Assuing

, prove the Poincare lemma for p=1.

Poincare lemma for p=1 would be, if a 1-form

satisfies

, then

locally has the form

for some scalar field (0-form)

I think it is implied that we should prove Poincare lemma for p=1 in two dimensions, since our original assumption in the statement of the exercise was in two dimensions. However, I am doing the problem in n dimensions.

The one form

for

dimensions

The exterior derivative of

only if

for all

and

.

This means

Let us define functions

and

like this

That is, they are the anti-derivative of

and

. I think we can do this. (Is this where the "locality" of the lemma comes into play?)

So back to this equation

Integrate both sides over

or

The indefinite integral adds a constant.

Integrate both sides over

or

which adds another constant.

Of course, we could have done the integrations the other way around, doing

first and then

, which would yield (where

and

are constants)

Substituting one equation into the other

Which could only be generally true if

and

and

(right?)

So,

And if we make

(or even if we left it, I think), then there exists

(or

)

Which has the properties

and

Since this is true of any two

s in the 1-form

it is true for all of them

Which is what we were trying to get to.

I would like to see if my thinking is correct, so I will say this.

We showed that if a 1-form

exists in n-dimensions, and if a special relationship between the partial derivatives of

in any two dimensions, then the 1-form was a gradient of a scalar field in that two dimensional plane. Then the existence

as the gradient of the scalar field in all dimensions is built up from different planes where this relationship holds for the different two partial derivatives of

. The entire n-dimensional scalar field

is made up of those planes, sort of stacking them to get to three dimensions, stacking the three dimensional spaces to get to four, etc. and

is its gradient.

I will probably revisit this exercise.