laplace operator on a laplace












2












$begingroup$


What is the result when Laplace operator is applied on Laplace operator, is it



$$nabla^2(nabla^2 r)= nabla^4r = frac{partial^4 r}{partial x^4} + frac{partial^4 r}{partial y^4} + frac{partial^4 r}{partial z^4} $$



or something else with more terms ?



Thanks






pde







share|cite|improve this question











$endgroup$












  • $begingroup$
    I think we need to define what $cdot$ means in this context.
    $endgroup$
    – manooooh
    Dec 12 '18 at 9:16


















2












$begingroup$


What is the result when Laplace operator is applied on Laplace operator, is it



$$nabla^2(nabla^2 r)= nabla^4r = frac{partial^4 r}{partial x^4} + frac{partial^4 r}{partial y^4} + frac{partial^4 r}{partial z^4} $$



or something else with more terms ?



Thanks






pde







share|cite|improve this question











$endgroup$












  • $begingroup$
    I think we need to define what $cdot$ means in this context.
    $endgroup$
    – manooooh
    Dec 12 '18 at 9:16
















2












2








2





$begingroup$


What is the result when Laplace operator is applied on Laplace operator, is it



$$nabla^2(nabla^2 r)= nabla^4r = frac{partial^4 r}{partial x^4} + frac{partial^4 r}{partial y^4} + frac{partial^4 r}{partial z^4} $$



or something else with more terms ?



Thanks






pde







share|cite|improve this question











$endgroup$




What is the result when Laplace operator is applied on Laplace operator, is it



$$nabla^2(nabla^2 r)= nabla^4r = frac{partial^4 r}{partial x^4} + frac{partial^4 r}{partial y^4} + frac{partial^4 r}{partial z^4} $$



or something else with more terms ?



Thanks






pde




pde






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 12 '18 at 9:33







newstudent

















asked Dec 12 '18 at 9:15









newstudentnewstudent

366




366












  • $begingroup$
    I think we need to define what $cdot$ means in this context.
    $endgroup$
    – manooooh
    Dec 12 '18 at 9:16




















  • $begingroup$
    I think we need to define what $cdot$ means in this context.
    $endgroup$
    – manooooh
    Dec 12 '18 at 9:16


















$begingroup$
I think we need to define what $cdot$ means in this context.
$endgroup$
– manooooh
Dec 12 '18 at 9:16






$begingroup$
I think we need to define what $cdot$ means in this context.
$endgroup$
– manooooh
Dec 12 '18 at 9:16












1 Answer
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$begingroup$

The way you put it is
$$
nabla^2left( frac{partial^2 r}{partial x^2} + frac{partial^2 r}{partial y^2} + frac{partial^2 r}{partial z^2}right)=
frac{partial^4 r}{partial x^4} + frac{partial^4 r}{partial y^4} + frac{partial^4 r}{partial z^4}+2frac{partial^4r}{partial x^2partial y^2}
+2frac{partial^4r}{partial x^2partial z^2}
+2frac{partial^4r}{partial y^2partial z^2}.
$$






share|cite|improve this answer









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    $begingroup$

    The way you put it is
    $$
    nabla^2left( frac{partial^2 r}{partial x^2} + frac{partial^2 r}{partial y^2} + frac{partial^2 r}{partial z^2}right)=
    frac{partial^4 r}{partial x^4} + frac{partial^4 r}{partial y^4} + frac{partial^4 r}{partial z^4}+2frac{partial^4r}{partial x^2partial y^2}
    +2frac{partial^4r}{partial x^2partial z^2}
    +2frac{partial^4r}{partial y^2partial z^2}.
    $$






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      The way you put it is
      $$
      nabla^2left( frac{partial^2 r}{partial x^2} + frac{partial^2 r}{partial y^2} + frac{partial^2 r}{partial z^2}right)=
      frac{partial^4 r}{partial x^4} + frac{partial^4 r}{partial y^4} + frac{partial^4 r}{partial z^4}+2frac{partial^4r}{partial x^2partial y^2}
      +2frac{partial^4r}{partial x^2partial z^2}
      +2frac{partial^4r}{partial y^2partial z^2}.
      $$






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        The way you put it is
        $$
        nabla^2left( frac{partial^2 r}{partial x^2} + frac{partial^2 r}{partial y^2} + frac{partial^2 r}{partial z^2}right)=
        frac{partial^4 r}{partial x^4} + frac{partial^4 r}{partial y^4} + frac{partial^4 r}{partial z^4}+2frac{partial^4r}{partial x^2partial y^2}
        +2frac{partial^4r}{partial x^2partial z^2}
        +2frac{partial^4r}{partial y^2partial z^2}.
        $$






        share|cite|improve this answer









        $endgroup$



        The way you put it is
        $$
        nabla^2left( frac{partial^2 r}{partial x^2} + frac{partial^2 r}{partial y^2} + frac{partial^2 r}{partial z^2}right)=
        frac{partial^4 r}{partial x^4} + frac{partial^4 r}{partial y^4} + frac{partial^4 r}{partial z^4}+2frac{partial^4r}{partial x^2partial y^2}
        +2frac{partial^4r}{partial x^2partial z^2}
        +2frac{partial^4r}{partial y^2partial z^2}.
        $$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 12 '18 at 9:57









        JonJon

        4,43211223




        4,43211223






























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