Question on vector identity proof in order to derive maxwell stress tensor












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In the process of deriving the Maxwell stress tensor we have proven the following vector identity
begin{align}
(vec{B} cdot vec{nabla})vec{B} + vec{B} times (vec{nabla} times vec{B}) &= (B_j partial_jB_i + epsilon_{ijk}B_jepsilon_{klm}partial_lB_m)vec{e}_i \ &= (B_jpartial_jB_i + (delta_{il}delta_{jm} - delta_{im}delta_{jl})B_jpartial_lB_m)vec{e}_i \ &= (B_jpartial_jB_i + B_jpartial_iB_j - B_jpartial_jB_i)vec{e}_i \ &= B_jpartial_iB_jvec{e}_i \ &= frac{1}{2}vec{nabla} (vec{B}^2)end{align}



I am wondering about what has happened in order to get the final "=".
Can anyone explain it exactly?










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    1












    $begingroup$


    In the process of deriving the Maxwell stress tensor we have proven the following vector identity
    begin{align}
    (vec{B} cdot vec{nabla})vec{B} + vec{B} times (vec{nabla} times vec{B}) &= (B_j partial_jB_i + epsilon_{ijk}B_jepsilon_{klm}partial_lB_m)vec{e}_i \ &= (B_jpartial_jB_i + (delta_{il}delta_{jm} - delta_{im}delta_{jl})B_jpartial_lB_m)vec{e}_i \ &= (B_jpartial_jB_i + B_jpartial_iB_j - B_jpartial_jB_i)vec{e}_i \ &= B_jpartial_iB_jvec{e}_i \ &= frac{1}{2}vec{nabla} (vec{B}^2)end{align}



    I am wondering about what has happened in order to get the final "=".
    Can anyone explain it exactly?










    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      In the process of deriving the Maxwell stress tensor we have proven the following vector identity
      begin{align}
      (vec{B} cdot vec{nabla})vec{B} + vec{B} times (vec{nabla} times vec{B}) &= (B_j partial_jB_i + epsilon_{ijk}B_jepsilon_{klm}partial_lB_m)vec{e}_i \ &= (B_jpartial_jB_i + (delta_{il}delta_{jm} - delta_{im}delta_{jl})B_jpartial_lB_m)vec{e}_i \ &= (B_jpartial_jB_i + B_jpartial_iB_j - B_jpartial_jB_i)vec{e}_i \ &= B_jpartial_iB_jvec{e}_i \ &= frac{1}{2}vec{nabla} (vec{B}^2)end{align}



      I am wondering about what has happened in order to get the final "=".
      Can anyone explain it exactly?










      share|cite|improve this question











      $endgroup$




      In the process of deriving the Maxwell stress tensor we have proven the following vector identity
      begin{align}
      (vec{B} cdot vec{nabla})vec{B} + vec{B} times (vec{nabla} times vec{B}) &= (B_j partial_jB_i + epsilon_{ijk}B_jepsilon_{klm}partial_lB_m)vec{e}_i \ &= (B_jpartial_jB_i + (delta_{il}delta_{jm} - delta_{im}delta_{jl})B_jpartial_lB_m)vec{e}_i \ &= (B_jpartial_jB_i + B_jpartial_iB_j - B_jpartial_jB_i)vec{e}_i \ &= B_jpartial_iB_jvec{e}_i \ &= frac{1}{2}vec{nabla} (vec{B}^2)end{align}



      I am wondering about what has happened in order to get the final "=".
      Can anyone explain it exactly?







      vectors physics tensors






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      share|cite|improve this question













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      edited Dec 23 '18 at 17:49







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      asked Dec 23 '18 at 17:44









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          Well, $$B_jpartial_iB_jvec{e}_i=B_jvec{nabla}B_j=frac{1}{2}vec{nabla}left(B_jB_jright)=frac{1}{2}vec{nabla}left(vec{B}^2right).$$






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          • $begingroup$
            Well, to be honest it remains unclear. I guess I don't see something very simple but what has been used to get the second "="?
            $endgroup$
            – offline
            Dec 23 '18 at 17:58






          • 1




            $begingroup$
            @offline The product rule, or chain rule, whichever you prefer. Perhaps for clarity I should have worked with $partial_i$s and $vec{e}_i$s for longer.
            $endgroup$
            – J.G.
            Dec 23 '18 at 18:04










          • $begingroup$
            Ohhh, thank you! Problem solved.
            $endgroup$
            – offline
            Dec 23 '18 at 18:07











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          $begingroup$

          Well, $$B_jpartial_iB_jvec{e}_i=B_jvec{nabla}B_j=frac{1}{2}vec{nabla}left(B_jB_jright)=frac{1}{2}vec{nabla}left(vec{B}^2right).$$






