Question

In Cartesian coordinates, verify that ∇•(MA) = A •∇M + M∇•A where A = xyz(ax + ay + az) and
M= 3xy + 4zx by carrying out the indicated derivatives.

Answer 1

$$A \cdot∇M=(xyz(ax + ay + az))∇(3xy + 4zx)\\=(xyz(ax + ay + az))(3y+4z,3x,4x)$$

$$M \cdot∇A=(3xy + 4zx)∇(xyz(ax + ay + az))\\=a(3xy + 4zx)\cdot\\(yz(2x+y+z),xz(x+2y+z),yx(x+y+2z))$$

$$∇(MA)=∇(xyz(ax + ay + az)(3xy + 4zx))=a(3xy + 4zx)\cdot\\(yz(2x+y+z),xz(x+2y+z),yx(x+y+2z))+\\(xyz(ax + ay + az))(3y+4z,3x,4x)$$

Thus,

$$∇(MA)=A \cdot∇M+M \cdot∇A$$