∇×(f(r)A(r))=( | ∂ ∂y | (fAz)− | ∂ ∂z | (fAy))ex +( | ∂ ∂z | (fAx)− | ∂ ∂x | (fAz))ey +( | ∂ ∂x | (fAy)− | ∂ ∂y | (fAx))ez |
= ( | ∂f ∂y | Az− | ∂f ∂z | Ay+f | ∂Az ∂y | −f | ∂Ay ∂z | )ex |
+( | ∂f ∂z | Ax− | ∂f ∂x | Az+f | ∂Ax ∂z | −f | ∂Az ∂x | )ey |
+( | ∂f ∂x | Ay− | ∂f ∂y | Ax+f | ∂Ay ∂x | −f | ∂Ax ∂y | )ez |
= ( | ∂f ∂y | Az− | ∂f ∂z | Ay)ex +( | ∂f ∂z | Ax− | ∂f ∂x | Az)ey +( | ∂f ∂x | Ay− | ∂f ∂y | Ax)ez |
+f(( | ∂Az ∂y | − | ∂Ay ∂z | )ex +( | ∂Ax ∂z | − | ∂Az ∂x | )ey+( | ∂Ay ∂x | − | ∂Ax ∂y | )ez) |
=(∇f)×A+f(∇×A)