A piecewise smooth curve in $mathbbR^n$ is a null set in $mathbbR^n$Prove that the image of a curve has zero contentNull velocity and piecewise smooth pathDefinitions of multiple Riemann integrals and boundednessPartitions of unity from PMA RudinUsing Green's Theorem to Express the Integral $I=int_C (Pdx+Qdy)$ as an expression of $I_i=int _C_i (Pdx+Qdy)$Question about Manifolds (without boundaries)$partial U$ is a smooth curve (i.e., the image of a smooth path $mathbfc$ with $mathbfc' not= 0$)?Smooth map with null differential at each point are constant on the connected component of the domainAbout change of coordinates of vector fields in smooth manifold theorySmooth curve segments and smooth charts

What problems would a superhuman have who's skin is constantly hot?

Can the alpha, lambda values of a glmnet object output determine whether ridge or Lasso?

Can I use a violin G string for D?

For which categories of spectra is there an explicit description of the fibrant objects via lifting properties?

Why aren't there more Gauls like Obelix?

What is the generally accepted pronunciation of “topoi”?

MySQL importing CSV files really slow

How can I manipulate the output of Information?

Recommendation letter by significant other if you worked with them professionally?

How do we create new idioms and use them in a novel?

Has a sovereign Communist government ever run, and conceded loss, on a fair election?

What is Tony Stark injecting into himself in Iron Man 3?

What is better: yes / no radio, or simple checkbox?

Doesn't allowing a user mode program to access kernel space memory and execute the IN and OUT instructions defeat the purpose of having CPU modes?

Is it possible that a question has only two answers?

Trig Subsitution When There's No Square Root

Professor forcing me to attend a conference, I can't afford even with 50% funding

Does a difference of tense count as a difference of meaning in a minimal pair?

What do you call someone who likes to pick fights?

Street obstacles in New Zealand

What's the 'present simple' form of the word "нашла́" in 3rd person singular female?

Help find my computational error for logarithms

Why do we say ‘pairwise disjoint’, rather than ‘disjoint’?

Why is there an extra space when I type "ls" in the Desktop directory?



A piecewise smooth curve in $mathbbR^n$ is a null set in $mathbbR^n$


Prove that the image of a curve has zero contentNull velocity and piecewise smooth pathDefinitions of multiple Riemann integrals and boundednessPartitions of unity from PMA RudinUsing Green's Theorem to Express the Integral $I=int_C (Pdx+Qdy)$ as an expression of $I_i=int _C_i (Pdx+Qdy)$Question about Manifolds (without boundaries)$partial U$ is a smooth curve (i.e., the image of a smooth path $mathbfc$ with $mathbfc' not= 0$)?Smooth map with null differential at each point are constant on the connected component of the domainAbout change of coordinates of vector fields in smooth manifold theorySmooth curve segments and smooth charts













0












$begingroup$


Definitions:




  • A piecewise smooth curve in $mathbbR^2$ is a subset of $mathbbR^2$ that can be written as $boldsymbolvarphi_1[a_1,b_1] cup boldsymbolvarphi_2[a_2,b_2]cup dotscup boldsymbolvarphi_N [a_N,b_N]$, with each $boldsymbolvarphi_i:[a_i,b_i] to mathbbR^2$ a parameterization of a smooth curve, and where each end point $boldsymbolvarphi_i(b_i)$ corresponds with the next starting point $boldsymbolvarphi_i+1(a_i+1)$.



  • A subset $X subseteq mathbbR^2$ is a null set in $mathbbR^2$ if for each $varepsilon > 0$ there is a sequence $R_1,R_2,dots$ of rectangles with sides parallel to the coordinate axes such that



    1. $X subseteq R_1 cup R_2 cup dots$


    2. $|R_1|+|R_2| + dots < varepsilon$, with $|R_i|$ the area of rectangle $R_i$.




My notes proceed with stating that a piecewise smooth curve in $mathbbR^n$ is a null set in $mathbbR^n$. I would like to understand to intuiton behind this theorem. Is there an easily understandable explanation behind it?



Thanks in advance.










share|cite|improve this question









$endgroup$
















    0












    $begingroup$


    Definitions:




    • A piecewise smooth curve in $mathbbR^2$ is a subset of $mathbbR^2$ that can be written as $boldsymbolvarphi_1[a_1,b_1] cup boldsymbolvarphi_2[a_2,b_2]cup dotscup boldsymbolvarphi_N [a_N,b_N]$, with each $boldsymbolvarphi_i:[a_i,b_i] to mathbbR^2$ a parameterization of a smooth curve, and where each end point $boldsymbolvarphi_i(b_i)$ corresponds with the next starting point $boldsymbolvarphi_i+1(a_i+1)$.



