I would like to ask how to prove these two formulas.for T is a scalar and r is radial component of a vector in spherical coordinate.(1) r ⋅ [ ( r ⋅ ∇ ) ∇ T + 2 ∇ T ] = r ⋅ ∇ T + r ⋅ ∇ ( r ⋅ ∇ T )(2) ∇ 2 ( T r ) = r ∇ 2 T + T ∇ 2 r + 2 ∇ T ⋅ ∇ r

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I would like to ask how to prove these two formulas.for T is a scalar and r is radial component of a vector in spherical coordinate.(1)              r        ⋅    [    (    r    ⋅    ∇    )    ∇    T  +  2  ∇  T  ]  =            r        ⋅    ∇    T  +      r    ⋅    ∇    (    r    ⋅    ∇    T  )(2)              ∇      2        (    T          r        )    =            r            ∇    2    T  +  T      ∇    2              r        +  2  ∇  T  ⋅  ∇            r

wietselau · Accepted Answer

For (1), note that                              r                ⋅        [        (                  r                ⋅        ∇        )        ∇        T        +        2        ∇        T        ]                            =                  x          i                [        (                  x          j                          ∂          j                )                  ∂          i                T        +        2                  ∂          i                T        ]                                          =                  x          i                (                  x          j                          ∂          j                )                  ∂          i                T        +        2                  x          i                          ∂          i                T                                          =                  x          j                          ∂          j                          x          i                          ∂          i                T        −                  x          j                (                  ∂          j                          x          i                )                  ∂          i                T        +        2                  x          i                          ∂          i                T                                          =                  x          j                          ∂          j                          x          i                          ∂          i                T        −                  x          j                          δ                      i            j                                    ∂          i                T        +        2                  x          i                          ∂          i                T                                          =                  x          j                          ∂          j                          x          i                          ∂          i                T        −                  x          i                          ∂          i                T        +        2                  x          i                          ∂          i                T                                          =                  x          j                          ∂          j                          x          i                          ∂          i                T        +                  x          i                          ∂          i                T                                          =                  r                ⋅        ∇        (                  r                ⋅        ∇        T        )        +                  r                ⋅        ∇        T            (2) is similar.

I would like to ask how to prove these two formulas. for T is a scalar and r is radial component of a vector in spherical coordinate. (1) r [(r grad)grad T+2grad T]=r * grad T+r * grad(r * grad T) (2) grad^2(Tr)=r grad^2 T+T grad^2 r+2 grad T * grad r

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