common formulas in calculus of differential forms

common formulas in calculus of differential forms

1

Euclidean forms

To begin with we have the

total differential

for scalars

f

:

D

→

ℝ

where

D

is a domain in

ℝ

n

:

d

⁢

f

=

∑

s

∂

⁡

f

∂

⁡

x

s

⁢

d

⁢

x

s

or by the

Einstein summation convention

d

⁢

f

=

∂

⁡

f

∂

⁡

x

s

⁢

d

⁢

x

s

which are a special case of the so-called

Euclidean

1-forms.
Here we reconize the covariant form of the

gradient

of

f

in contravaiant ”state”:

∇

⁡

f

=

∂

⁡

f

∂

⁡

x

s

being the

components

of

d

⁢

f

.

Here the symbols

d

⁢

x

s

are

linear functionals

ℝ

n

→

ℝ

dual to the

derivations

∂

∂

⁡

x

s

, that is

d

⁢

x

s

⁢

(

∂

∂

⁡

x

t

)

=

δ

t

s

this coincides with the calculation

d

⁢

x

s

⁢

(

∂

∂

⁡

x

t

)

=

∂

⁡

x

s

∂

⁡

x

t

=

δ

t

s

.

If

X

is a

vector field

and

f

a scalar field then one has for the

directional derivative

X

⁢

f

=

X

s

⁢

∂

∂

⁡

x

s

⁢

f

=

X

s

⁢

d

⁢

f

⁢

(

∂

∂

⁡

x

s

)

=

d

⁢

f

⁢

(

X

)

For a pair of functions

g

,

f

:

D

→

ℝ

we can check Leibniz’s rule

d

⁢

(

f

⁢

g

)

=

g

⁢

d

⁢

f

+

f

⁢

d

⁢

g

Let

Ω

0

⁢

(

D

)

=

C

∞

⁢

(

D

)

be the set of 0-forms in

D

and let

Ω

1

⁢

(

D

)

=

{

w

=

w

s

⁢

d

⁢

x

s

:

w

s

∈

Ω

0

}

(where

w

s

⁢

d

⁢

x

s

=

∑

s

w

s

⁢

d

⁢

x

s

) be the set of 1-forms in

D

.

Then the operator

d

can be seen as a

linear operator

d

:

Ω

0

⁢

(

D

)

→

Ω

1

⁢

(

D

)

.

This can be generalized by defining

Ω

k

⁢

(

D

)

to be the set of k-forms; that is,
expressions of the type:

A

s

1

⁢

…

⁢

s

k

⁢

d

⁢

x

s

1

∧

⋯

∧

d

⁢

x

s

k

where

A

s

1

⁢

…

⁢

s

k

are in

Ω

0

⁢

(

D

)

i.e. they are scalars and
they are multi-indexed sums. Further, the symbols

d

⁢

x

s

1

∧

⋯

∧

d

⁢

x

s

k

are the

wedge products

of the

d

⁢

x

s

.

So

d

:

Ω

k

⁢

(

D

)

→

Ω

k

+

1

⁢

(

D

)

is calculated by

d

⁢

(

A

s

1

⁢

…

⁢

s

k

⁢

d

⁢

x

s

1

∧

⋯

∧

d

⁢

x

s

k

)

=

d

⁢

(

A

s

1

⁢

…

⁢

s

k

)

∧

d

⁢

x

s

1

∧

⋯

∧

d

⁢

x

s

k

For example, if

A

=

A

s

⁢

d

⁢

x

s

then

d

⁢

A

=

d

⁢

A

s

∧

d

⁢

x

s

, hence

d

⁢

A

=

∂

⁡

A

s

∂

⁡

x

t

⁢

d

⁢

x

t

∧

d

⁢

x

s

which is rearranged as

d

⁢

A

=

(

∂

⁡

A

s

∂

⁡

x

t

-

∂

⁡

A

t

∂

⁡

x

s

)

⁢

d

⁢

x

t

∧

d

⁢

x

s

,

and for two forms, if

B

=

B

s

⁢

t

⁢

d

⁢

x

s

∧

d

⁢

x

t

then

d

⁢

B

=

∂

⁡

B

s

⁢

t

∂

⁡

x

u

⁢

d

⁢

x

u

∧

d

⁢

x

s

∧

d

⁢

x

t

.

