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1 Preliminaries
1.7 Basic Conceptions of Tensors
The tensor is a generalization of the concept of vector. An important feature of the
tensor is that the physical quantities and geometrical quantities expressed by it have
nothing to do with the choice of the coordinate system. But, in order to conveniently
research the tensor in certain coordinate systems, it will be determined by the set of
its components, and these components relate to the coordinate systems. This section
mainly discusses the algebraic operation and differential operation of the Cartesian
second-order tensor, they are the most basic contents of tensor analysis, and are also
the necessary tools of research on mechanical variational principle.
1.7.1 Rotation Transformations of Rectangle Coordinates
In the rectangular coordinate system Ox yz, a vector a can be represented with its
three components a x , a y and a z . Due to the coordinate system is artificially chosen, of
course another coordinate system Ox
y
z
can also be chosen, the three components
of a become a x , a y and a z . The same vector a, in different coordinate systems can
be represented in different components.
Let Ox 1 x 2 x 3 and Ox
1 x
2 x
3 be an old and a new right-handed rectangular coordinate systems respectively. e 1 , e 2 , e 3 and e
1 , e
2 , e
3 are the unit vectors on the axes in
the two coordinate systems respectively, then there is
e i · e j = e
i · e
j = δ i j
(1.7.1)
There are the following relationships between the unit vectors in the old and new
coordinates
⎧
⎨
⎩
e
1 = α 11 e 1 + α 12 e 2 + α 13 e 3
e
2 = α 21 e 1 + α 22 e 2 + α 23 e 3
e
3 = α 31 e 1 + α 32 e 2 + α 33 e 3
(1.7.2)
where, α i j = e
i · e j is the cosine of included angle of the different axes in the two
coordinate systems, namely the direction cosine, it is called the coefficient of transformation, The first index expresses the new coordinates, The second index expresses
the old coordinates. Making use of Einstein summation convention, Eq. (1.7.2) can
be written as
e
i = α i j e j , e i = α ji e
j
(1.7.3)
Let the three components of the vector a in the old coordinate system be a 1 , a 2 ,
a 3 , the three components in the new coordinate system are a
1 , a
2 , a
3 . In the old and
the new coordinate system, the vector a can be represented as
1 Preliminaries
1.7 Basic Conceptions of Tensors
The tensor is a generalization of the concept of vector. An important feature of the
tensor is that the physical quantities and geometrical quantities expressed by it have
nothing to do with the choice of the coordinate system. But, in order to conveniently
research the tensor in certain coordinate systems, it will be determined by the set of
its components, and these components relate to the coordinate systems. This section
mainly discusses the algebraic operation and differential operation of the Cartesian
second-order tensor, they are the most basic contents of tensor analysis, and are also
the necessary tools of research on mechanical variational principle.
1.7.1 Rotation Transformations of Rectangle Coordinates
In the rectangular coordinate system Ox yz, a vector a can be represented with its
three components a x , a y and a z . Due to the coordinate system is artificially chosen, of
course another coordinate system Ox
y
z
can also be chosen, the three components
of a become a x , a y and a z . The same vector a, in different coordinate systems can
be represented in different components.
Let Ox 1 x 2 x 3 and Ox
1 x
2 x
3 be an old and a new right-handed rectangular coordinate systems respectively. e 1 , e 2 , e 3 and e
1 , e
2 , e
3 are the unit vectors on the axes in
the two coordinate systems respectively, then there is
e i · e j = e
i · e
j = δ i j
(1.7.1)
There are the following relationships between the unit vectors in the old and new
coordinates
⎧
⎨
⎩
e
1 = α 11 e 1 + α 12 e 2 + α 13 e 3
e
2 = α 21 e 1 + α 22 e 2 + α 23 e 3
e
3 = α 31 e 1 + α 32 e 2 + α 33 e 3
(1.7.2)
where, α i j = e
i · e j is the cosine of included angle of the different axes in the two
coordinate systems, namely the direction cosine, it is called the coefficient of transformation, The first index expresses the new coordinates, The second index expresses
the old coordinates. Making use of Einstein summation convention, Eq. (1.7.2) can
be written as
e
i = α i j e j , e i = α ji e
j
(1.7.3)
Let the three components of the vector a in the old coordinate system be a 1 , a 2 ,
a 3 , the three components in the new coordinate system are a
1 , a
2 , a
3 . In the old and
the new coordinate system, the vector a can be represented as