A change in gravitational potential (ΔV) is equal to the negative of work by gravity on a unit mass,
Δ
V
=

E
Δ
r
Δ
V
=

E
Δ
r
For infinitesimal change, we can write the equation,
⇒
ⅆ
V
=

E
ⅆ
r
⇒
ⅆ
V
=

E
ⅆ
r
⇒
E
=
−
ⅆ
V
ⅆ
r
⇒
E
=
−
ⅆ
V
ⅆ
r
Thus, if we know potential function, we can find corresponding field strength. In words, gravitational field strength is equal to the negative potential gradient of the gravitational field. We should be slightly careful here. This is a relationship between a vector and scalar quantity. We have taken the advantage by considering field in one direction only and expressed the relation in scalar form, where sign indicates the direction with respect to assumed positive reference direction. In three dimensional region, the relation is written in terms of a special vector operator called “grad”.
Further, we can see here that gravitational field – a vector – is related to gravitational potential (scalar) and position in scalar form. We need to resolve this so that evaluation of the differentiation on the right yields the desired vector force. As a matter of fact, we handle this situation in a very unique way. Here, the differentiation in itself yields a vector. In three dimensions, we define an operator called “grad” as :
grad
=
∂
∂
x
i
+
∂
∂
y
j
+
∂
∂
z
k
grad
=
∂
∂
x
i
+
∂
∂
y
j
+
∂
∂
z
k
where "
∂
∂
x
∂
∂
x
” is partial differentiation operator with respect to "x". This is same like normal differentiation except that it considers other dimensions (y,z) constant. In terms of “grad”,
E
=

grad
V
E
=

grad
V