alternate characterization of curl

alternate characterization of curl

Let

𝐅

be a smooth

vector field

on (an open subset of)

ℝ

3

.

We show that

curl

⁡

𝐅

defined using the coordinate-free definition given on the parent entry (

http://planetmath.org/curl

)
is the same as the curl defined
by

∇

×

𝐅

in

Cartesian coordinates

.

The case for spherical surfaces

This will be done by directly computing the limit

𝐋

of

surface
integrals

defining

curl

⁡

𝐅

⁢

(

𝐩

)

,
using spheres

S

2

⁢

(

r

,

𝐩

)

centered at

𝐩

of radius

r

. The

formula

is:

curl

⁡

𝐅

⁢

(

𝐩

)

=

𝐋

=

lim

r

→

0

⁡

3

4

⁢

π

⁢

r

3

⁢

∬

S

2

⁢

(

r

,

𝐩

)

𝐧

×

𝐅

⁢

𝑑

A

=

lim

r

→

0

⁡

3

⁢

r

2

4

⁢

π

⁢

r

3

⁢

∬

S

2

𝐧

×

𝐅

⁢

(

r

⁢

𝐧

+

𝐩

)

⁢

𝑑

A

,

where

𝐧

is the outward

unit normal

to the surface (at each point of the surface), and

S

2

is the

unit sphere

at the origin.

We simplify the last

integral

.
Expanding

𝐅

⁢

(

r

⁢

𝐧

+

𝐩

)

in a first-degree

Taylor polynomial

about

𝐩

, we have

∬

S

2

𝐧

×

𝐅

⁢

(

r

⁢

𝐧

+

𝐩

)

⁢

𝑑

A

=

∬

S

2

𝐧

×

𝐅

⁢

(

𝐩

)

⁢

𝑑

A

+

∬

S

2

𝐧

×

D

⁡

𝐅

⁢

(

𝐩

)

⁢

r

⁢

𝐧

⁢

𝑑

A

+

∬

S

2

𝐧

×

o

⁢

(

∥

r

⁢

𝐧

∥

)

⁢

𝑑

A

.

The integral

∬

S

2

𝐧

×

𝐅

⁢

(

𝐩

)

⁢

𝑑

A

vanishes

by

symmetry

of the sphere,
while

∥

∬

S

2

𝐧

×

o

⁢

(

∥

r

⁢

𝐧

∥

)

⁢

𝑑

A

∥

≤

∬

S

2

∥

𝐧

∥

⁢

o

⁢

(

r

)

⁢

𝑑

A

=

o

⁢

(

r

)

.

Combining these facts, we obtain

𝐋

=

lim

r

→

0

⁡

[

0

+

3

4

⁢

π

⁢

r

⁢

∬

S

2

𝐧

×

D

⁡

𝐅

⁢

(

𝐩

)

⁢

r

⁢

𝐧

⁢

𝑑

A

+

o

⁢

(

1

)

]

=

3

4

⁢

π

⁢

∬

S

2

𝐧

×

D

⁡

𝐅

⁢

(

𝐩

)

⁢

𝐧

⁢

𝑑

A

.

Notice that

𝐋

depends only on the

derivative

of

𝐅

at

𝐩

.

We want to evaluate the last integral in Cartesian coordinates.
Let

𝐞

k

be an

orthonormal basis

of

ℝ

3

oriented positively, and let

B

be the matrix of the derivative

D

⁡

𝐅

⁢

(

p

)

in this basis.
Then the

k

th

coordinate

of

𝐋

with respect to the
same basis
is

(

∬

S

2

𝐧

×

B

⁢

𝐧

⁢

𝑑

A

)

⋅

𝐞

k

=

∬

S

2

(

𝐧

×

B

⁢

𝐧

)

⋅

𝐞

k

⁢

𝑑

A

The

k

th coordinate of the integrand is

(

𝐧

×

B

⁢

𝐧

)

⋅

𝐞

k

=

n

i

⁢

(

B

⁢

𝐧

)

j

⁢

ϵ

i

⁢

j

⁢

k

=

n

i

⁢

B

l

j

⁢

n

l

⁢

ϵ

i

⁢

j

⁢

k

,

where to lessen the writing, we employ
the

Einstein summation convention

, along
with the

Levi-Civita permutation symbol

ϵ

i

⁢

j

⁢

k

, and

B

l

j

denotes the entry at the

j

th row,

l

th column of

B

.

In the summation above, if a summmand has

i

≠

l

,
then the integral of that summand over the sphere is zero, by symmetry.
This means that in the summation the index

l

may be set to

i

,
and thus

(

∬

S

2

𝐧

×

B

⁢

𝐧

⁢

𝑑

A

)

⋅

𝐞

k

=

∬

S

2

n

i

⁢

B

i

j

⁢

n

i

⁢

ϵ

i

⁢

j

⁢

k

⁢

𝑑

A

=

B

i

j

⁢

ϵ

i

⁢

j

⁢

k

⁢

∬

S

2

(

n

i

)

2

⁢

𝑑

A

.

Now there is a formula for the evaluation of integrals
of

polynomials

over

S

m

-

1

⊂

ℝ

m

, in terms of the gamma function;
in our case (

m

=

3

) the formula reads:

∬

S

2

(

n

i

)

2

⁢

𝑑

A

=

2

⁢

Γ

⁢

(

3

2

)

⁢

Γ

⁢

(

1

2

)

⁢

Γ

⁢

(

1

2

)

Γ

⁢

(

3

2

+

1

2

+

1

2

)

=

2

⁢

Γ

⁢

(

3

2

)

⁢

π

⁢

π

3

2

⁢

Γ

⁢

(

3

2

)

=

4

⁢

π

3

.

