1.7 Basic Conceptions of Tensors
65
Theorem 1.7.3 If the inner product of set A i 1 i 2 ...i m j 1 j 2 ... j n of array and an arbitrary
tensor B j 1 j 2 ... j n of order n
A i 1 i 2 ...i m j 1 j 2 ... j n B j 1 j 2 ... j n = C i 1 i 2 ...i m
is identically a tensor of order m, then A i 1 i 2 ...i m j 1 j 2 ... j n must be a tensor of order m + n.
The method of proof of the theorem is similar to one of Theorem 1.7.2. Theorems
1.7.2 and 1.7.3 are called the tensor recognition theorem, recognition theorem of
tensor or quotient theorem of tensors, sometimes they are also called the indirect
test for tensor character.
If v and w are real numbers, with v = 0, then the quotient is defined as u =
w
v
, it
can also be written as w = uv. Similarly, C = A · B can be interpreted as A is the
quotient of C and B, because the division of the vectors and tensors is not defined
C
B
is nonsense. Because a second order tensor can be interpreted as the quotient of
two first-order tensors, Theorem 1.7.3 is called the quotient theorem of tensors.
Example 1.7.2 Prove that the permutation symbol is a third order tensor.
Proof The cross product of two vectors can be expressed as
a × b = a i e i × b j e j = a i b j ε i jk e k = c k e k = C
where, c k = a i b j ε i jk ; a i b j is arbitrary second order tensor formed by the dyad of
arbitrary vectors a i and b j , and c k is a known first order tensor. It can be known from
the quotient theorem that the permutation symbol ε i jk is bound to be a third order
tensor. Quod erat demonstrandum.
1.7.5 Principal Axes, Characteristic Values and Invariants
of Second Order Tensors
Let T be a second order tensor, for arbitrary nonzero vector a, do the right inner
product of the tensor and the vector, there is
T · a = b
(1.7.21)
If the vector b and vector a have the same direction, namely
b = λa
(1.7.22)
then the direction of the vector a is called the characteristic direction or direction
of principal axis of the tensor T , λ is called the characteristic value, Eigenvalue
or principal value of the tensor T . It can be known from the tensor recognition
65
Theorem 1.7.3 If the inner product of set A i 1 i 2 ...i m j 1 j 2 ... j n of array and an arbitrary
tensor B j 1 j 2 ... j n of order n
A i 1 i 2 ...i m j 1 j 2 ... j n B j 1 j 2 ... j n = C i 1 i 2 ...i m
is identically a tensor of order m, then A i 1 i 2 ...i m j 1 j 2 ... j n must be a tensor of order m + n.
The method of proof of the theorem is similar to one of Theorem 1.7.2. Theorems
1.7.2 and 1.7.3 are called the tensor recognition theorem, recognition theorem of
tensor or quotient theorem of tensors, sometimes they are also called the indirect
test for tensor character.
If v and w are real numbers, with v = 0, then the quotient is defined as u =
w
v
, it
can also be written as w = uv. Similarly, C = A · B can be interpreted as A is the
quotient of C and B, because the division of the vectors and tensors is not defined
C
B
is nonsense. Because a second order tensor can be interpreted as the quotient of
two first-order tensors, Theorem 1.7.3 is called the quotient theorem of tensors.
Example 1.7.2 Prove that the permutation symbol is a third order tensor.
Proof The cross product of two vectors can be expressed as
a × b = a i e i × b j e j = a i b j ε i jk e k = c k e k = C
where, c k = a i b j ε i jk ; a i b j is arbitrary second order tensor formed by the dyad of
arbitrary vectors a i and b j , and c k is a known first order tensor. It can be known from
the quotient theorem that the permutation symbol ε i jk is bound to be a third order
tensor. Quod erat demonstrandum.
1.7.5 Principal Axes, Characteristic Values and Invariants
of Second Order Tensors
Let T be a second order tensor, for arbitrary nonzero vector a, do the right inner
product of the tensor and the vector, there is
T · a = b
(1.7.21)
If the vector b and vector a have the same direction, namely
b = λa
(1.7.22)
then the direction of the vector a is called the characteristic direction or direction
of principal axis of the tensor T , λ is called the characteristic value, Eigenvalue
or principal value of the tensor T . It can be known from the tensor recognition