SetDirectory["c:\\myDirectory"]

<< continuous3DHomotopyOperator.m

example1 = Derivative[{1, 2, 0}, 0][u[1]][{x, y, z}, t]+Derivative[{2, 0, 0}, 0][u[1]][{x, y, z}, t]

vector1= homotopyOperatorMD[example1,  u,  
        1,  {x, y, z}]

<< Calculus`VectorAnalysis`

SetCoordinates[Cartesian[x, y, z]]

divvector1= Div[
        vector1]

testvector1  =  Factor[example1  -  divvector1]

example2 = 2 Derivative[{0, 1, 0}, 0][u[1]][{x, y, z}, t] Derivative[{0, 1, 0}, 0][u[2]][{x, y, z}, t]+2 u[1][{x, y, z}, t] Derivative[{0, 2, 0}, 0][u[2]][{x, y, z}, t]+Derivative[{1, 0, 2}, 0][u[2]][{x, y, z}, t]+2 Derivative[{1, 0, 0}, 0][u[1]][{x, y, z}, t] Derivative[{2, 0, 0}, 0][u[1]][{x, y, z}, t]

vector2  =  
      homotopyOperatorMD[example2,  u,  2,  {x, y, z}]

divvector2  =  Div[vector2]

testvector2  =  Factor[example2  -  divvector2]

example3 = Derivative[{0, 1, 1}, 0][u[1]][{x, y, z}, t]-Sin[u[1][{x, y, z}, t]] Derivative[{1, 0, 0}, 0][u[1]][{x, y, z}, t]+Cos[u[2][{x, y, z}, t]] Derivative[{0, 1, 0}, 0][u[2]][{x, y, z}, t] Derivative[{1, 0, 0}, 0][u[1]][{x, y, z}, t]+Sin[u[2][{x, y, z}, t]] Derivative[{1, 1, 0}, 0][u[1]][{x, y, z}, t]

vector3 = homotopyOperatorMD[example3,  u,  
        2,  {x, y, z}]

divvector3  =  Div[vector3]

testvector3  =  
      Factor[example3  -  divvector3]

example4 = Derivative[{0, 1}, 0][u[1]][{x, y}, t] Derivative[{1, 0}, 0][u[1]][{x, y}, t]+u[1][{x, y}, t] Derivative[{1, 1}, 0][u[1]][{x, y}, t]+Derivative[{2, 1}, 0][u[1]][{x, y}, t]

vector4= homotopyOperatorMD[example4,  u,  
        1,  {x, y}]

divvector4  =  Div[Flatten[{vector4, 0}]]

testvector4  =  
      Factor[example4  -  divvector4]

example5 = Derivative[{0, 1}, 0][u[1]][{x, y}, t] Derivative[{1, 0}, 0][u[2]][{x, y}, t]+2 Derivative[{0, 1}, 0][u[1]][{x, y}, t] Derivative[{1, 1}, 0][u[1]][{x, y}, t]+u[1][{x, y}, t] Derivative[{1, 1}, 0][u[2]][{x, y}, t]

vector5= homotopyOperatorMD[example5,  u,  
        2,  {x, y}]

divvector5  =  
      Div[Flatten[{vector5, 0}]]

testvector5   =  
      Factor[example5 -  divvector5]