# Vector calculus identity proof.

1. Sep 19, 2009

### rock.freak667

1. The problem statement, all variables and given/known data

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:

2. Relevant 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

3. 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)

2. Sep 19, 2009

### gabbagabbahey

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

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

3. Sep 19, 2009

### rock.freak667

I got it now!