differential form

differential form

1

Notation and Preliminaries.

Let

M

be an

n

-dimensional

differential manifold

. Let

T

⁢

M

denote
the manifold’s

tangent bundle

,

C

∞

⁢

(

M

)

the

algebra

of

smooth
functions

, and

V

⁢

(

M

)

the Lie algebra of smooth

vector fields

. The

directional derivative

makes

C

∞

⁢

(

M

)

into a

V

⁢

(

M

)

module. Using local coordinates, the directional derivative

operation

can be expressed as

v

⁢

(

f

)

=

v

i

⁢

∂

i

⁡

f

,

v

∈

V

⁢

(

M

)

,

f

∈

C

∞

⁢

(

M

)

.

2

Definitions.

Differential forms.

Let

A

be a

C

∞

⁢

(

M

)

module. An

ℝ

-

linear mapping

α

:

V

⁢

(

M

)

→

A

is said to be

tensorial

if it is a

C

∞

⁢

(

M

)

-

homomorphism

, in other words, if it

satisfies

α

⁢

(

f

⁢

v

)

=

f

⁢

α

⁢

(

v

)

for all
for all vector fields

v

∈

V

⁢

(

M

)

and

functions

f

∈

C

∞

⁢

(

M

)

.
More generally, a multilinear map

α

:

V

⁢

(

M

)

×

…

×

V

⁢

(

M

)

→

A

is called tensorial if it satisfies

α

⁢

(

f

⁢

u

,

…

,

v

)

=

⋯

=

α

⁢

(

u

,

…

,

f

⁢

v

)

=

f

⁢

α

⁢

(

u

,

…

,

v

)

for all vector fields

u

,

…

,

v

and all functions

f

∈

C

∞

⁢

(

M

)

.

We now define a

differential

1-form to be a tensorial linear mapping
from

V

⁢

(

M

)

to

C

∞

⁢

(

M

)

. More generally, for

k

=

0

,

1

,

2

,

…

,

we
define a differential

k

-form to be a tensorial

multilinear

,

antisymmetric

,

mapping

from

V

⁢

(

M

)

×

⋯

×

V

⁢

(

M

)

(

k

times) to

C

∞

⁢

(

M

)

. Using slightly fancier

language

, the above amounts
to saying that a

1

-form is a

section

of the

cotangent bundle

T

*

⁢

M

=

Hom

⁡

(

T

⁢

M

,

ℝ

)

, while a differential

k

-form as a section of

Hom

⁡

(

Λ

k

⁢

T

⁢

M

,

ℝ

)

.

Henceforth, we let

Ω

k

⁢

(

M

)

denote the

C

∞

⁢

(

M

)

-module of
differential

k

-forms. In particular, a differential

0

-form is the
same thing as a function. Since the

tangent spaces

of

M

are

n

-dimensional

vector spaces

, we also have

Ω

k

⁢

(

M

)

=

0

for

k

>

n

.
We let

Ω

⁢

(

M

)

=

⊕

k

=

0

n

Ω

k

⁢

(

M

)

denote the vector space of all

differential forms

. There is a natural

operator

, called the

exterior product

, that endows

Ω

⁢

(

M

)

with
the

structure

of a graded algebra. We describe this operation below.

Exterior and Interior Product.

Let

v

∈

V

⁢

(

M

)

be a vector field and

α

∈

Ω

k

⁢

(

M

)

a
differential form. We define

ι

v

⁢

(

ω

)

, the interior product
of

v

and

α

, to be the differential

k

-

1

form given by

ι

v

⁢

(

α

)

⁢

(

u

1

,

…

,

u

k

-

1

)

=

α

⁢

(

v

,

v

1

,

…

,

v

k

-

1

)

,

v

1

,

…

,

v

k

-

1

∈

V

⁢

(

M

)

.

The
interior product of a vector field with a

0

-form is defined to be
zero.

Let

α

∈

Ω

k

⁢

(

M

)

and

β

∈

Ω

ℓ

⁢

(

M

)

be differential
forms. We define the exterior, or wedge product

α

∧

β

∈

Ω

k

+

ℓ

⁢

(

M

)

to be the
unique differential form such that

ι

v

⁢

(

α

∧

β

)

=

ι

v

⁢

(

α

)

∧

β

+

(

-

1

)

k

⁢

α

∧

ι

v

⁢

(

β

)

for all vector fields

v

∈

V

⁢

(

M

)

. Equivalently, we could have defined

(

α

∧

β

)

⁢

(

v

1

,

…

,

v

k

+

ℓ

)

=

∑

π

sgn

⁡

(

π

)

⁢

α

⁢

(

v

π

1

,

…

,

v

π

k

)

⁢

β

⁢

(

v

π

k

+

1

,

…

,

v

π

k

+

ℓ

)

,

where the sum is taken
over all

permutations

π

of

{

1

,

2

,

…

,

k

+

ℓ

}

such that

π

1

<

π

2

<

⋯

⁢

π

k

and

π

k

+

1

<

⋯

<

π

k

+

ℓ

, and where

sgn

⁡

π

=

±

1

according to whether

π

is an even
or

odd permutation

.

