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(Fig. 2.2). If we take the vector d x along the plumb line, in the direction of
increasing height H , then its length will be
d x = dH
(2-19)
and its direction is opposite to the gravity vector g , which points downward,
so that the angle between d x and g is 180 . Using the definition of the scalar
product (i.e., for two vectors a and b it is defined as a · b = a b cos ω ,
where ω is the angle between the two vectors), we get
d x = gdH cos 180 =
g
·
gdH
(2-20)
accordingly, so that Eq. (2-15) becomes
dW =
gdH.
(2-21)
This equation relates the height H to the potential W and will be basic for
the theory of height determination (Chap. 4). It shows clearly the insepara-
ble interrelation that characterizes geodesy - the interrelation between the
geometrical concepts ( H ) and the dynamic concepts ( W ).
Another form of Eq. (2-21) is
∂W
∂H .
g =
(2-22)
It shows that gravity is the negative vertical gradient of the potential W ,or
the negative vertical component of the gradient vector grad W .
Since geodetic measurements (theodolite measurements, leveling, but
also satellite techniques etc.) are almost exclusively referred to the system
of level surfaces and plumb lines, the geoid plays an essential part. Thus,
we see why the aim of physical geodesy has been formulated as the de-
termination of the level surfaces of the earth's gravity field . In a still more
abstract but equivalent formulation, we may also say that physical geodesy
aims at the determination of the potential function W ( x, y, z ). At a first
glance, the reader is probably perplexed about this definition, which is due
to Bruns (1878), but its meaning is easily understood: If the potential W is
given as a function of the coordinates x, y, z , then we know all level surfaces
including the geoid; they are given by the equation
W ( x, y, z ) = constant.
(2-23)
2.3
Curvature of level surfaces and plumb lines
The formula for the curvature of a curve y = f ( x )is
κ = 1
y
(1 + y 2 ) 3 / 2 ,
=
(2-24)
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