Help evaluating this surface integral, how to evaluate $dS$ in this?












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Evaluating this surface integral



$int_{S}(a^2x^2+b^2y^2+c^2z^2)^{1/2}dS$ over the ellipsoid $S:ax^2+by^2+cz^2=1$



I am well aware of evaluating surface integral over any given sphere $x^2+y^2+z^2=a^2$ using the transformation $R(u,v)=langle acos(u)sin(v), asin(u)sin(v), acos(v) rangle$ and obtaining $dS =|R_u times Rv| = a^2sin(v)$.



So for the given surface integral, I want to be able to convert the ellipsoid to a sphere and then use the above transformation.



The transformation that seems reasonable to me for this task would be $R(u,v)=langle frac1acos(u)sin(v), frac1bsin(u)sin(v), frac1ccos(v)rangle$. But evaluating $dS=|R_utimes R_v|$ is not yielding any simple manageable form.



So I thought, I would first use the transformation to convert into sphere first using



$x=X/a, ;y=Y/b,;z=Z/c$ But I don't know how to evaluate $dS$ in this case since it has three unknowns and something like $|R_u times R_v|$ doesn't works.



Can you please help me how can solve this problem?






vectors vector-analysis surfaces surface-integrals multiple-integral







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    0












    $begingroup$


    Evaluating this surface integral



    $int_{S}(a^2x^2+b^2y^2+c^2z^2)^{1/2}dS$ over the ellipsoid $S:ax^2+by^2+cz^2=1$



    I am well aware of evaluating surface integral over any given sphere $x^2+y^2+z^2=a^2$ using the transformation $R(u,v)=langle acos(u)sin(v), asin(u)sin(v), acos(v) rangle$ and obtaining $dS =|R_u times Rv| = a^2sin(v)$.



    So for the given surface integral, I want to be able to convert the ellipsoid to a sphere and then use the above transformation.



    The transformation that seems reasonable to me for this task would be $R(u,v)=langle frac1acos(u)sin(v), frac1bsin(u)sin(v), frac1ccos(v)rangle$. But evaluating $dS=|R_utimes R_v|$ is not yielding any simple manageable form.



    So I thought, I would first use the transformation to convert into sphere first using



    $x=X/a, ;y=Y/b,;z=Z/c$ But I don't know how to evaluate $dS$ in this case since it has three unknowns and something like $|R_u times R_v|$ doesn't works.



    Can you please help me how can solve this problem?






    vectors vector-analysis surfaces surface-integrals multiple-integral







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Evaluating this surface integral



      $int_{S}(a^2x^2+b^2y^2+c^2z^2)^{1/2}dS$ over the ellipsoid $S:ax^2+by^2+cz^2=1$



      I am well aware of evaluating surface integral over any given sphere $x^2+y^2+z^2=a^2$ using the transformation $R(u,v)=langle acos(u)sin(v), asin(u)sin(v), acos(v) rangle$ and obtaining $dS =|R_u times Rv| = a^2sin(v)$.



      So for the given surface integral, I want to be able to convert the ellipsoid to a sphere and then use the above transformation.



      The transformation that seems reasonable to me for this task would be $R(u,v)=langle frac1acos(u)sin(v), frac1bsin(u)sin(v), frac1ccos(v)rangle$. But evaluating $dS=|R_utimes R_v|$ is not yielding any simple manageable form.



      So I thought, I would first use the transformation to convert into sphere first using



      $x=X/a, ;y=Y/b,;z=Z/c$ But I don't know how to evaluate $dS$ in this case since it has three unknowns and something like $|R_u times R_v|$ doesn't works.



      Can you please help me how can solve this problem?






      vectors vector-analysis surfaces surface-integrals multiple-integral







      share|cite|improve this question









      $endgroup$




      Evaluating this surface integral



      $int_{S}(a^2x^2+b^2y^2+c^2z^2)^{1/2}dS$ over the ellipsoid $S:ax^2+by^2+cz^2=1$



      I am well aware of evaluating surface integral over any given sphere $x^2+y^2+z^2=a^2$ using the transformation $R(u,v)=langle acos(u)sin(v), asin(u)sin(v), acos(v) rangle$ and obtaining $dS =|R_u times Rv| = a^2sin(v)$.



      So for the given surface integral, I want to be able to convert the ellipsoid to a sphere and then use the above transformation.



      The transformation that seems reasonable to me for this task would be $R(u,v)=langle frac1acos(u)sin(v), frac1bsin(u)sin(v), frac1ccos(v)rangle$. But evaluating $dS=|R_utimes R_v|$ is not yielding any simple manageable form.



