Gravity field outside the earth - Physical Geodesy - page 236

The components of γ along the coordinate lines are, by (2-131) and (2-132),

GM
u 2 + E 2 +

6 − ω 2 u cos 2 β ,

ω 2 a 2 E
u 2 + E 2 q 0 2 sin 2 β −

q

1
w

∂U
∂u

1
w

1

γ u = = −

−

q 0 + ω 2 u 2 + E 2 sin β cos β,

ω 2 a 2
√ u 2 + E 2

1
w √ u 2 + E 2

∂U
∂β

1
w

q

γ β = = −

1
√ u 2 + E 2 cos β

∂U
∂λ

γ λ = =0 .

(6-12)

To get the components of γ in the xyz -system, we compute

∂U
∂u

= ∂U
∂x

∂x

∂u + ∂U

∂y

∂u + ∂U

∂z
∂u ,

etc. (6-13)

∂y ∂z

The partial derivatives of x, y, z with respect to u, β, λ are obtained by
differentiating equations (6-6); we find

∂U
∂u =

u
√ u 2 + E 2 cos β cos λ ∂U

u
√ u 2 + E 2 cos β sin λ ∂U

∂y

+sin β ∂U
∂z ∂x +

,

u 2 + E 2 sin β sin λ ∂U
∂y

− √ u 2 + E 2 sin β cos λ ∂U

∂U
∂β

+ u cos β ∂U
∂z

=

∂x −

,

∂x + u 2 + E 2 cos β cos λ ∂U ∂λ = − √ u 2 + E 2 cos β sin λ ∂U

∂U

∂y .

(6-14)

Introducing the components

γ x = ∂U

1
w

∂U
∂u ,

∂x ,

··· ; γ u = ··· , (6-15)

we obtain

u
w √ u 2 + E 2

u
w √ u 2 + E 2

1
w

γ u =

cos β cos λγ x + cos β sin λγ y + sin βγ z ,

1
w

1
w

u
w √ u 2 + E 2

γ β = − sin β cos λγ x − sin β sin λγ y + cos βγ z ,

γ λ = − sin λγ x +cos λγ y .

(6-16)
These are the formulas of an orthogonal rectangular coordinate transforma-
tion. The inverse transformation is obtained by interchanging the rows and