    • A subset $X subseteq mathbbR^2$ is a null set in $mathbbR^2$ if for each $varepsilon > 0$ there is a sequence $R_1,R_2,dots$ of rectangles with sides parallel to the coordinate axes such that



      1. $X subseteq R_1 cup R_2 cup dots$


      2. $|R_1|+|R_2| + dots < varepsilon$, with $|R_i|$ the area of rectangle $R_i$.




    My notes proceed with stating that a piecewise smooth curve in $mathbbR^n$ is a null set in $mathbbR^n$. I would like to understand to intuiton behind this theorem. Is there an easily understandable explanation behind it?



    Thanks in advance.










    share|cite|improve this question









    $endgroup$














      0












      0








      0





      $begingroup$


      Definitions:




      • A piecewise smooth curve in $mathbbR^2$ is a subset of $mathbbR^2$ that can be written as $boldsymbolvarphi_1[a_1,b_1] cup boldsymbolvarphi_2[a_2,b_2]cup dotscup boldsymbolvarphi_N [a_N,b_N]$, with each $boldsymbolvarphi_i:[a_i,b_i] to mathbbR^2$ a parameterization of a smooth curve, and where each end point $boldsymbolvarphi_i(b_i)$ corresponds with the next starting point $boldsymbolvarphi_i+1(a_i+1)$.



      • A subset $X subseteq mathbbR^2$ is a null set in $mathbbR^2$ if for each $varepsilon > 0$ there is a sequence $R_1,R_2,dots$ of rectangles with sides parallel to the coordinate axes such that



        1. $X subseteq R_1 cup R_2 cup dots$


        2. $|R_1|+|R_2| + dots < varepsilon$, with $|R_i|$ the area of rectangle $R_i$.




      My notes proceed with stating that a piecewise smooth curve in $mathbbR^n$ is a null set in $mathbbR^n$. I would like to understand to intuiton behind this theorem. Is there an easily understandable explanation behind it?



      Thanks in advance.










      share|cite|improve this question









      $endgroup$




      Definitions:




      • A piecewise smooth curve in $mathbbR^2$ is a subset of $mathbbR^2$ that can be written as $boldsymbolvarphi_1[a_1,b_1] cup boldsymbolvarphi_2[a_2,b_2]cup dotscup boldsymbolvarphi_N [a_N,b_N]$, with each $boldsymbolvarphi_i:[a_i,b_i] to mathbbR^2$ a parameterization of a smooth curve, and where each end point $boldsymbolvarphi_i(b_i)$ corresponds with the next starting point $boldsymbolvarphi_i+1(a_i+1)$.



      • A subset $X subseteq mathbbR^2$ is a null set in $mathbbR^2$ if for each $varepsilon > 0$ there is a sequence $R_1,R_2,dots$ of rectangles with sides parallel to the coordinate axes such that



        1. $X subseteq R_1 cup R_2 cup dots$


        2. $|R_1|+|R_2| + dots < varepsilon$, with $|R_i|$ the area of rectangle $R_i$.




      My notes proceed with stating that a piecewise smooth curve in $mathbbR^n$ is a null set in $mathbbR^n$. I would like to understand to intuiton behind this theorem. Is there an easily understandable explanation behind it?



      Thanks in advance.







      multivariable-calculus






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked yesterday









      ZacharyZachary

      1799




      1799




















          1 Answer
          1






          active

          oldest

          votes


















          0












          $begingroup$

          The following is not a proof; it just helps the intuition.



          It is sufficient to consider a smooth curve, since a finite union of null sets is again a null set.



          A smooth curve $phi:>[a,b]tomathbb R^2$ has finite length $L>0$. Given an $Ngg1$ you should be able to cover it with $N$ squares of side length $4Lover N$, say. These squares have a total area $$Ncdotleft(4Lover Nright)^2=16L^2over N .$$
          Given an $epsilon>0$ we can make the right hand side $<epsilon$ by choosing $N$ large enough.



          One therefore can say that the basic reason for the intended theorem is the fact that the dimension of $mathbb R^2$ is $1$ greater than the dimension of $mathbb R$, and similarly when $mathbb R^2$ is replaced by $mathbb R^n$.






          share|cite|improve this answer









          $endgroup$












            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',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader:
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            ,
            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%2f3141020%2fa-piecewise-smooth-curve-in-mathbbrn-is-a-null-set-in-mathbbrn%23new-answer', 'question_page');

            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            0












            $begingroup$

            The following is not a proof; it just helps the intuition.



            It is sufficient to consider a smooth curve, since a finite union of null sets is again a null set.



            A smooth curve $phi:>[a,b]tomathbb R^2$ has finite length $L>0$. Given an $Ngg1$ you should be able to cover it with $N$ squares of side length $4Lover N$, say. These squares have a total area $$Ncdotleft(4Lover Nright)^2=16L^2over N .$$
            Given an $epsilon>0$ we can make the right hand side $<epsilon$ by choosing $N$ large enough.