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Well, to be honest it remains unclear. I guess I don't see something very simple but what has been used to get the second "="?
            $endgroup$
            – offline
            Dec 23 '18 at 17:58






          • 1




            $begingroup$
            @offline The product rule, or chain rule, whichever you prefer. Perhaps for clarity I should have worked with $partial_i$s and $vec{e}_i$s for longer.
            $endgroup$
            – J.G.
            Dec 23 '18 at 18:04










          • $begingroup$
            Ohhh, thank you! Problem solved.
            $endgroup$
            – offline
            Dec 23 '18 at 18:07
















          1












          $begingroup$

          Well, $$B_jpartial_iB_jvec{e}_i=B_jvec{nabla}B_j=frac{1}{2}vec{nabla}left(B_jB_jright)=frac{1}{2}vec{nabla}left(vec{B}^2right).$$






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Well, to be honest it remains unclear. I guess I don't see something very simple but what has been used to get the second "="?
            $endgroup$
            – offline
            Dec 23 '18 at 17:58






          • 1




            $begingroup$
            @offline The product rule, or chain rule, whichever you prefer. Perhaps for clarity I should have worked with $partial_i$s and $vec{e}_i$s for longer.
            $endgroup$
            – J.G.
            Dec 23 '18 at 18:04










          • $begingroup$
            Ohhh, thank you! Problem solved.
            $endgroup$
            – offline
            Dec 23 '18 at 18:07














          1












          1








          1





          $begingroup$

          Well, $$B_jpartial_iB_jvec{e}_i=B_jvec{nabla}B_j=frac{1}{2}vec{nabla}left(B_jB_jright)=frac{1}{2}vec{nabla}left(vec{B}^2right).$$






          share|cite|improve this answer











          $endgroup$



          Well, $$B_jpartial_iB_jvec{e}_i=B_jvec{nabla}B_j=frac{1}{2}vec{nabla}left(B_jB_jright)=frac{1}{2}vec{nabla}left(vec{B}^2right).$$







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 23 '18 at 18:04

























          answered Dec 23 '18 at 17:54









          J.G.J.G.

          29.4k22846




          29.4k22846












          • $begingroup$
            Well, to be honest it remains unclear. I guess I don't see something very simple but what has been used to get the second "="?
            $endgroup$
            – offline
            Dec 23 '18 at 17:58






          • 1




            $begingroup$
            @offline The product rule, or chain rule, whichever you prefer. Perhaps for clarity I should have worked with $partial_i$s and $vec{e}_i$s for longer.
            $endgroup$
            – J.G.
            Dec 23 '18 at 18:04










          • $begingroup$
            Ohhh, thank you! Problem solved.
            $endgroup$
            – offline
            Dec 23 '18 at 18:07


















          • $begingroup$
            Well, to be honest it remains unclear. I guess I don't see something very simple but what has been used to get the second "="?
            $endgroup$
            – offline
            Dec 23 '18 at 17:58






          • 1




            $begingroup$
            @offline The product rule, or chain rule, whichever you prefer. Perhaps for clarity I should have worked with $partial_i$s and $vec{e}_i$s for longer.
            $endgroup$
            – J.G.
            Dec 23 '18 at 18:04










          • $begingroup$
            Ohhh, thank you! Problem solved.
            $endgroup$
            – offline
            Dec 23 '18 at 18:07
















          $begingroup$
          Well, to be honest it remains unclear. I guess I don't see something very simple but what has been used to get the second "="?
          $endgroup$
          – offline
          Dec 23 '18 at 17:58




          $begingroup$
          Well, to be honest it remains unclear. I guess I don't see something very simple but what has been used to get the second "="?
          $endgroup$
          – offline
          Dec 23 '18 at 17:58




          1




          1




          $begingroup$
          @offline The product rule, or chain rule, whichever you prefer. Perhaps for clarity I should have worked with $partial_i$s and $vec{e}_i$s for longer.
          $endgroup$
          – J.G.
          Dec 23 '18 at 18:04




          $begingroup$
          @offline The product rule, or chain rule, whichever you prefer. Perhaps for clarity I should have worked with $partial_i$s and $vec{e}_i$s for longer.
          $endgroup$
          – J.G.
          Dec 23 '18 at 18:04












          $begingroup$
          Ohhh, thank you! Problem solved.
          $endgroup$
          – offline
          Dec 23 '18 at 18:07




          $begingroup$
          Ohhh, thank you! Problem solved.
          $endgroup$
          – offline
          Dec 23 '18 at 18:07


















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