Answer to Question #170694 in Physics for chetana

Question

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

Answer 1

$$\nabla \cdot(\bold{A}\times \bold{B})=\bold{B}\cdot(\nabla \times\bold{A})-\bold{A}\cdot(\nabla\times \bold{B})$$

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

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

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

Thus,

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