Relationship between F and U; between g and φ

To understand how g is related to φ:

Similarly, compare g = - G M r 2 and φ = - GM r in the above table.
If we differentiate φ = - GM r with respect to r, we will get ⅆ ϕ ⅆ r = - G M - r 2 , which has the same expression as g.
Hence, mathematically ⅆ ϕ ⅆ r = G M r 2 = - g
To understand the meaning of g = - ⅆ ϕ ⅆ r observe the two graphs carefully, on the right side where r is positive, the gradient of φ vs r graph is positive but the value of g will be negative. And on the left side where r is negative, the gradient of φ vs r graph is negative but the value of g is positive. Thus, g = - ⅆ ϕ ⅆ r

Similarly, it can be concluded that by multiplying both sides by test mass m.
m g = - ⅆ m ϕ ⅆ r
thus F = - ⅆ U ⅆ r

Summary

symbol	g = - G M r 2	φ = - GM r
name	Field strength	Potential
units	N kg^-1 or m s^-2	J kg^-1
meaning	Force per unit mass	Potential energy per unit mass
quantity	vector	scalar
equation	\|g\| = G M r 2 towards the centre of the source mass	ϕ = - G M r
relationship to mass	Force, F = G M 1 M 2 r 2 = mg	Potential energy, U = -m G M r = mφ
graph
computer model if M = 500.	-6.67500/(abs(r)r)	-6.67*500/abs(r)
relationship between g and φ	g = - ⅆ ϕ ⅆ t g = - ⅆ ϕ ⅆ t
relationship between F and U	F = - ⅆ U ⅆ t F = - ⅆ U ⅆ t