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Construct an integral map

  1. Jul 25, 2007 #1
    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. jcsd
  3. Jul 25, 2007 #2

    olgranpappy

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    you mean "r"?

    What text did this problem come from?
     
    Last edited: Jul 25, 2007
  4. Jul 25, 2007 #3
    Sorry

    Yes it is a real number r. not R. I don't know which text book has this problem.
     
  5. Jul 25, 2007 #4

    olgranpappy

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    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?
     
  6. Jul 25, 2007 #5
    ??

    I think the problem is well defined. Integral is defined on R^n, the n dimensional real space.
     
    Last edited: Jul 25, 2007
  7. Jul 25, 2007 #6

    olgranpappy

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    ...but f is not defined in all of R^n...
     
  8. Jul 25, 2007 #7
    ??

    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
  9. Jul 25, 2007 #8

    olgranpappy

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    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?
     
  10. Jul 25, 2007 #9
    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
  11. Jul 25, 2007 #10

    olgranpappy

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    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.
     
  12. Jul 26, 2007 #11
    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.
     
  13. Jul 26, 2007 #12
    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.
     
  14. Jul 26, 2007 #13

    olgranpappy

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    okay, then the answer is "wrong."

    I.e.,

    [tex]
    \int f w \neq \int w
    [/tex]

    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
  15. Jul 26, 2007 #14

    Dick

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    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.
     
  16. Jul 26, 2007 #15
    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??
     
  17. Jul 26, 2007 #16

    olgranpappy

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    I edited my post above. can you explain the star.
     
  18. Jul 26, 2007 #17

    Dick

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    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.
     
  19. Jul 26, 2007 #18

    Dick

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    olgranpappy, if you missed my post, he's looking for a change of variables formula.
     
  20. Jul 26, 2007 #19
    I don't understand why the proof of integral(df)=integral(f'(x)dx) is related to my question. Could you explain more?
     
  21. Jul 26, 2007 #20
    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
     
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