Intuition behind relation between intensity of gravitational field and gravitational potential I found out that the gravitational field intensity, $\vec{I}$ is the negative derivative of the gravitational potential $V$ wrt to distance $r$ from (point) mass. Can someone provide some intuition for this relation? Mathematically, I can see that the expressions of gravitational field and $dV\over dr$ are equal, but I would like some intuition for why this is true, or rather what that differential even means/represents/signifies.
 A: For gravity near the surface of the Earth, $V(r)=gh(r)$, where $h(r)$ is the height of the surface of the earth as a function of position, $r$. So the potential is (proportional to) the height of a hill. The gradient or slope of the hill, $\frac{dh}{dr}$, tells you how fast the hill is changing with $r$, and the gradient of the potential is simply proportional to the gradient of the hill's height. When the slope is large, there is a large gravitational force on you (think of sliding down a steep hill). When the slope is small, there is a small gravitational force (think of resting on a summit).
When thinking of the gravitational potential from a point mass, $V(r)\sim 1/r$, you can use the same analogy to get an intuitive grasp of the potential, even though there is no hill. Far away from the point mass, the potential is very flat, and the force is small. As you approach the point mass, the potential gets steeper and steeper (which we can measure using the gradient, or slope, of the potential), and the force toward the point mass gets larger and larger as well.
A: Another intuition-based answer: you can think of gravitational potential as the potential energy of a test mass at a certain point of space. If a test mass is falling towards a massive body (in the direction of the gravitational field), it will accelerate and "use up" some of its gravitational potential energy by turning it into kinetic energy. If it is somehow lifted away from the massive body (opposite the direction of the gravitational field) it will be gaining potential energy. Additionally, the stronger the field is, the more the object will accelerate and the faster the potential energy will be used up.
This is hardly a rigorous calculation, but it should give you an idea of why it's reasonable that the gravitational field strength at a point should be related to the rate of change of the potential. It also helps explain why there is a negative sign in the derivative, because moving with the field causes potential to go down and moving against it causes potential to go up.