            One therefore can say that the basic reason for the intended theorem is the fact that the dimension of $mathbb R^2$ is $1$ greater than the dimension of $mathbb R$, and similarly when $mathbb R^2$ is replaced by $mathbb R^n$.






            share|cite|improve this answer









            $endgroup$

















              0












              $begingroup$

              The following is not a proof; it just helps the intuition.



              It is sufficient to consider a smooth curve, since a finite union of null sets is again a null set.



              A smooth curve $phi:>[a,b]tomathbb R^2$ has finite length $L>0$. Given an $Ngg1$ you should be able to cover it with $N$ squares of side length $4Lover N$, say. These squares have a total area $$Ncdotleft(4Lover Nright)^2=16L^2over N .$$
              Given an $epsilon>0$ we can make the right hand side $<epsilon$ by choosing $N$ large enough.



              One therefore can say that the basic reason for the intended theorem is the fact that the dimension of $mathbb R^2$ is $1$ greater than the dimension of $mathbb R$, and similarly when $mathbb R^2$ is replaced by $mathbb R^n$.






              share|cite|improve this answer









              $endgroup$















                0












                0








                0





                $begingroup$

                The following is not a proof; it just helps the intuition.



                It is sufficient to consider a smooth curve, since a finite union of null sets is again a null set.



                A smooth curve $phi:>[a,b]tomathbb R^2$ has finite length $L>0$. Given an $Ngg1$ you should be able to cover it with $N$ squares of side length $4Lover N$, say. These squares have a total area $$Ncdotleft(4Lover Nright)^2=16L^2over N .$$
                Given an $epsilon>0$ we can make the right hand side $<epsilon$ by choosing $N$ large enough.



                One therefore can say that the basic reason for the intended theorem is the fact that the dimension of $mathbb R^2$ is $1$ greater than the dimension of $mathbb R$, and similarly when $mathbb R^2$ is replaced by $mathbb R^n$.






                share|cite|improve this answer









                $endgroup$



                The following is not a proof; it just helps the intuition.



                It is sufficient to consider a smooth curve, since a finite union of null sets is again a null set.



                A smooth curve $phi:>[a,b]tomathbb R^2$ has finite length $L>0$. Given an $Ngg1$ you should be able to cover it with $N$ squares of side length $4Lover N$, say. These squares have a total area $$Ncdotleft(4Lover Nright)^2=16L^2over N .$$
                Given an $epsilon>0$ we can make the right hand side $<epsilon$ by choosing $N$ large enough.



                One therefore can say that the basic reason for the intended theorem is the fact that the dimension of $mathbb R^2$ is $1$ greater than the dimension of $mathbb R$, and similarly when $mathbb R^2$ is replaced by $mathbb R^n$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered yesterday









                Christian BlatterChristian Blatter

                175k8115327




                175k8115327



























                    draft saved

                    draft discarded
















































                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid


                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.

                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function ()
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3141020%2fa-piecewise-smooth-curve-in-mathbbrn-is-a-null-set-in-mathbbrn%23new-answer', 'question_page');

                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    Popular posts from this blog

                    How should I support this large drywall patch? Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern) Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?How do I cover large gaps in drywall?How do I keep drywall around a patch from crumbling?Can I glue a second layer of drywall?How to patch long strip on drywall?Large drywall patch: how to avoid bulging seams?Drywall Mesh Patch vs. Bulge? To remove or not to remove?How to fix this drywall job?Prep drywall before backsplashWhat's the best way to fix this horrible drywall patch job?Drywall patching using 3M Patch Plus Primer

                    random experiment with two different functions on unit interval Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Random variable and probability space notionsRandom Walk with EdgesFinding functions where the increase over a random interval is Poisson distributedNumber of days until dayCan an observed event in fact be of zero probability?Unit random processmodels of coins and uniform distributionHow to get the number of successes given $n$ trials , probability $P$ and a random variable $X$Absorbing Markov chain in a computer. Is “almost every” turned into always convergence in computer executions?Stopped random walk is not uniformly integrable

                    Lowndes Grove History Architecture References Navigation menu32°48′6″N 79°57′58″W / 32.80167°N 79.96611°W / 32.80167; -79.9661132°48′6″N 79°57′58″W / 32.80167°N 79.96611°W / 32.80167; -79.9661178002500"National Register Information System"Historic houses of South Carolina"Lowndes Grove""+32° 48' 6.00", −79° 57' 58.00""Lowndes Grove, Charleston County (260 St. Margaret St., Charleston)""Lowndes Grove"The Charleston ExpositionIt Happened in South Carolina"Lowndes Grove (House), Saint Margaret Street & Sixth Avenue, Charleston, Charleston County, SC(Photographs)"Plantations of the Carolina Low Countrye