64 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS
is continuous and non-degenerate. The inverse of (2.3.10) is denoted by
(2.3.11) (bij) = (aij)−^1.
We remark that if the matrices in (2.3.10) and (2.3.11) are not continuous, then the trans-
formation (2.3.8) are not permitted.
Definition 2.29(General Tensors).Let T be a set of quantities defined on the Riemann space
{M,gij}, and T have nr+scomponents in each coordinate system x∈M:
T={Tji 11 ······jirs(x)} for x∈M.If under the coordinate transformation (2.3.9), the components of T transform as
(2.3.12) T ̃ji 11 ······jirs=aik^11 ···aikrrblj^11 ···bljssTl 1 k^1 ······lskr,
then T is called a(r,s)type of kth-order general tensor with k=r+s, where aijand bml are
as in (2.3.10) and (2.3.11). The(r, 0 )type tensors are called contra-variant tensors, and the
( 0 ,s)type tensors are called covariant tensors.
On a Riemannian space{M,gij}, the metric{gij}and its inverse{gij}are second-order
covariant and contra-variant symmetric tensors, i.e. theysatisfy
(2.3.13)
gij=gji, gij=gji,
̃gij=blibkjglk, g ̃ij=ailakjglk,under the transformation (2.3.9). For first-order tensors, we have
(2.3.14)
A ̃ 1
..
.
A ̃n
= (bij)T
A 1
..
.
An
,
A ̃^1
..
.
A ̃n
= (aij)
A^1
..
.
An
and for seconder-order tensors, we have
(2.3.15) (T ̃ij) = (bkl)T(Tij)(bkl), (T ̃ij) = (akl)(Tij)(akl)T.
2.General Invariants.General Invariants are derived from contractions of general ten-
sors. We infer from (2.3.14) that
A ̃kA ̃k = (A ̃ 1 ,···,A ̃n)
A ̃ 1
..
.
A ̃n
= (A 1 ,···,An)(bij)(aij)
A^1
..
.
An
= (A 1 ,···,An)
A^1
..
.
An
=AkAk, by(bij) = (aij)−^1 ,