In Chapter 10 we go back and prove the basic facts about finite-dimensional vector spaces and their linear transformations. The treatment here is a straightforward generalization, in the main, of the results obtained in the first four chapters in the two-dimensional case. The one new algorithm is that of row reduction. Two important new concepts (somewhat hard to get used to at first) are introduced: those of the dual space and the quotient space. These concepts will prove crucial in what follows.

Introduction

We have worked extensively with two-dimensional vector spaces, but so far always with one of two specific models in mind. A vector space V was either the set of displacements in an affine plane, or it was ℝ2, the set of ordered pairs of real numbers. By introducing coordinates, we were able to identify any two-dimensional vector space with ℝ2 and thereby to represent any linear transformation of the space by a 2 × 2 matrix.

We shall now begin to view more general vector spaces from an abstract and axiomatic point of view. The advantage of this approach is that it will permit us to consider vector spaces that are not defined either in geometrical terms or as n-tuples of real numbers. It will turn out that any such vector space containing only a finite number of linearly independent elements can be identified with ℝn for some integer n so that eventually we shall return to the study of ℝn and the use of matrices to represent linear transformations.

In Chapter 1 we explain the relation between the multiplication law of 2 × 2 matrices and the geometry of straight lines in the plane. We develop the algebra of 2 × 2 matrices and discuss the determinant and its relation to area and orientation. We define the notion of an abstract vector space, in general, and explain the concepts of basis and change of basis for one- and two-dimensional vector spaces.

Affine planes and vector spaces

The familiar Euclidean plane of high-school plane geometry arose early in the history of mathematics because its properties are readily discovered by physical experiments with a tabletop or blackboard. Through our experience in using rulers and protractors, we are inclined to accept ‘length’ and ‘angle’ as concepts which are as fundamental as ‘point’ and ‘line’. We frequently have occasion, though, both in pure mathematics and in its applications to physics and other disciplines, to consider planes for which straight lines are defined but in which no general notion of length is defined, or in which the usual Euclidean notion of length is not appropriate. Such a plane may be represented on a sheet of paper, but the physical distance between two points on the paper, as measured by a ruler, or the angle between two lines, as measured by a protractor, need have no significance.

The purpose of this book is to describe an approach to maximum and minimum principles which is both straightforward and unified. A particular aim is to identify and illustrate the structure of the theory, and to show how that structure leads to general formulae for the construction of upper and lower bounds associated with those principles. Such bounds are important in applications.

I have found that fresh and fruitful insights have arisen repeatedly during the working out of the ideas developed in this approach. It is offered to the reader in the hope that, as he absorbs the viewpoint, he will benefit from similar experiences.

The treatment is designed to be accessible in the first three chapters to final year undergraduates in mathematics and science, and throughout to contain material which will interest postgraduates and research workers in those subjects. Some elementary prior knowledge of the calculus of variations will be helpful, but otherwise the book is self-contained. Anyone knowing more than this minimum should be able to read Chapter 3 first, with occasional references back to Chapter 1.1 have given a central role to a pair of inner product spaces and, by emphasizing this simple idea, I have been able to avoid the need for any sophisticated functional analysis. The reader who does possess more technical knowledge may use it to add to what is here; he will recognise topics which I have omitted. The reader without such extra knowledge will be at no disadvantage in reading the book.

Introduction

The general ideas which have been developed in the first three chapters are now available to be used in a wider class of problems in applied mathematics than we have so far indicated. We can expect to go beyond the original objective, which was to find upper and lower bounds for the solution value of functional representing, for example, an overall energy expenditure in boundary value problems.

In this chapter we start to explore such extensions.

We begin §4.2 with a discussion of bounds on pre-assigned linear functionals. This is related to the question of pointwise bounds. Some rather different viewpoints in the literature are brought together and generalized in §§4.2(i)–(iii). We carry out some preliminary detailed calculations in §§4.2(iv) and (v). Then we discuss so-called ‘bivariational’ bounds, which require some new hypotheses.