Now if we have a map between two domains

F

:

D

→

E

and

F

=

(

F

1

,

…

,

F

n

)

, we can pullback forms as

F

*

:

Ω

k

⁢

(

E

)

→

Ω

k

⁢

(

D

)

, beginnig with the observation that at basics

d

⁢

x

k

, we pullback it as

F

*

⁢

(

d

⁢

x

k

)

=

d

⁢

(

x

k

∘

F

)

=

d

⁢

F

k

=

∂

⁡

F

k

∂

⁡

x

s

⁢

d

⁢

x

s

then, if we want

ω

↦

F

*

⁢

(

ω

)

,
where

ω

=

ω

s

1

⁢

…

⁢

s

k

⁢

d

⁢

x

s

1

∧

⋯

∧

d

⁢

x

s

k

,
we are going to receive

F

*

⁢

(

ω

)

=

ω

s

1

⁢

…

⁢

s

k

∘

f

⁢

∂

⁡

F

s

1

∂

⁡

x

t

1

⁢

⋯

⁢

∂

⁡

F

s

k

∂

⁡

x

t

k

⁢

d

⁢

x

t

1

∧

⋯

∧

d

⁢

x

t

k

Here the

t

i

-sums
must be taken between all indexes obeying

1

≤

t

1

<

t

2

<

⋯

<

t

k

≤

n

.

So if

ω

∈

Ω

n

⁢

(

D

)

,

F

*

⁢

(

ω

)

=

ω

1

⁢

…

⁢

n

∘

F

⁢

det

⁡

(

F

′

)

⁢

d

⁢

x

1

∧

⋯

∧

d

⁢

x

n

We also have

F

*

⁢

(

v

∧

w

)

=

F

*

⁢

(

v

)

∧

F

*

⁢

(

w

)

Obviously there are no

n

+

1

,

n

+

2

,

…

forms in

D

and usually one set

Ω

k

⁢

(

D

)

=

0

if

k

≥

n

.

2

The de Rham complex.

The collection of mappings

0

⟶

Ω

0

⁢

(

D

)

⟶

d

Ω

1

⁢

(

D

)

⟶

d

⋯

⟶

d

Ω

n

⁢

(

D

)

⟶

0

give us a chain complex due that

d

⁢

d

=

0

, so one can measure how much this differs from exactness via its homology

H

k

⁢

(

D

)

=

ker

⁡

(

d

)

im

⁡

(

d

)

called the cohomological

k

-group for

D

.

Some
with the fear of being confused with the giving of the same name to the operator

Ω

k

⁢

(

D

)

⟶

d

Ω

k

+

1

⁢

(

D

)

, would like to write

Ω

k

⁢

(

D

)

⟶

d

k

Ω

k

+

1

⁢

(

D

)

and then one should modify the above conventions with

d

k

+

1

⁢

d

k

=

0

and

H

k

⁢

(

D

)

=

ker

⁡

(

d

k

)

im

⁡

(

d

k

-

1

)

3

Manifold’s Forms.

One had seen that for mappings

F

:

D

→

E

between

ℝ

n

’s domains behave as

F

*

:

Ω

k

⁢

(

E

)

→

Ω

k

⁢

(

D

)

.
Then we can assign k-forms in each chart

(

U

,

Φ

)

of a n-manifold

M

by means of the coordinated functions

u

i

=

x

i

∘

Φ

on the neighborhood

U

.
Then

d

⁢

u

i

=

d

⁢

(

x

i

∘

Φ

)

=

Φ

*

⁢

d

⁢

x

i

which will be the duals of the derivations

∂

∂

⁡

u

j

.

Observe that if

Φ

*

:

Ω

0

⁢

(

ϕ

⁢

(

U

)

)

→

Ω

0

⁢

(

U

)

then

Φ

⁢

(

g

)

=

g

∘

Φ

is a scalar in

U

.

If

Φ

*

:

Ω

1

⁢

(

ϕ

⁢

(

U

)

)

→

Ω

1

⁢

(

U

)

then

Φ

*

⁢

(

w

s

⁢

d

⁢

x

s

)

=

w

s

∘

Φ

⁢

Φ

*

⁢

(

d

⁢

x

s

)

=

w

s

∘

Φ

⁢

d

⁢

u

s

For

k

-forms

w

s

1

⁢

s

2

⁢

…

⁢

s

k

⁢

d

⁢

u

s

1

∧

⋯

∧

d

⁢

u

s

k

=

w

s

1

⁢

s

2

⁢

…

⁢

s

k

∘

Φ

-

1

∘

Φ

⁢

d

⁢

(

x

s

1

∘

Φ

)

∧

⋯

∧

d

⁢

(

x

s

k

∘

Φ

)

=

Φ

*

⁢

(

w

s

1

⁢

s

2

⁢

…

⁢

s

k

∘

Φ

-

1

)

⁢

Φ

*

⁢

(

d

⁢

x

s

1

∧

⋯

∧

d

⁢

x

s

k

)

=

Φ

*

⁢

(

w

s

1

⁢

s

2

⁢

…

⁢

s

k

∘

Φ

-

1

⁢

d

⁢

x

s

1

∧

⋯

∧

d

⁢

x

s

k

)

where

w

s

1

⁢

s

2

⁢

…

⁢

s

k

∘

Φ

-

1

⁢

d

⁢

x

s

1

∧

⋯

∧

d

⁢

x

s

k

is a

k

-form in

Φ

⁢

(

U

)

.

