How does the magnitude of the gradient relate to the magnitude of each of it's components and the divergence? The slope of a gradient is a magnitude and this is reasonable if I imagine 2-D.  With 3-D the gradient has 3 slopes to deal with. There is one slope for each on the components of the gradient at that point holding the other two components constant, true ? 
So the scalar of each of the gradients components IS a magnitude for that particular unit direction. Now when you put them together with the i,j, k unit vectors to form the gradient vector then what is the magnitude of that vector? 
If I use the standard magnitude definition I get a square root of the squared terms of each of it's components, but is this correct or am I missing the boat here?
What motivates me to ask is that by definition divergence  adds all three scalars of the gradient's components together in order to input a vector ( i.e. the gradient vector ) to produce a scalar quantity that is supposed to represent net flow in or out, so it cannot be a magnitude of the gradient. 
I need to relate the magnitude of the gradient to the divergence and am lost!
 A: "So the scalar of each of the gradients components IS a magnitude for that particular unit direction. Now when you put them together with the i,j, k unit vectors to form the gradient vector then what is the magnitude of that vector? 
If I use the standard magnitude definition I get a square root of the squared terms of each of it's components, but is this correct or am I missing the boat here?:"
Yes, that is correct.
"What motivates me to ask is that by definition divergence adds all three scalars of the gradient's components together in order to input a vector ( i.e. the gradient vector ) to produce a scalar quantity that is supposed to represent net flow in or out, so it cannot be a magnitude of the gradient."
The divergence is NOT the magnitude of the gradient.  The gradient vector points in the direction of greatest "flow".  The magnitude of the gradient is the flow in the direction of that gradient vector.  The divergence is the total flow in all directions out of some "small bubble" around the point.