(If you do not know this formula, the integral in our case
can be computed directly using

spherical coordinates

.)
Therefore the

k

th

component

of

𝐋

is

𝐋

⋅

𝐞

k

=

3

4

⁢

π

⁢

B

i

j

⁢

ϵ

i

⁢

j

⁢

k

⁢

∬

S

2

(

n

i

)

2

⁢

𝑑

A

=

B

i

j

⁢

ϵ

i

⁢

j

⁢

k

=

∂

⁡

F

j

∂

⁡

x

i

|

𝐩

⁢

ϵ

i

⁢

j

⁢

k

.

But this is just

(

∇

×

𝐅

⁢

(

𝐩

)

)

⋅

𝐞

k

.

The case for arbitrary surfaces

Although we have only
computed

𝐋

=

curl

⁡

𝐅

⁢

(

𝐩

)

=

lim

V

→

0

⁡

1

V

⁢

∬

S

𝐧

×

𝐅

⁢

𝑑

A

only for spheres

S

=

S

2

⁢

(

r

,

𝐩

)

,
this formula holds for arbitrary closed surfaces

S

that shrink nicely to

𝐩

.
It is hardly obvious, especially since our computation before depended
on the symmetry of the sphere extensively.

To show the general result,
consider the

triple scalar product

(

𝐯

×

𝐅

)

⋅

𝐞

k

.
This is a

linear functional

in the vector

𝐯

,
so there exists
a unique

vector function

𝐠

k

such that

(

𝐯

×

𝐅

)

⋅

𝐞

k

=

𝐠

k

⋅

𝐯

for all

𝐯

∈

ℝ

3

.
We can find the components of this

𝐠

k

by evaluating the

functional

at

𝐯

=

𝐞

i

:

g

k

i

=

𝐠

k

⋅

𝐞

i

=

(

𝐞

i

×

𝐅

)

⋅

𝐞

k

=

det

⁡

(

𝐞

i

,

𝐅

,

𝐞

k

)

=

F

j

⁢

ϵ

i

⁢

j

⁢

k

.

The reason for considering such expressions is that, putting

𝐯

=

𝐧

, we have

∬

S

(

𝐧

×

𝐅

)

⋅

𝐞

k

⁢

𝑑

A

=

∬

S

𝐠

k

⋅

𝐧

⁢

𝑑

A

=

∬

S

𝐠

k

⋅

𝑑

𝐀

.

So we have converted the original integral into an ordinary surface integral.
And this surface integral can be changed into a volume integral, by using the

divergence theorem

:

∬

S

𝐠

k

⋅

𝑑

𝐀

=

∭

M

div

⁡

𝐠

k

⁢

d

⁢

V

=

∭

M

∂

⁡

F

j

∂

⁡

x

i

⁢

ϵ

i

⁢

j

⁢

k

⁢

𝑑

V

,

where

M

is the volume whose boundary is

S

.
Hence

𝐋

⋅

𝐞

k

=

lim

V

→

0

⁡

1

V

⁢

∬

S

(

𝐧

×

𝐅

)

⋅

𝐞

k

⁢

𝑑

A

=

lim

V

→

0

⁡

1

V

⁢

∭

M

∂

⁡

F

j

∂

⁡

x

i

⁢

ϵ

i

⁢

j

⁢

k

⁢

𝑑

V

=

∂

⁡

F

j

∂

⁡

x

i

|

𝐩

⁢

ϵ

i

⁢

j

⁢

k

=

(

∇

×

𝐅

⁢

(

𝐩

)

)

⋅

𝐞

k

.

Definition in terms of differential forms

We mention, in passing,
a computational, yet coordinate-free, alternative to the definition of the curl,
using

differential forms

.
If

ω

is a 1-form on

ℝ

3

such that

ω

⁢

(

𝐯

)

=

⟨

𝐅

,

𝐯

⟩

,
then the curl of

𝐅

is defined as the vector function

𝐠

=

g

k

⁢

e

k

such that

d

⁢

ω

⁢

(

𝐮

,

𝐯

)

=

⟨

𝐠

,

𝐮

×

𝐯

⟩

.

In Cartesian coordinates, we have

ω

=

F

1

⁢

d

⁢

x

1

+

F

2

⁢

d

⁢

x

2

+

F

3

⁢

d

⁢

x

3

d

⁢

ω

=

g

1

⁢

d

⁢

x

2

∧

d

⁢

x

3

+

g

2

⁢

d

⁢

x

3

∧

d

⁢

x

1

+

g

3

⁢

d

⁢

x

1

∧

d

⁢

x

2

,

If we take the

exterior derivative

of the first equation for

ω

, and then equate components
with the second equation for

d

⁢

ω

,
we find that

g

k

=

(

∇

×

𝐅

)

⋅

𝐞

k

,
so our new definition is

equivalent

to the others.

Title

alternate characterization of curl

Canonical name

AlternateCharacterizationOfCurl

Date of creation

2013-03-22 15:29:10

Last modified on

2013-03-22 15:29:10

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

9

Author

stevecheng (10074)

Entry type

Derivation

Classification

msc 53-01

Related topic

curl

Related topic

nabla

Related topic

FirstOrderOperatorsInRiemannianGeometry

Related topic

Curl

Related topic

NablaNabla

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