTeXML