What is the meaning of notation $ nabla u + nabla u^T$?












1












$begingroup$


In my exercises it appears ($u:mathbb{R}^3rightarrowmathbb{R}^3$):



$$ nabla u + nabla u^T$$



What is the meaning of this notation when writing it in terms of partials of $u_i$?






notation vector-analysis







share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    In my exercises it appears ($u:mathbb{R}^3rightarrowmathbb{R}^3$):



    $$ nabla u + nabla u^T$$



    What is the meaning of this notation when writing it in terms of partials of $u_i$?






    notation vector-analysis







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      In my exercises it appears ($u:mathbb{R}^3rightarrowmathbb{R}^3$):



      $$ nabla u + nabla u^T$$



      What is the meaning of this notation when writing it in terms of partials of $u_i$?






      notation vector-analysis







      share|cite|improve this question











      $endgroup$




      In my exercises it appears ($u:mathbb{R}^3rightarrowmathbb{R}^3$):



      $$ nabla u + nabla u^T$$



      What is the meaning of this notation when writing it in terms of partials of $u_i$?






      notation vector-analysis




      notation vector-analysis






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 19 '18 at 20:51









      Jean Marie

      30.4k42153




      30.4k42153










      asked Dec 19 '18 at 19:24









      davidivadfuldavidivadful

      13110




      13110






















          2 Answers
          2






          active

          oldest

          votes


















          3












          $begingroup$

          Typically, when $u:Bbb R^3 to Bbb R^3$, we define
          $$
          newcommand{pwrt}[2]{frac{partial #1}{partial #2}}\
          nabla u = pmatrix{pwrt {u_1}{x_1} & pwrt {u_1}{x_2} & pwrt {u_1}{x_3}\
          pwrt {u_2}{x_1} & pwrt {u_2}{x_2} & pwrt {u_2}{x_3}\
          pwrt {u_3}{x_1} & pwrt {u_3}{x_2} & pwrt {u_3}{x_3}\}
          $$
          In some contexts, this is referred to instead as the Jacobian of $u$.



          $M^T$ refers to the transpose of the matrix $M$.






          share|cite|improve this answer









          $endgroup$





















            2












            $begingroup$

            $M=nabla u ; ;$ is a $(3times 3)$ matrix
            with
            $$M_{i,j}=frac{partial u_i}{partial x_j}$$



            and



            $$nabla u^T=M^T=N$$
            with
            $$N_{i,j}=M_{j,i}=frac{partial u_j}{partial x_i}$$ is the transpose matrix of $M$.






            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%2f3046786%2fwhat-is-the-meaning-of-notation-nabla-u-nabla-ut%23new-answer', 'question_page');
              }
              );

              Post as a guest















              Required, but never shown

























              2 Answers
              2






              active

              oldest

              votes








              2 Answers
              2






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes









              3












              $begingroup$

              Typically, when $u:Bbb R^3 to Bbb R^3$, we define
              $$
              newcommand{pwrt}[2]{frac{partial #1}{partial #2}}\
              nabla u = pmatrix{pwrt {u_1}{x_1} & pwrt {u_1}{x_2} & pwrt {u_1}{x_3}\
              pwrt {u_2}{x_1} & pwrt {u_2}{x_2} & pwrt {u_2}{x_3}\
              pwrt {u_3}{x_1} & pwrt {u_3}{x_2} & pwrt {u_3}{x_3}\}
              $$
              In some contexts, this is referred to instead as the Jacobian of $u$.



              $M^T$ refers to the transpose of the matrix $M$.






              share|cite|improve this answer









              $endgroup$


















                3












                $begingroup$

                Typically, when $u:Bbb R^3 to Bbb R^3$, we define
                $$
                newcommand{pwrt}[2]{frac{partial #1}{partial #2}}\
                nabla u = pmatrix{pwrt {u_1}{x_1} & pwrt {u_1}{x_2} & pwrt {u_1}{x_3}\
                pwrt {u_2}{x_1} & pwrt {u_2}{x_2} & pwrt {u_2}{x_3}\
                pwrt {u_3}{x_1} & pwrt {u_3}{x_2} & pwrt {u_3}{x_3}\}
                $$
                In some contexts, this is referred to instead as the Jacobian of $u$.



                $M^T$ refers to the transpose of the matrix $M$.






                share|cite|improve this answer









                $endgroup$
















                  3












                  3








                  3





                  $begingroup$

                  Typically, when $u:Bbb R^3 to Bbb R^3$, we define
                  $$
                  newcommand{pwrt}[2]{frac{partial #1}{partial #2}}\
                  nabla u = pmatrix{pwrt {u_1}{x_1} & pwrt {u_1}{x_2} & pwrt {u_1}{x_3}\
                  pwrt {u_2}{x_1} & pwrt {u_2}{x_2} & pwrt {u_2}{x_3}\
                  pwrt {u_3}{x_1} & pwrt {u_3}{x_2} & pwrt {u_3}{x_3}\}
                  $$
                  In some contexts, this is referred to instead as the Jacobian of $u$.



                  $M^T$ refers to the transpose of the matrix $M$.






                  share|cite|improve this answer









                  $endgroup$



                  Typically, when $u:Bbb R^3 to Bbb R^3$, we define
                  $$
                  newcommand{pwrt}[2]{frac{partial #1}{partial #2}}\
                  nabla u = pmatrix{pwrt {u_1}{x_1} & pwrt {u_1}{x_2} & pwrt {u_1}{x_3}\
                  pwrt {u_2}{x_1} & pwrt {u_2}{x_2} & pwrt {u_2}{x_3}\
                  pwrt {u_3}{x_1} & pwrt {u_3}{x_2} & pwrt {u_3}{x_3}\}
                  $$
                  In some contexts, this is referred to instead as the Jacobian of $u$.



                  $M^T$ refers to the transpose of the matrix $M$.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 19 '18 at 19:30









                  OmnomnomnomOmnomnomnom

                  128k791185




                  128k791185























                      2












                      $begingroup$

                      $M=nabla u ; ;$ is a $(3times 3)$ matrix
                      with
                      $$M_{i,j}=frac{partial u_i}{partial x_j}$$



                      and



                      $$nabla u^T=M^T=N$$
                      with
                      $$N_{i,j}=M_{j,i}=frac{partial u_j}{partial x_i}$$ is the transpose matrix of $M$.






                      share|cite|improve this answer











                      $endgroup$


















                        2












                        $begingroup$

                        $M=nabla u ; ;$ is a $(3times 3)$ matrix
                        with
                        $$M_{i,j}=frac{partial u_i}{partial x_j}$$



                        and



                        $$nabla u^T=M^T=N$$
                        with
                        $$N_{i,j}=M_{j,i}=frac{partial u_j}{partial x_i}$$ is the transpose matrix of $M$.






                        share|cite|improve this answer











                        $endgroup$
















                          2












                          2








                          2





                          $begingroup$

                          $M=nabla u ; ;$ is a $(3times 3)$ matrix
                          with
                          $$M_{i,j}=frac{partial u_i}{partial x_j}$$



                          and



                          $$nabla u^T=M^T=N$$
                          with
                          $$N_{i,j}=M_{j,i}=frac{partial u_j}{partial x_i}$$ is the transpose matrix of $M$.






                          share|cite|improve this answer











                          $endgroup$



                          $M=nabla u ; ;$ is a $(3times 3)$ matrix
                          with
                          $$M_{i,j}=frac{partial u_i}{partial x_j}$$



                          and



                          $$nabla u^T=M^T=N$$
                          with
                          $$N_{i,j}=M_{j,i}=frac{partial u_j}{partial x_i}$$ is the transpose matrix of $M$.







                          share|cite|improve this answer














                          share|cite|improve this answer



                          share|cite|improve this answer








                          edited Dec 19 '18 at 20:05

























                          answered Dec 19 '18 at 19:31









                          hamam_Abdallahhamam_Abdallah

                          38k21634




                          38k21634






























                              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%2f3046786%2fwhat-is-the-meaning-of-notation-nabla-u-nabla-ut%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

                              Probability when a professor distributes a quiz and homework assignment to a class of n students.

                              Image
                              0 0 $begingroup$ Need help with this problem. Suppose our lazy professor collects a quiz and a homework assignment from a class of n students one day, then distributes both the quizzes and the homework assignments back to the class in a random fashion for grading. Each student receives one quiz and one homework assignment to grade. (a) What is the probability that every student receives someone else's quiz to grade, and someone else's homework to grade? (b) What is the probability that no student receives both their own quiz and their own homework assignment to grade? In this case, some students may receive their own quiz, and others may receive their own homework assignment. (c) Compute the limiting probability as n approaches infinity in each case. ...
                              Read more

                              Aardman Animations

                              Image
                              Aardman Animations, Ltd. Aardman Animations Tipo privada Atividade Animação em stop-motion , Animação em CGI Fundação 1972 Fundador(es) Peter Lord David Sproxton Sede Bristol, Inglaterra  Reino Unido Proprietário(s) Dreamworks Pessoas-chave Peter Lord David Sproxton Nick Park Divisões Aardman Features Aardman Digital Aardman Commercials Aardman Broadcast Aardman International Aardman Rights Aardman Effects Aardman 3-D Systems Aardman Nathan Love AardBoiled Website oficial aardman.com ...
                              Read more

                              Are they similar matrix

                              Image
                              0 $begingroup$ Do $begin{bmatrix} 0&i&0\0&0&1\0&0&0 end{bmatrix} $ and $begin{bmatrix} 0&0&0\-i&0&0\0&1&0 end{bmatrix} $ are similar.Is this True/false Clearly both are nilpotent and one is conjucate transpose of other but how to know if they are similar.i'm stuck. Please help me linear-algebra vector-spaces eigenvalues-eigenvectors generalizedeigenvector share | cite | improve this question edited Dec 10 '18 at 6:38 Vasanth Kris asked Dec 10 '18 at 6:19 ...
                              Read more