Geology Reference
In-Depth Information
1.1.3 Vectors and co-ordinate transformations
Consider how the components of the vector
V
transform under a transformation of
co-ordinates. Suppose we adopt a new set of curvilinear co-ordinates (
u
1
,
u
2
,
u
3
)
in place of (
u
1
,
u
2
,
u
3
). The unitary base vectors
b
i
are expressible as
∂
u
j
∂
u
j
b
j
∂
u
j
∂
r
∂
u
i
=
∂
r
b
i
=
∂
u
i
=
∂
u
i
,
(1.24)
using the chain rule for partial derivatives. The vectors
∂
r
∂
u
j
b
j
=
(1.25)
form a new triplet of unitary base vectors in the new co-ordinate system. The
vector
V
may be expressed by its contravariant components in either co-ordinate
system as
i
b
i
=
v
j
b
j
.
V
=
v
(1.26)
Substituting for
b
i
from (1.24), we find that
i
b
i
=
∂
u
j
i
b
j
=
v
j
b
j
.
V
=
v
∂
u
i
v
(1.27)
Thus, the transformation law for the contravariant components of the vector
V
is
∂
u
j
∂
u
i
v
v
j
i
=
.
(1.28)
The new covariant components of the vector
V
may be found from its new
contravariant components, using the second of relations (1.16) and the metrical
coe
cient g
kj
in the new co-ordinate system, where
g
kj
=
b
k
·
b
j
,
(1.29)
with
b
i
being a unitary base vector in the new co-ordinate system. Using the chain
rule for partial derivatives, the base vectors in the new co-ordinate system can be
related to those in the original co-ordinate system by
∂
u
l
∂
u
l
b
l
∂
u
l
∂
r
∂
u
k
=
∂
r
b
k
=
∂
u
k
=
∂
u
k
,
(1.30)
∂
u
m
∂
u
m
b
m
∂
u
m
∂
r
∂
u
j
=
∂
r
b
j
=
∂
u
j
=
∂
u
j
.
(1.31)
Then,
b
m
∂
u
l
∂
u
k
∂
u
m
∂
u
k
∂
u
m
∂
u
l
g
kj
=
b
k
·
b
j
=
b
l
·
∂
u
j
=
∂
u
j
g
lm
.
(1.32)
Search WWH ::
Custom Search