Why if $m(varphi(A))leq int_A|det varphi'|$ doesn't hold, there is $varepsilon>0$ s.t. $m(varphi(A))>int_A|det varphi'|+varepsilon m(A)$?

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
-1
down vote

favorite












Let $A$ a closed cube, $m$ the Lebesgue measure on $mathbb R^n$ and $varphi:Uto mathbb R^n$ a $mathcal C^1$ application (but this point is not important for my question). I recal that $varphi'$ is the Jacobian matrix. Why if $$m(varphi(A))leq int_A|det varphi'|$$ doesn't hold for a closed cube of side $a$, then there is $varepsilon>0$ s.t. $$m(varphi(A))>int_A|det varphi'|+varepsilon m(A) ?$$



For me it would be only : there is a cube of side $a$ s.t. $$m(varphi(A))>int_A |det varphi'|,$$
I don't understand the thing with the term $varepsilon m(A)$.







share|cite|improve this question
























    up vote
    -1
    down vote

    favorite












    Let $A$ a closed cube, $m$ the Lebesgue measure on $mathbb R^n$ and $varphi:Uto mathbb R^n$ a $mathcal C^1$ application (but this point is not important for my question). I recal that $varphi'$ is the Jacobian matrix. Why if $$m(varphi(A))leq int_A|det varphi'|$$ doesn't hold for a closed cube of side $a$, then there is $varepsilon>0$ s.t. $$m(varphi(A))>int_A|det varphi'|+varepsilon m(A) ?$$



    For me it would be only : there is a cube of side $a$ s.t. $$m(varphi(A))>int_A |det varphi'|,$$
    I don't understand the thing with the term $varepsilon m(A)$.







    share|cite|improve this question






















      up vote
      -1
      down vote

      favorite









      up vote
      -1
      down vote

      favorite











      Let $A$ a closed cube, $m$ the Lebesgue measure on $mathbb R^n$ and $varphi:Uto mathbb R^n$ a $mathcal C^1$ application (but this point is not important for my question). I recal that $varphi'$ is the Jacobian matrix. Why if $$m(varphi(A))leq int_A|det varphi'|$$ doesn't hold for a closed cube of side $a$, then there is $varepsilon>0$ s.t. $$m(varphi(A))>int_A|det varphi'|+varepsilon m(A) ?$$



      For me it would be only : there is a cube of side $a$ s.t. $$m(varphi(A))>int_A |det varphi'|,$$
      I don't understand the thing with the term $varepsilon m(A)$.







      share|cite|improve this question












      Let $A$ a closed cube, $m$ the Lebesgue measure on $mathbb R^n$ and $varphi:Uto mathbb R^n$ a $mathcal C^1$ application (but this point is not important for my question). I recal that $varphi'$ is the Jacobian matrix. Why if $$m(varphi(A))leq int_A|det varphi'|$$ doesn't hold for a closed cube of side $a$, then there is $varepsilon>0$ s.t. $$m(varphi(A))>int_A|det varphi'|+varepsilon m(A) ?$$



      For me it would be only : there is a cube of side $a$ s.t. $$m(varphi(A))>int_A |det varphi'|,$$
      I don't understand the thing with the term $varepsilon m(A)$.









      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Aug 25 at 9:39









      Peter

      547113




      547113




















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          1
          down vote













          Hint



          $$aleq biff forall varepsilon>0, a<b+varepsilon iff exists Asubset mathbb R^N : forall varepsilon>0, a <b+varepsilon m(A).$$






          share|cite|improve this answer




















          • Thank you but I really don't see in what it helps...
            – Peter
            Aug 25 at 10:02










          • What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
            – Surb
            Aug 25 at 12:19










          Your Answer




          StackExchange.ifUsing("editor", function ()
          return StackExchange.using("mathjaxEditing", function ()
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          );
          );
          , "mathjax-editing");

          StackExchange.ready(function()
          var channelOptions =
          tags: "".split(" "),
          id: "69"
          ;
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function()
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled)
          StackExchange.using("snippets", function()
          createEditor();
          );

          else
          createEditor();

          );

          function createEditor()
          StackExchange.prepareEditor(
          heartbeatType: 'answer',
          convertImagesToLinks: true,
          noModals: false,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          );



          );













           

          draft saved


          draft discarded


















          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2893943%2fwhy-if-m-varphia-leq-int-a-det-varphi-doesnt-hold-there-is-vareps%23new-answer', 'question_page');

          );

          Post as a guest






























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          1
          down vote













          Hint



          $$aleq biff forall varepsilon>0, a<b+varepsilon iff exists Asubset mathbb R^N : forall varepsilon>0, a <b+varepsilon m(A).$$






          share|cite|improve this answer




















          • Thank you but I really don't see in what it helps...
            – Peter
            Aug 25 at 10:02










          • What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
            – Surb
            Aug 25 at 12:19














          up vote
          1
          down vote













          Hint



          $$aleq biff forall varepsilon>0, a<b+varepsilon iff exists Asubset mathbb R^N : forall varepsilon>0, a <b+varepsilon m(A).$$






          share|cite|improve this answer




















          • Thank you but I really don't see in what it helps...
            – Peter
            Aug 25 at 10:02










          • What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
            – Surb
            Aug 25 at 12:19












          up vote
          1
          down vote










          up vote
          1
          down vote









          Hint



          $$aleq biff forall varepsilon>0, a<b+varepsilon iff exists Asubset mathbb R^N : forall varepsilon>0, a <b+varepsilon m(A).$$






          share|cite|improve this answer












          Hint



          $$aleq biff forall varepsilon>0, a<b+varepsilon iff exists Asubset mathbb R^N : forall varepsilon>0, a <b+varepsilon m(A).$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Aug 25 at 9:41









          Surb

          36.6k84376




          36.6k84376











          • Thank you but I really don't see in what it helps...
            – Peter
            Aug 25 at 10:02










          • What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
            – Surb
            Aug 25 at 12:19
















          • Thank you but I really don't see in what it helps...
            – Peter
            Aug 25 at 10:02










          • What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
            – Surb
            Aug 25 at 12:19















          Thank you but I really don't see in what it helps...
          – Peter
          Aug 25 at 10:02




          Thank you but I really don't see in what it helps...
          – Peter
          Aug 25 at 10:02












          What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
          – Surb
          Aug 25 at 12:19




          What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
          – Surb
          Aug 25 at 12:19

















           

          draft saved


          draft discarded















































           


          draft saved


          draft discarded














          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2893943%2fwhy-if-m-varphia-leq-int-a-det-varphi-doesnt-hold-there-is-vareps%23new-answer', 'question_page');

          );

          Post as a guest













































































          這個網誌中的熱門文章

          How to combine Bézier curves to a surface?

          Carbon dioxide

          Why am i infinitely getting the same tweet with the Twitter Search API?