Page 158 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 14: Notation and Summation Convention

second-order tensor and the matrix of the (3 × 1) components of the original vector. These Notes
components are with respect to a rectangular Cartesian coordinate system, hence, the term
Cartesian tensor analysis. Examples from classical mechanics and stress analysis are as follows.
The angular momentum vector, h, of a rigid body about its mass center is given by h = J, where
J is the inertia tensor of the body about its mass center and  is the angular velocity vector. In this
equation the components of the vectors, h and  can be represented by (3 × 1) matrices and the
tensor J by a (3 × 3) matrix with matrix multiplication implied. A further example is the relation
t = n, between the stress vector t acting on a material area element and the unit normal n to the
element, where s is the Cauchy stress tensor. The relations h = J and t = n are examples of
coordinate-free symbolic notation, and the corresponding matrix relations refer to a particular
coordinate system.
We will meet further examples of the operator properties of second order tensors in the study of
continuum mechanics and thermodynamics. Tensors of order greater than two can be regarded
as operators operating on lower-order tensors. Components of tensors of order greater than
two cannot be expressed in matrix form.
It is very important to note that physical laws are independent of any particular coordinate
system. Consequently, equations describing physical laws, when referred to a particular
coordinate system, must transform in definite manner under transformation of coordinate
systems. This leads to the concept of a tensor, that is, a quantity that does not depend on the
choice of coordinate system. The simplest tensor is a scalar, a zeroth-order tensor. A scalar is
represented by a single component that is invariant under coordinate transformation. Examples
of scalars are the density of a material and temperature.
Higher-order tensors have components relative to various coordinate systems, and these
components transform in a definite way under transformation of coordinate systems. The velocity
v of a particle is an example of a first-order tensor; henceforth we denote vectors, in symbolic
notation, by lowercase bold letters. We can express v by its components relative to any convenient
coordinate system, but since v has no preferential relationship to any particular coordinate
system, there must be a definite relationship between components of v in different coordinate
systems. Intuitively, a vector may be regarded as a directed line segment, in a three-dimensional
Euclidean point space E , and the set of directed line segments in E , of classical vectors, is a
3
3
vector space V . That is, a classical vector is the difference of two points in E . A vector, according
3
3
to this concept, is a first-order tensor.
There are many physical laws for which a second-order tensor is an operator associating one
vector with another. Remember that physical laws must be independent of a coordinate system;
it is precisely this independence that motivates us to study tensors.
14.1 Rectangular Cartesian Coordinate Systems

The simplest type of coordinate system is a rectangular Cartesian system, and this system is
particularly useful for developing most of the theory to be presented in this text.
A rectangular Cartesian coordinate system consists of an orthonormal basis of unit vectors
(e , e , e ) and a point 0 which is the origin. Right-handed Cartesian coordinate systems are
3
2
1
considered, and the axes in the (e , e , e ) directions are denoted by 0x , 0x , and 0x , respectively,
2
1
1
3
2
3
rather than the more usual 0x, 0y, and 0z. A right-handed system is such that a 90º right-handed
screw rotation along the 0x direction rotates 0x to 0x , similarly a right-handed rotation about
3
1
2
0x rotates 0x to 0x , and a right-handed rotation about 0x rotates 0x to 0x .
3
2
2
3
1
1
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