Answer to Question #170694 in Physics for chetana

Question #170694

Check product rule (iv) (by calculating each term separately) for the functions A = x X+2y Y+3d; B = 3yx - 2xY.


1
Expert's answer
2021-03-10T17:14:54-0500
(A×B)=B(×A)A(×B)\nabla \cdot(\bold{A}\times \bold{B})=\bold{B}\cdot(\nabla \times\bold{A})-\bold{A}\cdot(\nabla\times \bold{B})

(A×B)=(x,2y,3z)×(3y,2x,0)=(6xz,9yz,2x26y2)(6xz,9yz,2x26y2)=6z+9z=15z(\bold{A}\times \bold{B})=(x,2y,3z)\times(3y,-2x,0)\\=(6xz,9yz,-2x^2-6y^2)\\ \nabla \cdot(6xz,9yz,-2x^2-6y^2)=6z+9z=15z

B(×A)=(3y,2x,0)(0,0,0)=0\bold{B}\cdot(\nabla \times\bold{A})=(3y,-2x,0)\cdot (0,0,0)=0

A(×B)=(x,2y,3z)(0,0,5)=15z\bold{A}\cdot(\nabla\times \bold{B})=(x,2y,3z)\cdot(0,0,-5)=-15z

Thus,


15z=0(15z)15z=0-(-15z)


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