proof of closed differential forms on a simple connected domain

proof of closed differential forms on a simple connected domain

lemma 1.

Let

γ

0

and

γ

1

be two regular homotopic curves in

D

with
the same end-points.
Let

σ

:

[

0

,

1

]

×

[

0

,

1

]

→

D

be the homotopy between

γ

0

and

γ

1

i.e.

σ

⁢

(

0

,

t

)

=

γ

0

⁢

(

t

)

,

σ

⁢

(

1

,

t

)

=

γ

1

⁢

(

t

)

.

Notice that we may (and shall) suppose that

σ

is regular too. In fact

σ

⁢

(

[

0

,

1

]

×

[

0

,

1

]

)

is a

compact subset

of

D

. Being

D

open this compact set has

positive

distance

from the boundary

∂

⁡

D

. So we could
regularize

σ

by mollification leaving its image in

D

.

Let

ω

⁢

(

x

,

y

)

=

a

⁢

(

x

,

y

)

⁢

d

⁢

x

+

b

⁢

(

x

,

y

)

⁢

d

⁢

y

be our

closed differential form

and let

σ

⁢

(

s

,

t

)

=

(

x

⁢

(

s

,

t

)

,

y

⁢

(

s

,

t

)

)

.
Define

F

⁢

(

s

)

=

∫

0

1

a

⁢

(

x

⁢

(

s

,

t

)

,

y

⁢

(

s

,

t

)

)

⁢

x

t

⁢

(

s

,

t

)

+

b

⁢

(

x

⁢

(

s

,

t

)

,

y

⁢

(

s

,

t

)

)

⁢

y

t

⁢

(

s

,

t

)

⁢

d

⁢

t

;

we only have to prove that

F

⁢

(

1

)

=

F

⁢

(

0

)

.

We have

F

′

⁢

(

s

)

=

d

d

⁢

s

⁢

∫

0

1

a

⁢

x

t

+

b

⁢

y

t

⁢

d

⁢

t

=

∫

0

1

a

x

⁢

x

s

⁢

x

t

+

a

y

⁢

y

s

⁢

x

t

+

a

⁢

x

t

⁢

s

+

b

x

⁢

x

s

⁢

y

t

+

b

y

⁢

y

s

⁢

y

t

+

b

⁢

y

t

⁢

s

⁢

d

⁢

t

.

Notice now that being

a

y

=

b

x

we have

d

d

⁢

t

⁢

[

a

⁢

x

s

+

b

⁢

y

s

]

=

a

x

⁢

x

t

⁢

x

s

+

a

y

⁢

y

t

⁢

x

s

+

a

⁢

x

s

⁢

t

+

b

x

⁢

x

t

⁢

y

s

+

b

y

⁢

y

t

⁢

y

s

+

b

⁢

y

s

⁢

t

=

a

x

⁢

x

s

⁢

x

t

+

b

x

⁢

x

s

⁢

y

t

+

a

⁢

x

t

⁢

s

+

a

y

⁢

y

s

⁢

x

t

+

b

y

⁢

y

s

⁢

y

t

+

b

⁢

y

t

⁢

s

hence

F

′

⁢

(

s

)

=

∫

0

1

d

d

⁢

t

⁢

[

a

⁢

x

s

+

b

⁢

y

s

]

⁢

𝑑

t

=

[

a

⁢

x

s

+

b

⁢

y

s

]

0

1

.

Notice, however, that

σ

⁢

(

s

,

0

)

and

σ

⁢

(

s

,

1

)

are constant hence

x

s

=

0

and

y

s

=

0

for

t

=

0

,

1

. So

F

′

⁢

(

s

)

=

0

for all

s

and

F

⁢

(

1

)

=

F

⁢

(

0

)

.
∎

Lemma 2.

Let us fix a point

(

x

0

,

y

0

)

∈

D

and define a function

F

:

D

→

ℝ

by letting

F

⁢

(

x

,

y

)

be the

integral

of

ω

on any curve joining

(

x

0

,

y

0

)

with

(

x

,

y

)

. The

hypothesis

assures that

F

is well defined.
Let

ω

=

a

⁢

(

x

,

y

)

⁢

d

⁢

x

+

b

⁢

(

x

,

y

)

⁢

d

⁢

y

. We only have to prove that

∂

⁡

F

/

∂

⁡

x

=

a

and

∂

⁡

F

/

∂

⁡

y

=

b

.

Let

(

x

,

y

)

∈

D

and suppose that

h

∈

ℝ

is so small that for all

t

∈

[

0

,

h

]

also

(

x

+

t

,

y

)

∈

D

. Consider the increment

F

⁢

(

x

+

h

,

y

)

-

F

⁢

(

x

,

y

)

.
From the

definition

of

F

we know that

F

⁢

(

x

+

h

,

y

)

is equal to the integral of

ω

on a curve which starts from

(

x

0

,

y

0

)

goes to

(

x

,

y

)

and then goes to

(

x

+

h

,

y

)

along the straight

segment

(

x

+

t

,

y

)

with

t

∈

[

0

,

h

]

.
So we understand that

F

⁢

(

x

+

h

,

y

)

-

F

⁢

(

x

,

y

)

=

∫

0

h

a

⁢

(

x

+

t

,

y

)

⁢

𝑑

t

.

For the

integral mean value theorem

we know that the last integral is equal to

h

⁢

a

⁢

(

x

+

ξ

,

y

)

for some

ξ

∈

[

0

,

h

]

and hence letting

h

→

0

we have

F

⁢

(

x

+

h

,

y

)

-

F

⁢

(

x

,

y

)

h

=

a

⁢

(

x

+

ξ

,

y

)

→

a

⁢

(

x

,

y

)

h

→

0

that is

∂

⁡

F

⁢

(

x

,

y

)

/

∂

⁡

x

=

a

⁢

(

x

,

y

)

. With a similar

argument

(exchange

x

with

y

) we prove that also

∂

⁡

F

/

∂

⁡

y

=

b

⁢

(

x

,

y

)

.
∎

Theorem.

Just notice that if

D

is simply connected, then any two curves in

D

with the same end points are homotopic. Hence we can apply Lemma 1 and then Lemma 2 to obtain the desired result.
∎

Title

proof of closed differential forms on a simple connected domain

Canonical

name

ProofOfClosedDifferentialFormsOnASimpleConnectedDomain

Date of creation

2013-03-22 13:32:49

Last modified on

2013-03-22 13:32:49

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

9

Author

paolini (1187)

Entry type

Proof

Classification

msc 53-00

Related topic

SubstitutionNotation

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