Vector and Tensor Analysis with Applications

A. I. Borisenko, I. E. Tarapov, Richard A. Silverman

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486131900
• Topic: Matemáticas
• Subtopic: Análisis vectorial

Concise and readable, this text ranges from definition of vectors and discussion of algebraic operations on vectors to the concept of tensor and algebraic operations on tensors. It also includes a systematic study of the differential and integral calculus of vector and tensor functions of space and time. Worked-out problems and solutions. 1968 edition.

1

VECTOR ALGEBRA

In this chapter we define vectors and then discuss algebraic operations on , vectors. The vector concept will be generalized in a natural way in Chapter 2, leading to the concept of a tensor. Then in Chapter 3 we will consider algebraic operations on tensors.

1.1. Vectors and Scalars

Quantities which can be specified by giving just one number (positive, negative or zero) are called scalars. For example, temperature, density, mass and work are all scalars. Scalars can be compared only if they have the same physical dimensions. Two such scalars measured in the same system of units are said to be equal if they have the same magnitude (absolute value) and sign.

However, one must often deal with quantities, called vectors, whose specification requires a direction as well as a numerical value. For example, displacement, velocity, acceleration, force, the moment of a force, electric field strength and dielectric polarization are all vectors. Operations on vectors obey the rules of vector algebra, to be considered below.

Scalars and vectors hardly exhaust the class of quantities of interest in applied mathematics and physics. In fact, there are quantities of a more complicated structure than scalars or vectors, called tensors (of order 2 or higher), whose specification requires more than knowledge of a magnitude and a direction. For example, examination of the set of all stress vectors acting on all elements of area which can be drawn through a given point of an elastic body leads to the concept of the stress tensor (of order 2), while examination of the deformation of an arbitrary elementary volume of an elastic body leads to the concept of the deformation (or strain) tensor. We will defer further discussion of tensors until Chapter 2, concentrating our attention for now on vectors.

A vector A is represented by a directed line segment, whose direction and length coincide with the direction and magnitude (measured in the chosen system of units) of the quantity under consideration. Vector...