I can't figure out how to prove this using Index/Cartesian Tensor notation ∇ ⋅ ( a b ) = a ⋅ ∇ b + b ( ∇ ⋅ a )Please note.....on the left side, that is NOT a dot product between vectors a and b. I'm assuming the problem wants them multiplied together to make a vector " a 1 b 1 , a 2 b 2 , a 3 b 3 ".

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I can&#039;t figure out how to prove this using Index/Cartesian Tensor notation   ∇  ⋅  (  a  b  )  =  a  ⋅  ∇  b  +  b  (  ∇  ⋅  a  )Please note.....on the left side, that is NOT a dot product between vectors a and b. I&#039;m assuming the problem wants them multiplied together to make a vector &quot;      a    1        b    1    ,      a    2        b    2    ,      a    3        b    3  &quot;.

Kendrick Mendez · Accepted Answer

You want to prove       ∂    i    (      a    i        b    j    )  =      a    i        ∂    i        b    j    +      b    j        ∂    i        a    i   (summation over the repeated index i is implicit). So yes, you just need the product rule.

alexmjn · Accepted Answer

I&#039;m guessing that a is a vector field and b is a scalar field.   ∇  ⋅  (  a  b  ) stands for the divergence of the vector field ab.  a  =  (      a    1    ,  ⋯  ,      a    n    )  ,    a  b  =  (  b      a    1    ,  ⋯  ,  b      a    n    )                    ∇        ⋅        (        a        b        )        =                              ∂                      ∂                          x              1                                      (        b                  a          1                )        +        ⋯        +                  ∂                      ∂                          x              n                                      (        b                  a          n                )        =                              ∂            b                                ∂                          x              1                                                a          1                +        b                              ∂                          a              1                                            ∂                          x              1                                      +        ⋯        +                              ∂            b                                ∂                          x              n                                                a          n                +        b                              ∂                          a              n                                            ∂                          x              n                                                          =                            (                              ∂            b                                ∂                          x              1                                                a          1                +        ⋯        +                              ∂            b                                ∂                          x              n                                                a          n                )        +        (        b                              ∂                          a              1                                            ∂                          x              1                                      +        ⋯        b                              ∂                          a              n                                            ∂                          x              n                                      )                            =                    a        ⋅        ∇        b        +        b        (        ∇        ⋅        a        )

How to prove this using Index/Cartesian Tensor notation grad (bb(ab))=a grad bb(b)+bb(b)(grad * bb(a))

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