Find ∇ ⋅ ( r r 2 ) where r ( x , y , z ) = x i + y j + z k, r = x 2 + y 2 + z 2 In this case, why can we not bring the scalar function r − 2 outside the dot product?When I leave the function in place, I calculate a result of 1 r 2 . When I pull it out, I find 3 r 2

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Find   ∇  ⋅      (          r              r        2              )   where   r  (  x  ,  y  ,  z  )  =  x  i  +  y  j  +  z  k,   r  =            x      2        +          y      2        +          z      2      In this case, why can we not bring the scalar function       r          −      2       outside the dot product?When I leave the function in place, I calculate a result of       1          r              2            . When I pull it out, I find       3          r              2

cokeman206 · Accepted Answer

It&#039;s not just a dot product, it&#039;s a divergence operator, and the variables x,y,z appear in       r          −      2      , it&#039;s not a constant.However, you may use the product rule:  ∇  ⋅  (  ψ  F  )  =  ψ  ∇  ⋅  F  +  ∇  ψ  ⋅  FAnd with   ψ  =      r          −      2      , it&#039;s not difficult to prove that  ∇  ψ  =  −      2          r      4        r

faois3nh · Accepted Answer

∇  ⋅      (                            r          →                            r        2              )    =  ∇  ⋅      (          x                        x          2                +                  y          2                +                  z          2                                    i        →              +          y                        x          2                +                  y          2                +                  z          2                                    j        →              +          z                        x          2                +                  y          2                +                  z          2                                    k        →              )      =                    y        2            +              z        2            −              x        2                    (              x        2            +              y        2            +              z        2                    )        2              +                    x        2            +              z        2            −              y        2                    (              x        2            +              y        2            +              z        2                    )        2              +                    x        2            +              y        2            −              z        2                    (              x        2            +              y        2            +              z        2                    )        2              =                    x        2            +              y        2            +              z        2                    (              x        2            +              y        2            +              z        2                    )        2                =      1                  x        2            +              y        2            +              z        2              =      1          r      2        .

Find grad *((bbr)/(r^2)) where bbr(x,y,z)=x bbi+y bbj+z bbk, r=sqrt(x^2+y^2+z^2) In this case, why can we not bring the scalar function r^(-2) outside the dot product?

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