The Geoid and Different Height Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 132

Fig. 4.2 Centrifugal force
!

respect to each coordinate axis are the gravity components along the coordinate
axes. Therefore, we will use potential function in our discussion instead of gravity
vector. Gravitational force, centrifugal force, and gravity force all have their
corresponding potential functions.
The function of gravitational potential is a numeric function with respect to the
variables of coordinate axes x, y, and z. Its partial derivatives with respect to the
three coordinate axes correspond to the components F x , F y , F z of the gravitational
force ! in these three directions respectively, namely:

9
=
;

V

∂

x ¼

F x

∂

V

∂

y ¼

F y

: ð 4 : 6 Þ

∂

V
∂

∂

z ¼

F z

The gravitational potential function at an exterior point of the body can be
derived from a point mass gravitational potential function, which will be discussed
first.
As shown in Fig. 4.1 , m is the mass of the gravitationally attracting body at the
point (0,0,0); m 0 is the attracted mass at the point (x, y, z); the distance r between
them is given by:

p

x 2

r ¼ þ y 2 þ z 2 :

Take a numeric function