In a longer derivation I ran into the following quantity: ∇ [ ∇ ⋅ ( r 0 e i k ⋅ r ) ] ( i.e., the gradient of the divergence ) where k is a vector of constants and r is a position vector.Can someone help explaining how to calculate this? I am hoping it gives: ∇ [ ∇ ⋅ ( r 0 e i k ⋅ r ) ] = − k ( k ⋅ r 0 )

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In a longer derivation I ran into the following quantity:  ∇      [    ∇    ⋅          (                        r                          0                                            e                                      i                                k                    ⋅                      r                                    )        ]  ( i.e., the gradient of the divergence ) where k is a vector of constants and r is a position vector.Can someone help explaining how to calculate this? I am hoping it gives:  ∇      [    ∇    ⋅          (                        r                          0                                            e                                      i                                k                    ⋅                      r                                    )        ]    =  −      k        (          k        ⋅                  r                    0              )

Emmanuel Russo · Accepted Answer

Summing over repeated indices, the divergence is       r          0      i            ∂    i        e          i              k        j                    r        j              =      r          0      i        i      k    i        e          i              k        j                    r        j              =  i  (  k  ⋅      r    0    )      e          i              k        j                    r        j            . Applying       ∂    l   pulls down another   i      k    l   factor, so the gradient is   −  k  (  k  ⋅      r    0    )      e          i      k      ⋅      r      . Your desired result drops the exponential, which I suspect is a typo.

ter3k4w8x · Accepted Answer

∇                          [                                            ∇                ⋅                                  (                                                                                                      r                                                                    0                                                                                                                                e                                                                                              i                                                                          k                                                ⋅                                                  r                                                                                                                                            )                                                          ]                                      =        ∇                  [                                                                                                    (                                                              ∇                      ⋅                                                                        r                                                                          0                                                                                                          )                                    ⏞                                                                                                  =                                         0                                                                                                                               e                                                              i                                                  k                                ⋅                                  r                                                                  +                                          r                                            0                                      ⋅            ∇                          (                                                                                                  e                                                                              i                                                              k                                        ⋅                                          r                                                                                                                )                                          ]                =                  i                ∇                  (                                                              r                                            0                                      ⋅                          k                                                                              e                                                              i                                                  k                                ⋅                                  r                                                                                    )                                    =                                       i                                              r                                0                          ⋅                  k                        ∇                                      e                                              i                                      k                        ⋅                          r                                              =                  i                                              r                                0                          ⋅                  k                                  (                                                                          e                                                              i                                                  k                                ⋅                                  r                                                                                            i                                      k                                          )                =                                                                    −                                  k                                                  (                                                                              k                                        ⋅                                                                  r                                                                    0                                                                                                )                                                                                      e                                                                              i                                                              k                                        ⋅                                          r

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