# Construct an integral map

1. Jul 25, 2007

### daishin

1. The problem statement, all variables and given/known data
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.

2. Relevant 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?

3. The attempt at a solution

I don't even believe this result.

Last edited: Jul 25, 2007
2. Jul 25, 2007

### olgranpappy

you mean "r"?

What text did this problem come from?

Last edited: Jul 25, 2007
3. Jul 25, 2007

### daishin

Sorry

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

4. Jul 25, 2007

### 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?

5. Jul 25, 2007

### daishin

??

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

Last edited: Jul 25, 2007
6. Jul 25, 2007

### olgranpappy

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

7. Jul 25, 2007

### 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.

Last edited: Jul 25, 2007
8. Jul 25, 2007

### 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?

9. Jul 25, 2007

### 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?

Last edited: Jul 25, 2007
10. Jul 25, 2007

### olgranpappy

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.

11. Jul 26, 2007

### 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.

12. Jul 26, 2007

### 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.

13. Jul 26, 2007

### 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?

Last edited: Jul 26, 2007
14. Jul 26, 2007

### 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.

15. Jul 26, 2007

### 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??

16. Jul 26, 2007

### olgranpappy

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

17. Jul 26, 2007

### Dick

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.

18. Jul 26, 2007

### Dick

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

19. Jul 26, 2007

### daishin

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

20. Jul 26, 2007

### 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