Answer to Question #138698 in Electricity and Magnetism for Omar

Question #138698
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.
1
Expert's answer
2020-10-19T13:23:01-0400
AM=(xyz(ax+ay+az))(3xy+4zx)=(xyz(ax+ay+az))(3y+4z,3x,4x)A \cdot∇M=(xyz(ax + ay + az))∇(3xy + 4zx)\\=(xyz(ax + ay + az))(3y+4z,3x,4x)

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


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