Vector calculus identity proof.

[rock.freak667]

Homework Statement

Let f(x,y,z) be a function of three variables and G(x,y,z) be a vector field defined in 3D space. Prove the identity:

div(fG)= f*div(G)+G*grad(f)

Homework Equations

For F=Pi +Qj+Rk

div(F)=dF/dx + dQ/dy + dR/dz

grad(F)=dF/dx i + dQ/dy j + dR/dz k

The Attempt at a Solution

My problem starts with how do I find fG? Because I am thinking that f is a vector with i,j,k components and so is G. So fG should be the dot product of f and G, which gives a scalar, and one can't get the divergence of a scalar (Since this is an identity, I know somewhere I am missing some elementary fact)

 

[gabbagabbahey]

No, $f(x,y,z)$ is a scalar function, $\textbf{G}(x,y,z)$ is a vector field.

So, $$f\textbf{G}=f(x,y,z)G_x(x,y,z)\textbf{i}+f(x,y,z)G_y(x,y,z)\textbf{j}+f(x,y,z)G_z(x,y,z)\textbf{k}$$

 

[rock.freak667]

No, $f(x,y,z)$ is a scalar function, $\textbf{G}(x,y,z)$ is a vector field.

So, $$f\textbf{G}=f(x,y,z)G_x(x,y,z)\textbf{i}+f(x,y,z)G_y(x,y,z)\textbf{j}+f(x,y,z)G_z(x,y,z)\textbf{k}$$

I got it now!