1.7 Basic Conceptions of Tensors
63
Cartesian tensors is still the same order tensors, their components equal the sum (or
difference) of the components of the two same order Cartesian tensors.
(2) Multiplicative operation
Arbitrary tensors of the same space can be continually multiplied, instead of requiring
them to have the same structure, but the order can not be disorderly, can also not
have the same index, this kind of operation is called the outer product, exterior
product, external product or outer multiplication of a tensor. For instance, the
components of the two tensors are A i j and B lmn , the cross product is a fifth order
tensor, its composition is
C i jlmn = A i j B lmn
(1.7.18)
(3) Contraction of tensor
Suppose that there is a nth order tensor (n ≥ 2), If there are two same indexes, the
repeated indexes are summed, then a new tensor of order n− 2 can be obtained, this
kind of operation is called the contraction of a tensor. Whenever contract a pair of
indexes, the tensor order is minus 2. For instance, a second order tensor is T i j , let
i = j, then there is
T ii = T 11 + T 22 + T 33
(1.7.19)
It shows that the contraction of a second order tensor T ii is a scalar tensor, it is
the sum of the main diagonal elements of a second order square matrix.
Contract the outer product of two Cartesian tensors, to produce a new tensor,
this kind of operation is called the inner product, dot product, interior product,
internal product or inner multiplication of the Cartesian tensor. The inner product
of a tensor can be seen as the outer product and contraction of tensors. For instance
a i b i = a 1 b 1 + a 2 b 2 + a 3 b 3 = a · b
(1.7.20)
is a scalar tensor namely scalar, this is just the scalar product of two vectors.
Write a number of independent vectors side by side together, but they do not
do the inner product or cross product, which is called the dyad, it is outer product
operation of tensor. The dyad of two vectors is a second order tensor, the dyad of n
vectors is a nth order tensor.
Proof Suppose that there is a ab, in an old coordinate system its components are
a i b j , When the coordinate system is transformed, it can be known from the vector
transformation relation formula (1.7.6) that a
i = α il a l , b
j = α jm b m , so in a new
coordinate system, the components of the dyad ab are
a
i b
j = α il α jm a l b m
63
Cartesian tensors is still the same order tensors, their components equal the sum (or
difference) of the components of the two same order Cartesian tensors.
(2) Multiplicative operation
Arbitrary tensors of the same space can be continually multiplied, instead of requiring
them to have the same structure, but the order can not be disorderly, can also not
have the same index, this kind of operation is called the outer product, exterior
product, external product or outer multiplication of a tensor. For instance, the
components of the two tensors are A i j and B lmn , the cross product is a fifth order
tensor, its composition is
C i jlmn = A i j B lmn
(1.7.18)
(3) Contraction of tensor
Suppose that there is a nth order tensor (n ≥ 2), If there are two same indexes, the
repeated indexes are summed, then a new tensor of order n− 2 can be obtained, this
kind of operation is called the contraction of a tensor. Whenever contract a pair of
indexes, the tensor order is minus 2. For instance, a second order tensor is T i j , let
i = j, then there is
T ii = T 11 + T 22 + T 33
(1.7.19)
It shows that the contraction of a second order tensor T ii is a scalar tensor, it is
the sum of the main diagonal elements of a second order square matrix.
Contract the outer product of two Cartesian tensors, to produce a new tensor,
this kind of operation is called the inner product, dot product, interior product,
internal product or inner multiplication of the Cartesian tensor. The inner product
of a tensor can be seen as the outer product and contraction of tensors. For instance
a i b i = a 1 b 1 + a 2 b 2 + a 3 b 3 = a · b
(1.7.20)
is a scalar tensor namely scalar, this is just the scalar product of two vectors.
Write a number of independent vectors side by side together, but they do not
do the inner product or cross product, which is called the dyad, it is outer product
operation of tensor. The dyad of two vectors is a second order tensor, the dyad of n
vectors is a nth order tensor.
Proof Suppose that there is a ab, in an old coordinate system its components are
a i b j , When the coordinate system is transformed, it can be known from the vector
transformation relation formula (1.7.6) that a
i = α il a l , b
j = α jm b m , so in a new
coordinate system, the components of the dyad ab are
a
i b
j = α il α jm a l b m