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1 Preliminaries
It shows that the components of a dyad ab satisfy the definition Eq. (1.7.12) of a
second order tensor, this can be concluded that a dyad is just a second order tensor.
In the same way, it can be proved that the dyad of n vectors is just a tensor of order
n. Quod erat demonstrandum.
1.7.4 Quotient Laws of Tensors
The quotient laws of tensors is an indirect rule which is used to judge whether a
set amount constitute a tensor. This kind of judgement does not need to satisfy the
coordinate transformation, only need to discriminate in a fixed coordinate system.
The quotient laws of tensors can use the following two theorems to state.
Theorem 1.7.2 If the outer product of set A i 1 i 2 ...i m of array and an arbitray tensor
B j 1 j 2 ... j n of order n
A i 1 i 2 ...i m B j 1 j 2 ... j n = C i 1 i 2 ...i m j 1 j 2 ... j n
is identically a tensor of order m + n, then A i 1 i 2 ...i m is bound to be a tensor of order
m.
Proof Take m = 3, n = 2 as an example, let
A i jk B lm = C i jklm
where, B lm is a second order tensor; C i jklm is a fifth order tensor. The both sides of
the above expression are multiplied by a second order tensor B lm , we obtain
A i jk B lm B lm = C i jklm B lm
The right-handed side of the equality is the inner product of a fifth order tensor
and a second order equation which are contracted twice, the resulting tensor is a third
order tensor D i jk , the twice contractions on the left-handed side B lm B lm is a scalar,
let it be λ. Since B lm is arbitrary second order tensor, there always exist such a B lm
that λ = 0, so there is
λA i jk = D i jk = C i jklm B lm
This proves that A i jk is a third order tensor. The proof is obtained in the case of
m = 3, n = 2, if m and n are replaced by any other positive integer, the process of
proof of the theorem can be exactly the same. Quod erat demonstrandum.
1 Preliminaries
It shows that the components of a dyad ab satisfy the definition Eq. (1.7.12) of a
second order tensor, this can be concluded that a dyad is just a second order tensor.
In the same way, it can be proved that the dyad of n vectors is just a tensor of order
n. Quod erat demonstrandum.
1.7.4 Quotient Laws of Tensors
The quotient laws of tensors is an indirect rule which is used to judge whether a
set amount constitute a tensor. This kind of judgement does not need to satisfy the
coordinate transformation, only need to discriminate in a fixed coordinate system.
The quotient laws of tensors can use the following two theorems to state.
Theorem 1.7.2 If the outer product of set A i 1 i 2 ...i m of array and an arbitray tensor
B j 1 j 2 ... j n of order n
A i 1 i 2 ...i m B j 1 j 2 ... j n = C i 1 i 2 ...i m j 1 j 2 ... j n
is identically a tensor of order m + n, then A i 1 i 2 ...i m is bound to be a tensor of order
m.
Proof Take m = 3, n = 2 as an example, let
A i jk B lm = C i jklm
where, B lm is a second order tensor; C i jklm is a fifth order tensor. The both sides of
the above expression are multiplied by a second order tensor B lm , we obtain
A i jk B lm B lm = C i jklm B lm
The right-handed side of the equality is the inner product of a fifth order tensor
and a second order equation which are contracted twice, the resulting tensor is a third
order tensor D i jk , the twice contractions on the left-handed side B lm B lm is a scalar,
let it be λ. Since B lm is arbitrary second order tensor, there always exist such a B lm
that λ = 0, so there is
λA i jk = D i jk = C i jklm B lm
This proves that A i jk is a third order tensor. The proof is obtained in the case of
m = 3, n = 2, if m and n are replaced by any other positive integer, the process of
proof of the theorem can be exactly the same. Quod erat demonstrandum.