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!