Answer to Question #278282 in Calculus for Pial

Question

Given that U is a function of x, y and z and A a vector field, prove that:

∇⋅(UA)=(∇U)⋅A+U(∇⋅A).

Answer 1

$∇⋅(UA)=\frac{\partial}{\partial x}(UA_x)+\frac{\partial}{\partial y}(UA_y)+\frac{\partial}{\partial z}(UA_z)=$

$=\frac{\partial U}{\partial x}(A_x)+\frac{\partial U}{\partial y}(A_y)+\frac{\partial U}{\partial z}(A_z)+\frac{\partial A_x}{\partial x}(U)+\frac{\partial A_y}{\partial y}(U)+\frac{\partial A_z}{\partial z}(U)=$

$=(∇U)⋅A+U(∇⋅A)$