Laplacian of the Euclidean Norm












2












$begingroup$


Let $r:mathbb{R}^n rightarrow mathbb{R}$ be defined by $r(x)=|x|,$ where $large|x|=(x_1^2+dots+x_n^2)^{frac{1}{2}}.$ Compute $nabla^2r,$ the Laplacian of $r.$




I perform the following computations,
$$largepartial_{x_i}r(x)=frac{1}{2} cdot frac{2x_i}{(x_1^2+dots+x_n^2)^{frac{1}{2}}}=frac{x_i}{r(x)},$$
so that we get
$$largepartial_{x_i}partial_{x_i}r(x)=frac{r(x)-frac{x_i^2}{r(x)}}{r(x)^2},$$
and putting these together yields
begin{align*}largenabla^2r(x)
&=largepartial_{x_1}^2r(x)+dots+partial_{x_n}^2r(x) \
&=largefrac{r(x)-x_1^2cdot r(x)^{-1}}{r(x)^2}+dots+frac{r(x)-x_n^2cdot r(x)^{-1}}{r(x)^2}\
&=largefrac{ncdot r(x)-r(x)^{-1}cdot r(x)^2}{r(x)^2}\
&=frac{n-1}{r(x)}.\
end{align*}



Is the above computation correct? I checked the calculation several times, but still have a feeling that I made a mistake somewhere






calculus multivariable-calculus partial-derivative







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Looks right to me
    $endgroup$
    – Anthony Ter
    Feb 3 at 9:50






  • 1




    $begingroup$
    Take a look to this document which addresses a more general question ; it confirms your result (see the last line !) staff.science.uu.nl/~kolk0101/Books/Analysis/normlinearmap.pdf
    $endgroup$
    – Jean Marie
    Feb 3 at 10:10
















2












$begingroup$


Let $r:mathbb{R}^n rightarrow mathbb{R}$ be defined by $r(x)=|x|,$ where $large|x|=(x_1^2+dots+x_n^2)^{frac{1}{2}}.$ Compute $nabla^2r,$ the Laplacian of $r.$




I perform the following computations,
$$largepartial_{x_i}r(x)=frac{1}{2} cdot frac{2x_i}{(x_1^2+dots+x_n^2)^{frac{1}{2}}}=frac{x_i}{r(x)},$$
so that we get
$$largepartial_{x_i}partial_{x_i}r(x)=frac{r(x)-frac{x_i^2}{r(x)}}{r(x)^2},$$
and putting these together yields
begin{align*}largenabla^2r(x)
&=largepartial_{x_1}^2r(x)+dots+partial_{x_n}^2r(x) \
&=largefrac{r(x)-x_1^2cdot r(x)^{-1}}{r(x)^2}+dots+frac{r(x)-x_n^2cdot r(x)^{-1}}{r(x)^2}\
&=largefrac{ncdot r(x)-r(x)^{-1}cdot r(x)^2}{r(x)^2}\
&=frac{n-1}{r(x)}.\
end{align*}



Is the above computation correct? I checked the calculation several times, but still have a feeling that I made a mistake somewhere






calculus multivariable-calculus partial-derivative







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Looks right to me
    $endgroup$
    – Anthony Ter
    Feb 3 at 9:50






  • 1




    $begingroup$
    Take a look to this document which addresses a more general question ; it confirms your result (see the last line !) staff.science.uu.nl/~kolk0101/Books/Analysis/normlinearmap.pdf
    $endgroup$
    – Jean Marie
    Feb 3 at 10:10














2












2








2





$begingroup$


Let $r:mathbb{R}^n rightarrow mathbb{R}$ be defined by $r(x)=|x|,$ where $large|x|=(x_1^2+dots+x_n^2)^{frac{1}{2}}.$ Compute $nabla^2r,$ the Laplacian of $r.$




I perform the following computations,
$$largepartial_{x_i}r(x)=frac{1}{2} cdot frac{2x_i}{(x_1^2+dots+x_n^2)^{frac{1}{2}}}=frac{x_i}{r(x)},$$
so that we get
$$largepartial_{x_i}partial_{x_i}r(x)=frac{r(x)-frac{x_i^2}{r(x)}}{r(x)^2},$$
and putting these together yields
begin{align*}largenabla^2r(x)
&=largepartial_{x_1}^2r(x)+dots+partial_{x_n}^2r(x) \
&=largefrac{r(x)-x_1^2cdot r(x)^{-1}}{r(x)^2}+dots+frac{r(x)-x_n^2cdot r(x)^{-1}}{r(x)^2}\
&=largefrac{ncdot r(x)-r(x)^{-1}cdot r(x)^2}{r(x)^2}\
&=frac{n-1}{r(x)}.\
end{align*}



Is the above computation correct? I checked the calculation several times, but still have a feeling that I made a mistake somewhere






calculus multivariable-calculus partial-derivative







share|cite|improve this question











$endgroup$




Let $r:mathbb{R}^n rightarrow mathbb{R}$ be defined by $r(x)=|x|,$ where $large|x|=(x_1^2+dots+x_n^2)^{frac{1}{2}}.$ Compute $nabla^2r,$ the Laplacian of $r.$




I perform the following computations,
$$largepartial_{x_i}r(x)=frac{1}{2} cdot frac{2x_i}{(x_1^2+dots+x_n^2)^{frac{1}{2}}}=frac{x_i}{r(x)},$$
so that we get
$$largepartial_{x_i}partial_{x_i}r(x)=frac{r(x)-frac{x_i^2}{r(x)}}{r(x)^2},$$
and putting these together yields
begin{align*}largenabla^2r(x)
&=largepartial_{x_1}^2r(x)+dots+partial_{x_n}^2r(x) \
&=largefrac{r(x)-x_1^2cdot r(x)^{-1}}{r(x)^2}+dots+frac{r(x)-x_n^2cdot r(x)^{-1}}{r(x)^2}\
&=largefrac{ncdot r(x)-r(x)^{-1}cdot r(x)^2}{r(x)^2}\
&=frac{n-1}{r(x)}.\
end{align*}