In §§4.3 and 4.4 we give some discussion of bounds for initial value problems. In §4.5 we return briefly to comparison methods, in order to make contact with an approach which has been influential in solid mechanics, where information about a ‘hard’ problem is obtained by comparison with a notional ‘easy’ problem.

The idea of working in a pair of inner product spaces offers some fresh viewpoints in all these contexts, which already have their own substantial literature. The purpose in this chapter is to hint at what may be achieved by the systematic development of a body of detailed and substantial examples.

Introduction

The main purpose of this chapter is to give a very explicit account of the Legendre transformation, including information which is not widely known. We work in finite dimensions only, and to that extent the treatment is elementary. Basic properties of the transformation are laid out in §2.2. Detailed illustrations are given in §§2.3 and 2.4 for functions of one and two variables respectively.

The transformation may be viewed as a conversion of one continuous scalar function into another. Under quite light restrictions it turns out that the transformation is reversible, and then we say that each function is the dual of the other, or that each is the Legendre transform of the other. The reversible Legendre transformation is also called the Legendre dual transformation. It is important for both practical and theoretical reasons.

On the practical side we can often expect to solve at least some of the constraints governing a problem, whether it be only a stationary value problem or, more strongly, a problem offering an upper bound to be evaluated prior to minimization. The Legendre transformation is one of the devices which can effect the solution of such partial constraints. This can be explained in the notation of the very particular examples of the transformation already used to facilitate the proofs of Theorems 1.3(b) and 1.9(a) (see also Exercises 1.3). The dual functions there, for example J[u] and G[υ] in (1.34), have the properties υ = J′[u] and u = G′[υ] where the prime denotes the gradient.

Introduction

This is the core of the book. We show how to construct upper and lower bounds in a number of representative infinite dimensional problems.

The approach is to generalize Theorem 1.6, which was proved for the finite dimensional case. The basic result is Theorem 3.4. We set the problems in a pair of infinite dimensional inner product spaces. We construct some simple tools which allow this setting to mimic the finite dimensional development in Chapters 1 and 2.

These tools include a straightforward but unfamiliar type of inner product space element (§3.2 (ii)), a pair of linear operators T and T* which are adjoint to each other in the true sense, not merely the formal sense (§3.3 (ii) and (iii)), and the gradient of a functional whose domain is an inner product space (§3.4(ii) and (iii)).

Application to differential equations with one type of mixed boundary conditions is explained by the detailed working of an example (§3.7) and the extension of similar methods to other cases is indicated in §§3.8–3.10. A connection with variational inequalities is mentioned in §3.11. We conclude (§3.12) with an account of equations containing a nonnegative operator, typically decomposable as T*T. These admit not only simultaneous bounds but also, in some cases, alternative pairs of such bounds not unlike those already indicated for networks in §2.8. This is one situation in which integral equations become prominent, as we illustrate for a wave scattering problem in fluid mechanics.

Introduction

In this chapter we select topics from the mechanics of fluids and elastic and plastic solids, and use them to illustrate ideas introduced previously. In §5.2 we review some Legendre transformations of classical thermodynamics in the light of Chapter 2, exhibiting quartets and isolated singularities associated with phase changes. For compressible flow another Legendre transformation, involving a swallowtail flow stress function which plays the role of a complementary energy in the sense of Fig. 2.27, is displayed in Fig. 5.3.

Subsequently we begin each of a number of contexts with a statement of the general boundary value problem, in terms of the ab initio kinematic and force variables. Then we obtain, for example, stationary principles and upper and lower bound theorems at that level of generality. This is in contrast to our previous illustrations, which usually began with a differential or integral equation for a single unknown variable, presumed to have resulted from some combination and elimination of ab initio variables.

This chapter shows that theories of considerable algebraic complexity can be well organized within the general framework developed in Chapters 1–3. Many other examples in continuum mechanics could be given.

Thermodynamics

(i) Thermodynamic potential functions

This is the obvious classical illustration of Fig. 2.18. We use certain terminology and notation which have become familiar for the context, in order to aid the reader's comparison of our treatment with what he may already know.