Exterior derivative.

The

exterior derivative

is a
first-order

differential operator

d

:

Ω

*

⁢

(

M

)

→

Ω

*

⁢

(

M

)

, that can be defined as the unique linear mapping
satisfying

d

⁢

(

d

⁢

α

)

=

0

,

α

∈

Ω

k

⁢

(

M

)

;

ι

V

⁢

(

d

⁢

f

)

=

v

⁢

(

f

)

,

v

∈

V

⁢

(

M

)

,

f

∈

C

∞

⁢

(

M

)

;

d

⁢

(

α

∧

β

)

=

d

⁢

(

α

)

∧

β

+

(

-

1

)

k

⁢

α

∧

d

⁢

(

β

)

,

α

∈

Ω

k

⁢

(

M

)

,

β

∈

Ω

ℓ

⁢

(

M

)

.

3

Local coordinates.

Let

(

x

1

,

…

,

x

n

)

be a system of local coordinates on

M

, and
let

∂

1

,

…

,

∂

n

denote the corresponding frame of

coordinate vector

fields. In other words,

∂

i

(

x

j

)

=

δ

i

,

j

where the

right hand side

is the
usual

Kronecker delta

symbol. By the definition of the
exterior derivative,

ι

∂

i

(

d

x

j

)

=

δ

i

;

j

In other words, the 1-forms

d

⁢

x

1

,

…

,

d

⁢

x

n

form the dual

coframe

.

Locally, the

∂

i

freely generate

V

⁢

(

M

)

, meaning
that every vector field

v

∈

V

⁢

(

M

)

has the form

v

=

v

i

⁢

∂

i

,

where the coordinate

components

v

i

are uniquely determined as

v

i

=

v

⁢

(

x

i

)

.

Similarly, locally the

d

⁢

x

i

freely generate

Ω

1

⁢

(

M

)

. This
means that
every

one-form

α

∈

Ω

1

⁢

(

M

)

takes the form

α

=

α

i

⁢

d

⁢

x

i

,

where

α

i

=

ι

∂

i

⁢

(

α

)

.

More generally, locally

Ω

k

⁢

(

M

)

is a freely generated by the
differential

k

-forms

d

⁢

x

i

1

∧

⋯

∧

d

⁢

x

i

k

,

1

≤

i

1

<

i

2

<

⋯

<

i

k

≤

n

.

Thus, a differential form

α

∈

Ω

k

⁢

(

M

)

is given by

α

=

∑

i

1

<

…

<

i

k

α

i

1

⁢

…

⁢

i

k

⁢

d

⁢

x

i

1

∧

…

∧

d

⁢

x

i

k

,

(1)

=

1

k

!

⁢

α

i

1

⁢

…

⁢

i

k

⁢

d

⁢

x

i

1

∧

…

∧

d

⁢

x

i

k

,

where

α

i

1

⁢

…

⁢

i

k

=

α

⁢

(

∂

i

1

,

…

,

∂

i

k

)

.

Consequently, for vector fields

u

,

v

,

…

,

w

∈

V

⁢

(

M

)

, we have

α

⁢

(

u

,

v

,

…

,

w

)

=

α

i

1

⁢

i

2

⁢

…

⁢

i

k

⁢

u

i

1

⁢

v

i

2

⁢

⋯

⁢

w

i

k

.

In terms of local coordinates and the skew-symmetrization index
notation, the interior and exterior product, and the exterior
derivative take the following expressions:

(

ι

v

⁢

(

α

)

)

i

1

⁢

…

⁢

i

k

=

v

j

⁢

α

j

⁢

i

1

⁢

…

⁢

i

k

,

v

∈

V

⁢

(

M

)

,

α

∈

Ω

k

+

1

⁢

(

M

)

;

(2)

(

α

∧

β

)

i

1

⁢

…

⁢

i

k

+

ℓ

=

(

k

+

ℓ

k

)

⁢

α

[

i

1

…

i

k

⁢

β

i

k

+

1

…

i

k

+

ℓ

]

,

α

∈

Ω

k

⁢

(

M

)

,

β

∈

Ω

ℓ

⁢

(

M

)

;

(3)

(

d

⁢

α

)

i

0

⁢

i

1

⁢

…

⁢

i

k

=

(

k

+

1

)

⁢

∂

[

i

0

⁡

α

i

1

…

i

k

]

,

α

∈

Ω

k

⁢

(

M

)

.

(4)

Note that some authors prefer a different definition of the components
of a differential. According to this alternate convention, a factor
of

k

!

placed before the summation sign in
(

1

), and the leading factors are removed from
(

3

) and (

4

).

Title

differential form

Canonical name

DifferentialForm

Date of creation

2013-03-22 12:44:46

Last modified on

2013-03-22 12:44:46

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

28

Author

rmilson (146)

Entry type

Definition

Classification

msc 58A10

Defines

exterior derivative

Defines

1-form

Defines

exterior product

Defines

wedge product

Defines

interior product

Defines

tensorial

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