I feel like Ive been banging my head against the wall for three days now in which Ive been trying to understand the process of creating a vector field which depicts the gradient of a scalar field.

I understand the math but I just cant get it to "click" in my head, especially since Im having a hard time imagining this thing for a field Z(x,y,z), how the hell do you do that?

Just to see if Im clear on this, the gradient of the field is a vector field which tells me the the rate of change of the scalar field(may I call it the "primitive", like for curves?). This gradient is denoted with the "del" operator.

The construct that I think with is like a three dimesional coordinate system with a sperical form in it with the vectors of the gradient pointing towards the center of the sphere, the sphere being something like a star or something that gets progressivley denser as you move closer to its center, with this desity being Z(x,y,z)

Can anybody give me a hand with this? Some form of help visualizing it or ANYTHING at all, Im studying this using Feynmans lectures.