Answer to Question #138698 in Electricity and Magnetism for Omar
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
2020-10-19T13:23:01-0400
A⋅∇M=(xyz(ax+ay+az))∇(3xy+4zx)=(xyz(ax+ay+az))(3y+4z,3x,4x)
M⋅∇A=(3xy+4zx)∇(xyz(ax+ay+az))=a(3xy+4zx)⋅(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)
Thus,
∇(MA)=A⋅∇M+M⋅∇A
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