Is the above computation correct? I checked the calculation several times, but still have a feeling that I made a mistake somewhere






calculus multivariable-calculus partial-derivative




calculus multivariable-calculus partial-derivative






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Feb 3 at 10:11









Jean Marie

31.5k42355




31.5k42355










asked Feb 3 at 9:33









Gaby AlfonsoGaby Alfonso

1,2051418




1,2051418








  • 1




    $begingroup$
    Looks right to me
    $endgroup$
    – Anthony Ter
    Feb 3 at 9:50






  • 1




    $begingroup$
    Take a look to this document which addresses a more general question ; it confirms your result (see the last line !) staff.science.uu.nl/~kolk0101/Books/Analysis/normlinearmap.pdf
    $endgroup$
    – Jean Marie
    Feb 3 at 10:10














  • 1




    $begingroup$
    Looks right to me
    $endgroup$
    – Anthony Ter
    Feb 3 at 9:50






  • 1




    $begingroup$
    Take a look to this document which addresses a more general question ; it confirms your result (see the last line !) staff.science.uu.nl/~kolk0101/Books/Analysis/normlinearmap.pdf
    $endgroup$
    – Jean Marie
    Feb 3 at 10:10








1




1




$begingroup$
Looks right to me
$endgroup$
– Anthony Ter
Feb 3 at 9:50




$begingroup$
Looks right to me
$endgroup$
– Anthony Ter
Feb 3 at 9:50




1




1




$begingroup$
Take a look to this document which addresses a more general question ; it confirms your result (see the last line !) staff.science.uu.nl/~kolk0101/Books/Analysis/normlinearmap.pdf
$endgroup$
– Jean Marie
Feb 3 at 10:10




$begingroup$
Take a look to this document which addresses a more general question ; it confirms your result (see the last line !) staff.science.uu.nl/~kolk0101/Books/Analysis/normlinearmap.pdf
$endgroup$
– Jean Marie
Feb 3 at 10:10










1 Answer
1






active

oldest

votes


















2












$begingroup$

If $u(x)=v(|x|)$ then
$$Delta u=frac{n-1}{|x|}v'(|x|)+v''(|x|).$$
See, for example https://web.stanford.edu/class/math220b/handouts/laplace.pdf






share|cite|improve this answer









$endgroup$














    Your Answer








    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%2f3098355%2flaplacian-of-the-euclidean-norm%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









    2












    $begingroup$

    If $u(x)=v(|x|)$ then
    $$Delta u=frac{n-1}{|x|}v'(|x|)+v''(|x|).$$
    See, for example https://web.stanford.edu/class/math220b/handouts/laplace.pdf






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      If $u(x)=v(|x|)$ then
      $$Delta u=frac{n-1}{|x|}v'(|x|)+v''(|x|).$$
      See, for example https://web.stanford.edu/class/math220b/handouts/laplace.pdf






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        If $u(x)=v(|x|)$ then
        $$Delta u=frac{n-1}{|x|}v'(|x|)+v''(|x|).$$
        See, for example https://web.stanford.edu/class/math220b/handouts/laplace.pdf






        share|cite|improve this answer









        $endgroup$



        If $u(x)=v(|x|)$ then
        $$Delta u=frac{n-1}{|x|}v'(|x|)+v''(|x|).$$
        See, for example https://web.stanford.edu/class/math220b/handouts/laplace.pdf







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Feb 3 at 11:08









        Aleksas DomarkasAleksas Domarkas

        1,62317




        1,62317






























            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%2f3098355%2flaplacian-of-the-euclidean-norm%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

            android studio warns about leanback feature tag usage required on manifest while using Unity exported app?

            Image
            .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 0 This is a simple project showing cubes on Unity, exported to android project and imported on Android studio. On maifest file it showed following warning: Expecting <uses-feature android:name="android.software.leanback" android:required="false" /> tag I have no information about it. And also on default emptyview android project does not use/mention this feature. I know it is required by TV's and such but why its not added by default? Why is this? And Should I concern about it? android unity3d
            Read more

            SQL update select statement

            Image
            0 I'm trying to update an entry in my database (postgresql). I'm having a problem with updating the query: INSERT INTO prices (price, product_id) SELECT product_price, commn_id FROM products_temp as prod WHERE NOT EXISTS (SELECT * FROM prices od WHERE prod.commn_id = od.product_id) OR prod.product_price != ( SELECT price FROM prices as p WHERE p.product_id = prod.commn_id order by p.end desc LIMIT 1 ) Query works fine when i delete: OR prod.product_price != ( SELECT price FROM prices as p WHERE p.product_id = prod.commn_id order by p.end desc LIMIT 1 ) So it seems to me that it's looping through this operation. My question is, how can I fix it? sql postgresql
            Read more

            WPF add header to Image with URL pettitions [duplicate]

            Image
            up vote 0 down vote favorite This question already has an answer here: Image source URI requiring authorization 3 answers Add header to all* HTTP requests on a machine 1 answer first of all sorry for my english. Recently I added some security in the backend of my project, and now I need to send a validation code(apiKey) inside the headers to retrieve the images. I have a problem with ALL images in my project that I had attached in my UI. For example, before I had: <Image Height="90" Source="{Binding Video.Thumbnail}"/> Now I need to change my code to create a CUSTOM petition add the apiKey in the header:
            Read more