      So I thought, I would first use the transformation to convert into sphere first using



      $x=X/a, ;y=Y/b,;z=Z/c$ But I don't know how to evaluate $dS$ in this case since it has three unknowns and something like $|R_u times R_v|$ doesn't works.



      Can you please help me how can solve this problem?






      vectors vector-analysis surfaces surface-integrals multiple-integral




      vectors vector-analysis surfaces surface-integrals multiple-integral






      share|cite|improve this question













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      asked Jan 11 at 17:03









      AbhayAbhay

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          $begingroup$

          Let $f(x, y, z) = a x^2 + b y^2 + c z^2 - 1$. Notice that the integrand is $|nabla f|/2$. Therefore,
          $$int_{partial mathcal E} frac {|nabla f|} 2 ,dS =
          frac 1 2 int_{partial mathcal E} |nabla f|
          ,hat {mathbf n} cdot dmathbf S =
          frac 1 2 int_{partial mathcal E} |nabla f|
          ,frac {nabla f} {|nabla f|} cdot dmathbf S = \
          frac 1 2 int_{partial mathcal E} nabla f cdot dmathbf S =
          frac 1 2 int_{mathcal E} nabla^2 f ,dV = \
          (a + b + c) operatorname{Vol}(mathcal E) =
          frac {4 pi (a + b + c)} {3 sqrt {a b c ,}}.$$






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            $begingroup$

            Let $f(x, y, z) = a x^2 + b y^2 + c z^2 - 1$. Notice that the integrand is $|nabla f|/2$. Therefore,
            $$int_{partial mathcal E} frac {|nabla f|} 2 ,dS =
            frac 1 2 int_{partial mathcal E} |nabla f|
            ,hat {mathbf n} cdot dmathbf S =
            frac 1 2 int_{partial mathcal E} |nabla f|
            ,frac {nabla f} {|nabla f|} cdot dmathbf S = \
            frac 1 2 int_{partial mathcal E} nabla f cdot dmathbf S =
            frac 1 2 int_{mathcal E} nabla^2 f ,dV = \
            (a + b + c) operatorname{Vol}(mathcal E) =
            frac {4 pi (a + b + c)} {3 sqrt {a b c ,}}.$$






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              Let $f(x, y, z) = a x^2 + b y^2 + c z^2 - 1$. Notice that the integrand is $|nabla f|/2$. Therefore,
              $$int_{partial mathcal E} frac {|nabla f|} 2 ,dS =
              frac 1 2 int_{partial mathcal E} |nabla f|
              ,hat {mathbf n} cdot dmathbf S =
              frac 1 2 int_{partial mathcal E} |nabla f|
              ,frac {nabla f} {|nabla f|} cdot dmathbf S = \
              frac 1 2 int_{partial mathcal E} nabla f cdot dmathbf S =
              frac 1 2 int_{mathcal E} nabla^2 f ,dV = \
              (a + b + c) operatorname{Vol}(mathcal E) =
              frac {4 pi (a + b + c)} {3 sqrt {a b c ,}}.$$






              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$

                Let $f(x, y, z) = a x^2 + b y^2 + c z^2 - 1$. Notice that the integrand is $|nabla f|/2$. Therefore,
                $$int_{partial mathcal E} frac {|nabla f|} 2 ,dS =
                frac 1 2 int_{partial mathcal E} |nabla f|
                ,hat {mathbf n} cdot dmathbf S =
                frac 1 2 int_{partial mathcal E} |nabla f|
                ,frac {nabla f} {|nabla f|} cdot dmathbf S = \
                frac 1 2 int_{partial mathcal E} nabla f cdot dmathbf S =
                frac 1 2 int_{mathcal E} nabla^2 f ,dV = \
                (a + b + c) operatorname{Vol}(mathcal E) =
                frac {4 pi (a + b + c)} {3 sqrt {a b c ,}}.$$






                share|cite|improve this answer









                $endgroup$



                Let $f(x, y, z) = a x^2 + b y^2 + c z^2 - 1$. Notice that the integrand is $|nabla f|/2$. Therefore,
                $$int_{partial mathcal E} frac {|nabla f|} 2 ,dS =
                frac 1 2 int_{partial mathcal E} |nabla f|
                ,hat {mathbf n} cdot dmathbf S =
                frac 1 2 int_{partial mathcal E} |nabla f|
                ,frac {nabla f} {|nabla f|} cdot dmathbf S = \
                frac 1 2 int_{partial mathcal E} nabla f cdot dmathbf S =
                frac 1 2 int_{mathcal E} nabla^2 f ,dV = \
                (a + b + c) operatorname{Vol}(mathcal E) =
                frac {4 pi (a + b + c)} {3 sqrt {a b c ,}}.$$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Feb 2 at 15:37









                MaximMaxim

                5,5031219




                5,5031219






























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