Given a vector u = ( x , y , z ) ∈ R 3 and a 3 × 3 real matrix M, I woud like to know if there exists some formulas to express in other manner the two quantities: ∇ ∇ ⋅ ( M u ) and ∇ × ( M u ) in terms of M and u.Also, I want to know for which type of matrix M we have ∇ ∇ ⋅ ( M u ) = M ∇ ∇ ⋅ uand ∇ × ( M u ) = M ∇ × u .

Question

Given a vector   u  =  (  x  ,  y  ,  z  )  ∈            R        3   and a   3  ×  3 real matrix M, I woud like to know if there exists some formulas to express in other manner the two quantities:   ∇  ∇  ⋅  (  M  u  ) and   ∇  ×  (  M  u  ) in terms of M and u.Also, I want to know for which type of matrix M we have  ∇  ∇  ⋅  (  M  u  )  =  M  ∇  ∇  ⋅  uand  ∇  ×  (  M  u  )  =  M  ∇  ×  u  .

Andre Beck · Accepted Answer

With summation over repeated indices, let&#039;s write vectors as   u  =      u    k        e    k  , matrices as   M  =      M          i      j            e          i      j       etc. Then  ∇  ⋅  M  u  =      ∂    i    (  M  u      )    i    =      M          i      j            ∂    i        u    j  (I&#039;m assuming you want constant M), so  ∇  ∇  ⋅  M  u  =      ∂    k    (      M          i      j            ∂    i        u    j    )      e    k    =      M          i      j            ∂    i        ∂    k        u    j        e    k    .This can&#039;t be written as Mv for a vector v, but it can be written as M:v (: denotes summation over each index of M) for a rank-3 tensor  v  :=      ∂    i        ∂    k        u    j        e          i      j        ⊗      e    k    ,where   ⊗ denotes a tensor product. Similarly,  ∇  ×  (  M  u  )  =      ϵ          i      j      k            M          j      l            ∂    i        u    l        e    k    ,i.e. M:v with   v  :=      ϵ          i      j      k            ∂    i        u    l        e          j      l        ⊗      e    k

Given a vector u=(x,y,z) in RR^3 and a 3 xx 3 real matrix M, I woud like to know if there exists some formulas to express in other manner the two quantities: grad grad *(Mu) and grad xx (Mu) in terms of M and u.

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