Let $\mathbf { F } ( x , y , z ) = \left\langle \frac { z } { x - y } , \frac { z } { y - x } , \ln ( x - y ) \right\rangle$ , find $\nabla \cdot ( \nabla \times \mathbf { F } )$ at the point $( 1,3,1 )$ .
A) $2$
B) $4$

C) $6$
D) $1$
E) $- 2$
F)$- 4$
G)$- 6$
H)$0$

Question

Let $\mathbf { F } ( x , y , z ) = \left\langle \frac { z } { x - y } , \frac { z } { y - x } , \ln ( x - y ) \right\rangle$ , find $\nabla \cdot ( \nabla \times \mathbf { F } )$ at the point $( 1,3,1 )$ .
A) $2$ 
B) $4$

C) $6$ 
D) $1$
E) $- 2$ 
F)$- 4$ 
G)$- 6$ 
H)$0$

Quizplus · Accepted Answer

The divergence of the curl of any vector field is always zero due to the properties of vector calculus operations.

Let F(x,y,z)=⟨zx−y,zy−x,ln⁡(x−y)⟩\mathbf { F } ( x , y , z ) = \left\langle \frac { z } { x - y } , \frac { z } { y - x } , \ln ( x - y ) \right\rangleF(x,y,z)=⟨x−yz​,y−xz​,ln(x−y)⟩

Let $\mathbf { F } ( x , y , z ) = \left\langle \frac { z } { x - y } , \frac { z } { y - x } , \ln ( x - y ) \right\rangle$