4

Forms and connections

A

connection

is a bi-linear operator

∇

:

Γ

⁢

(

T

⁢

M

)

2

→

Γ

⁢

(

T

⁢

M

)

where

Γ

⁢

(

T

⁢

M

)

is the space of

differentiable

sections in the

tangent bundle

.

The Chistoffel symbols

Γ

i

⁢

j

s

are the components of

∇

∂

i

⁡

∂

j

through the equation

∇

∂

i

⁡

∂

j

=

Γ

i

⁢

j

s

⁢

∂

s

where the

∂

s

are the coordinated

tangent vectors

.

The curvature tensor is defined as

R

⁢

(

X

,

Y

)

⁢

Z

=

∇

X

⁡

∇

Y

⁡

Z

-

∇

Y

⁡

∇

X

⁡

Z

-

∇

[

X

,

Y

]

⁡

Z

which is a tri-linear map

Γ

⁢

(

T

⁢

M

)

3

→

Γ

⁢

(

T

⁢

M

)

, so the Riemann-Chistoffel symbols are defined by the components

R

s

i

⁢

j

⁢

k

of

R

⁢

(

∂

i

,

∂

j

)

⁢

∂

k

=

R

s

i

⁢

j

⁢

k

⁢

∂

s

With these one define the connection forms and the

curvature forms

as

∇

X

⁡

∂

j

=

ω

s

j

⁢

(

X

)

⁢

∂

s

and

R

⁢

(

X

,

Y

)

⁢

∂

j

=

Ω

s

j

⁢

(

X

,

Y

)

⁢

∂

s

these

ω

s

j

and

Ω

s

j

define a 1-form and a 2-form viewed as a sections

M

→

Ω

1

⁢

(

T

⁢

M

)

and

M

→

Ω

2

⁢

(

T

⁢

M

)

respectively.

Observe that

∇

∂

k

⁡

∂

j

=

ω

s

j

⁢

(

∂

k

)

⁢

∂

s

which compared with

∇

∂

k

⁡

∂

j

=

Γ

k

⁢

j

s

⁢

∂

s

, it implies

ω

s

j

⁢

(

∂

k

)

=

Γ

k

⁢

j

s

and for an arbitrary vector field

X

=

X

k

⁢

∂

k

(in the

tangent

coordinated basis)

ω

s

j

⁢

(

X

)

=

X

k

⁢

Γ

k

⁢

j

s

Let

X

1

,

X

2

,

…

,

X

n

be another frame field (the

∂

i

are the coordinated frame field) , i.e. a system of

n

-tangent vectors which are

linearly independent

in the

tangent space

, i.e, they span each

T

p

⁢

M

.

Define thru

∇

X

i

⁡

X

j

=

Γ

^

i

⁢

j

s

⁢

X

s

a an-holonomic

connection coefficients

and

R

⁢

(

X

i

,

X

j

)

⁢

X

k

=

R

s

^

i

⁢

j

⁢

k

⁢

X

s

as the an-holonomic.

Remember that in the coordinated frame field

[

∂

i

,

∂

j

]

=

0

, but since

∇

X

i

⁡

X

j

-

∇

X

j

⁡

X

i

=

[

X

i

,

X

j

]

this define the structural ”constants”

c

s

i

⁢

j

⁢

X

s

=

[

X

i

,

X

j

]

and the give relation

c

s

i

⁢

j

=

Γ

^

i

⁢

j

s

-

Γ

^

j

⁢

i

s

5

Cartan Structural Equations

The connection and the curvature forms satisfy the premiere

d

⁢

θ

i

=

-

ω

i

^

s

∧

θ

s

,
where the

θ

i

are the 1-forms dual to the

X

j

and the deuxieme

Ω

i

^

j

=

d

⁢

ω

i

^

j

+

ω

i

^

s

∧

ω

s

^

j

where the corresponding connection forms are calculated by

∇

Y

⁡

X

j

=

ω

s

^

j

⁢

(

Y

)

⁢

X

s

i.e.

ω

l

^

j

=

Γ

^

j

⁢

s

l

⁢

θ

s

.

All that fits perfectly to give

Ω

i

^

j

=

1

2

⁢

R

i

^

j

⁢

k

⁢

l

⁢

θ

k

∧

θ

l

with

k

<

l

.

This shows that the calculations of

R

i

^

j

⁢

k

⁢

l

are very easy objects to put into an algorithm (Debever).

Title

common formulas in calculus of differential forms

Canonical name

CommonFormulasInCalculusOfDifferentialForms

Date of creation

2013-03-22 15:51:28

Last modified on

2013-03-22 15:51:28

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

37

Author

juanman (12619)

Entry type

Topic

Classification

msc 58A12

Classification

msc 58A10

Related topic

Calculus

Related topic

TopicsOnCalculus

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