Geology Reference
In-Depth Information
2
V
is not defined directly
In orthogonal curvilinear co-ordinates, the operation
∇
but may be expressed as the di
ff
erence
∇
(
∇·
V
)
−∇×
(
∇×
V
).
(1.119)
We conclude this subsection with the specific orthogonal, curvilinear co-
ordinates we will most often use: the spherical polar system (
r
,θ,φ). A point
P
(
u
1
,
u
2
,
u
3
) is at radius
u
1
r
, co-latitude
u
2
=
θ and east longitude
u
3
=
=
φ and has
Cartesian co-ordinates (
x
,y,
z
) given by
x
=
r
sinθcosφ, y
=
r
sinθsinφ,
z
=
r
cosθ.
(1.120)
φ
in the directions of increasing
r
,θ and φ form a right-
handed system, and the metrical coe
θ
,
The unit vectors
r
,
cients are
h
1
=
1,
h
2
=
r
and
h
3
=
r
sinθ. Sub-
stitution in expression (1.113) for the curl yields
θ
/
r
sinθ
φ
/
r
r
/
r
2
sinθ
∇×
V
=
∂/∂
r
∂/∂θ
∂/∂φ
,
(1.121)
V
r
rV
θ
r
sinθ
V
φ
with (
V
r
,
V
θ
,
V
φ
) the spherical polar components of the vector
V
. Substitution in
expression (1.114) for the divergence gives
∂
r
r
2
V
r
sinθ
V
θ
∂
V
φ
∂φ
.
r
2
∂
1
1
r
sinθ
∂
∂θ
1
r
sinθ
∇·
V
=
+
+
(1.122)
From expression (1.115), the gradient of an arbitrary scalar field ψ becomes
θ
φ
r
sinθ
r
∂ψ
r
∂ψ
∂ψ
∂φ
,
∇
ψ
=
∂
r
+
∂θ
+
(1.123)
while, from expression (1.116), its Laplacian becomes
r
2
∂ψ
∂
r
sinθ
∂ψ
∂θ
2
r
2
∂
1
1
r
2
sinθ
∂
∂θ
1
r
2
sin
2
∂
ψ
∂φ
2
∇
ψ
=
+
+
2
∂
r
θ
2
2
2
∂
ψ
∂
r
2
+
2
r
∂ψ
r
2
∂
1
ψ
∂θ
cot
θ
r
2
∂ψ
1
r
2
sin
2
∂
ψ
∂φ
=
∂
r
+
2
+
∂θ
+
2
.
(1.124)
θ
1.1.8 Pseudo-tensors
There is a wide class of physical quantities that have many of the properties of true
tensors but are not invariant in all co-ordinate systems. They are called
pseudo-
tensors
.
Vectors are first-order tensors whose properties under co-ordinate transforma-
tions we have already examined in Subsection 1.1.3. Vectors following the regular
transformation laws for all co-ordinate systems are called
polar vectors
. Thus far,
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