Can a Proper Map Preserve Integral of Smooth n-Forms with Compact Support?

[daishin]

Homework Statement

Let f:R^n-->R^n be a C^oo proper map. Suppose there is a real number r such that f(x)=x for all x in R^n with |x|> r. Show that for every compactly supported smooth n-form w on R^n
integral of f*w = integral of w. Here, integral is defined on R^n.

Homework Equations

I think this problem implies that when the condition above holds, we cannot construct a map from R to R s.t integral of w =/= integral of f*w which is not obvious for me.
Could anybody provide a solution of the problem?

The Attempt at a Solution

I don't even believe this result.

 

[olgranpappy]

Homework Statement

Let f:R^n-->R^n be a C^oo proper map. Suppose there is a real number R

you mean "r"?

What text did this problem come from?

 

[daishin]

Sorry

Yes it is a real number r. not R. I don't know which textbook has this problem.

 

[olgranpappy]

it's just that the problem doesn't seem completely well-defined. The integral over what "volume"? over the ball itself? Then it seems trivial... is what you have given the entire statement of the problem?

 

[daishin]

??

I think the problem is well defined. Integral is defined on R^n, the n dimensional real space.

 

[olgranpappy]

I think the problem is well defined. Integral is defined on R^n, the n dimensional real space.

...but f is not defined in all of R^n...

 

[daishin]

??

Why not? Maybe you are thinking that the map f is defined on r^n-->r^n.
But f is defined between R^n, the n dimensional real space.

 

[olgranpappy]

No. No, that is not what I am thinking. Obviously, that doesn't make any sense.

What I mean is that you have only defined the function f for |x|>r in your original post.

Look at your original post and tell me, what is the value of f when x=0? How about for any region surrounding x=0 but with |x|<r. Yes, please... do tell: what is the value of f then?

It is not specified... so, then... If the integration region contains, say, some volume surrounding x=0 then how am I supposed to make any statement about the value of the integral?

 

[daishin]

Oh.. I see what you mean. But still the problem is well defined. Initially, we don't know what the value of f is for any region |x|<r. If you think this problem is simply wrong, then could you provide an example of any function f which satisfies the condition in the problem and that the integral of f*w =/= integral of w?

 

[olgranpappy]

Initially, we don't know what the value of f is for any region |x|<r.

Initially? I'm not sure what you mean by that, either you know f or you don't...

If f is not specified then, indeed, you can not do the integral in general.

 

[daishin]

If we check that there exists a C^oo proper map from R^n to R^n which satisfies the condition in the problem, then either the statement
[integral of f*w = integral of w for all such f] is right or wrong.
The condition is that there is a real number r such that f(x)=x for all x in R^n with |x|> r. The identity map from R^n to R^n satisfies the condition.
So either integral of f*w = integral of w is right or wrong.

 

[daishin]

So although initially(when we just looked at the problem) we don't know what f is, the problem is well defined. I don't think we can determine what function f is even after when we solve the problem. But I am not sure.

 

[olgranpappy]

okay, then the answer is "wrong."

I.e.,

$$\int f w \neq \int w$$

oh, wait a second... you have f*w... um... in that case, maybe it's correct. what's the star mean?

 

[Dick]

daishin's notation and intent is really opaque, but I believe what he is being asked for is just a proof of the change of variables formula. E.g. integral(df)=integral(f'(x)dx). f*w can't mean an ordinary product since f takes values in R^n, it has to mean the change in the n-form induced by the change of variables f. I've looked at this several times today and finally figured out what the OP is talking about.

 

[daishin]

Why?? Could you provide an example of f such that the integra of f*w is not equal to the integral of w?? By the way, do you know what f* means??

 

[olgranpappy]

I edited my post above. can you explain the star.

 

[Dick]

Why?? Could you provide an example of f such that the integra of f*w is not equal to the integral of w?? By the way, do you know what f* means??

f*w means f times w to us laymen. If it means something else you should clarify in your post and you would get much better advice in a much more timely fashion.

 

[Dick]

I edited my post above. can you explain the star.

olgranpappy, if you missed my post, he's looking for a change of variables formula.

 

[daishin]

I don't understand why the proof of integral(df)=integral(f'(x)dx) is related to my question. Could you explain more?

 

[daishin]

f* is a pull back if f is defind from M to N then there is a map f* from the differential forms on N to a differential forms on M

 

[Dick]

In one dimension f'(x)dx is the change in integration measure induced by the change of variables x->f(x). I'm assuming that's what your f*w means.

 

[olgranpappy]

olgranpappy, if you missed my post, he's looking for a change of variables formula.

I caught it. I really hate this math-notation. They need to hang indices off everything like us physicists. Anyways, I think his question is not exactly about a change of variables...

it looks like one is going to have to use the fact that because f is smooth and because it's equal to x everywhere outside some finite region then also f(x) is roughly equal to x almost everywhere, or everywhere as far as the measure in concerned such that the integrals are equal...

I obviously jumped into trying to give an answer too quickly since I still don't know what his star notation means. Cheers.

 

[Dick]

f* is a pull back if f is defind from M to N then there is a map f* from the differential forms on N to a differential forms on M

Yes, the pullback. Same thing. Can you think how to extend the calculus change of variables formula to an n-manifold and n-forms?

 

[Dick]

I caught it. I really hate this math-notation. They need to hang indices off everything like us physicists. Anyways, I think his question is not exactly about a change of variables...

it looks like one is going to have to use the fact that because f is smooth and because it's equal to x everywhere outside some finite region then also f(x) is roughly equal to x almost everywhere, or everywhere as far as the measure in concerned such that the integrals are equal...

I obviously jumped into trying to give an answer too quickly since I still don't know what his star notation means. Cheers.

It's never too quick to jump if nobody else is jumping. Watching you struggle with the problem cleared it up for me, finally. f*w may be standard notation in the OP's class but not here. Mention of the word pullback might have helped. If only everyone were more like 'us physicists'.

 

[daishin]

I shall come back tomorrow. Any help would be appreciated!

 

[Dick]

An integration of a differential form involves an integration over chains (coordinate volume elements) of a form. You have to show that for the change of variables x->f(x) that the change of the form cancels the change in the chains.