300
5 X-ray Pulsar-Based Navigation: Theories and Experiments
Meanwhile, the corresponding inverse transformation relation of expression
(5.15) is expressed as
x
j
= x
j
x
i
.
(5.17)
In this way, the functional relation of mutual transformation between the two
coordinate systems is established.
According to the definition of covariant basis vectors, the transformational
relations of the covariant basis vectors between two coordinate systems are
g i = β
j
i g j .
(5.18)
g
i
= β
i
j g
j
,
(5.19)
where β
j
i =
∂x
j
∂x i , known as covariant transformation coefficients; β
i
j =
∂x
i
∂x j , known
as contravariant transformation coefficients.
For any vector a in three-dimensional affine space, the different basis vectors
can be decomposed in different coordinate systems. In the new and old coordinate
systems, the different covariant components of the same vector are obtained by
decomposing the contravariant basis, while the different contravariant components
of the same vector are obtained by decomposing the covariant basis. Although the
components of different coordinate systems are different, the sum of decomposition
vectors will not change due to the different coordinate systems. Therefore, in the
different coordinate systems, the relations between the components of the same
vector can be determined by the transformations of the basis vector, that is
a i = β
j
i a j
a
i
= β
i
j a
j
.
(5.20)
Under the condition of satisfying expression (5.15), the components of any vector
in three-dimensional affine space have the transformational law described in expression (5.20). This law of coordinate transformation is extended to n-dimensional affine
space, to study the coordinate transformation of an ordered array. In n-dimensional
affine space, for a quantity with n
N components, each component can be transformed
by the following expression:
x
i
= x
i
x
j
.
i
, j = 1, 2, . . . , n
(5.21)
According to this transformation law, there is
T
j
1 ···j
l
i
1 ···i
m
= β
j
1
j 1
· · · β
j
l
j l
β
i 1
i
1
· · · β
i m
i
m
T
j 1 ···j l
i 1 ···i m
,
(5.22)
5 X-ray Pulsar-Based Navigation: Theories and Experiments
Meanwhile, the corresponding inverse transformation relation of expression
(5.15) is expressed as
x
j
= x
j
x
i
.
(5.17)
In this way, the functional relation of mutual transformation between the two
coordinate systems is established.
According to the definition of covariant basis vectors, the transformational
relations of the covariant basis vectors between two coordinate systems are
g i = β
j
i g j .
(5.18)
g
i
= β
i
j g
j
,
(5.19)
where β
j
i =
∂x
j
∂x i , known as covariant transformation coefficients; β
i
j =
∂x
i
∂x j , known
as contravariant transformation coefficients.
For any vector a in three-dimensional affine space, the different basis vectors
can be decomposed in different coordinate systems. In the new and old coordinate
systems, the different covariant components of the same vector are obtained by
decomposing the contravariant basis, while the different contravariant components
of the same vector are obtained by decomposing the covariant basis. Although the
components of different coordinate systems are different, the sum of decomposition
vectors will not change due to the different coordinate systems. Therefore, in the
different coordinate systems, the relations between the components of the same
vector can be determined by the transformations of the basis vector, that is
a i = β
j
i a j
a
i
= β
i
j a
j
.
(5.20)
Under the condition of satisfying expression (5.15), the components of any vector
in three-dimensional affine space have the transformational law described in expression (5.20). This law of coordinate transformation is extended to n-dimensional affine
space, to study the coordinate transformation of an ordered array. In n-dimensional
affine space, for a quantity with n
N components, each component can be transformed
by the following expression:
x
i
= x
i
x
j
.
i
, j = 1, 2, . . . , n
(5.21)
According to this transformation law, there is
T
j
1 ···j
l
i
1 ···i
m
= β
j
1
j 1
· · · β
j
l
j l
β
i 1
i
1
· · · β
i m
i
m
T
j 1 ···j l
i 1 ···i m
,
(5.22)