Mathematical analysis for economists/ R.G.D. Allen

The Use of Greek Letters in Mathematical Analysis
I. Numbers and Variables
1.1 Introduction -- 1.2 Numbers of various types -- 1.3 The real number system -- 1.4 Continuous and discontinuous variables -- 1.5 Quantities and their measurement -- 1.6 Units of measurement -- 1.7 Derived quantities -- 1.8 The location of points in space -- 1.9 Variable points and their co-ordinates -- Examples I—The measurement of quantities; graphical methods --

II. Functions and their Diagrammatic Representation
2.1 Definition and examples of functions -- 2.2 The graphs of functions -- 2.3 Functions and curves -- 2.4 Classification of functions -- 2.5 Function types -- 2.6 The symbolic representation of functions of any form -- 2.7 The diagrammatic method -- 2.8 The solution of equations in one variable -- 2.9 Simultaneous equations in two variables -- Examples II—Functions and graphs ; the solution of equations --

III. Elementary Analytical Geometry
3.1 Introduction -- 3.2 The gradient of a straight line -- 3.3 The equation of a straight line -- 3.4 The parabola -- 3.6 The rectangular hyperbola -- 3.6 The circle -- 3.7 Curve classes and curve systems -- 3.8 An economic problem in analytical geometry -- Examples III—The straight line ; curves and curve systems

IV. Limits and Continuity of Functions
4.1 The fundamental notion of a limit -- 4.2 Examples of the limit of a function -- 4.3 Definition of the limit of a single-valued function -- 4.4 Limiting and approximate values -- 4.5 Some properties of limits -- 4.6 The continuity of functions
4.7 Illustrations of continuity and discontinuity of functions -- 4.8 Multi-valued functions -- Examples IV—Limits of functions ; continuity of functions --

V. Functions and Diagrams in Economic Theory
5.1 Introduction -- 5.2 Demand functions and curves -- 5.3 Particular demand functions and curves -- 5.4 Total revenue functions and curves -- 5.5 Cost functions and curves -- 5.6 Other functions and curves in economic theory -- 5.7 Indifference curves for consumers' goods -- 5.8 Indifference curves for the flow of income over time -- Examples V—Economic functions and curves

VI. Derivatives and their Interpretation
6.1 Introduction -- 6.2 The definition of a derivative -- 6.3 Examples of the evaluation of derivatives -- 6.4 Derivatives and approximate values -- 6.5 Derivatives and tangents to curves -- 6.6 Second and higher order derivatives -- 6.7 The application of derivatives in the natural sciences -- 6.8 The application of derivatives in economic theory -- Examples VI—Evaluation and interpretation of derivatives

VII. The Technique of Derivation
7.1 Introduction -- 7.2 The power function and its derivative -- 7.3 Rules for the evaluation of derivatives -- 7.4 Examples of the evaluation of derivatives -- 7.5 The function of a function rule -- 7.6 The inverse function rule -- 7.7 The evaluation of second and higher order derivatives -- Examples VII—Practical derivation -

VIII. Applications op Derivatives
8.1 The sign and magnitude of the derivative -- 8.2 Maximum and minimum values -- 8.3 Applications of the second derivative -- 8.4 Practical methods of finding maximum and minimum values -- 8.5 A general problem of average and marginal values -- 8.6 Points of inflexion -- 8.7 Monopoly problems in economic theory-- 8.8 Problems of duopoly -- 8.9 A note on necessary and sufficient conditions -- Examples VIII—General applications of derivatives ; economic applications of derivatives -

IX. Exponential and Logarithmic Functions
9.1 Exponential functions -- 9.2 Logarithms and their properties -- 9.3 Logarithmic functions -- 9.4 Logarithmic scales and graphs -- 9.5 Examples of logarithmic plotting -- 9.6 Compound interest -- 9.7 Present values and capital values -- 9.8 Natural exponential and logarithmic functions -- Examples IX—Exponential and logarithmic functions; compound interest problems -

X. Logarithmic Derivation
10.1 Derivatives of exponential and logarithmic functions -- 10.2 Logarithmic derivation -- 10.3 A problem of capital and interest -- 10.4 The elasticity of a function -- 10.5 The evaluation of elasticities -- 10.6 The elasticity of demand -- 10.7 Normal conditions of demand -- 10.8 Cost elasticity and normal cost conditions -- Examples X—Exponential and logarithmic derivatives; elasticities and their applications --

XI. Fctnctions of Two ok Moke Vakiables
11.1 Functions of two variables -- 11.2 Diagrammatic representation of functions of two variables. - - 11.3 Plane sections of a surface -- 11.4 Functions of more than two variables -- 11.5 Non-measurable variables -- 11.6 Systems of equations -- 11.7 Functions of several variables in economic theory -- 11.8 The production function and constant product curves -- 11.9 The utility function and indifference curves -- Examples XI—Functions of two or more variables ; economic functions and surfaces -

XII. Partial Derivatives and their Applications
12.1 Partial derivatives of functions of two variables
12.2 Partial derivatives of the second and higher orders
12.3 The signs of partial derivatives
12.4 The tangent plane to a surface
12.5 Partial derivatives of functions of more than twovariables
12.6 Economic applications of partial derivatives
12.7 Homogeneous functions
12.8 Euler's Theorem and other properties of homogeneous functions
12.9 The linear homogeneous production function
Examples XII—Partial derivatives ; homogeneous functions ; economic applications of partial derivatives and homogeneous functions

XIII. Differentials and Differentiation
13.1 The variation of a function of two variables
13.2 The differential of a fimction of two variables
13.3 The technique of differentiation -
13.4 Differentiation of functions of fxmctions
13.5 Differentiation of implicit functions
13.6 The differential of a function of more than two variables
13.7 The substitution of factors in production
13.8 Substitution in other economic problems
13.9 Further consideration of duopoly problems
Examples XIII—^Differentiation ; economic applications of differentials

XIV. Problems op Maximum and Minimum Values
14.1 Partial stationary values
14.2 Maximum and minimum values of a function of two or more variables
14.3 Examples of maximum and minimum values
14.4 Monopoly and joint production
14.5 Production, capital and interest
14.6 Relative maximum and minimiun values
14.7 Examples of relative maximum and minimum values
14.8 The demand for factors of production
14.9 The demand for consumers' goods and for loans
Examples XIV—General maximum and minimum problems ; economic maximum and minimum problems

XV. Integrals of Functions of One Variable
15.1 The definition of a definite integral
15.2 Definite integrals as areas -
15.3 Indefinite integrals and inverse differentiation
15.4 The technique of integration
15.5 Definite integral and approximate integration
15.C The relation b, tween average and marginal concepts
15.7 Capital values
15.8 A problem of durable capital goods
15.9 Average and dispersion of a frequency distribution
Examples XV—Integration ; integrals in economic problems

XVI. Differential Equations
16.1 The nature of the problem
16.2 Linear differential equations and their integration
16.3 The general integral of a linear differential equation
16.4 Simultaneous linear d'Me -ential equations
16.5 Orthogonal curve c. i surface systems
16.6 Other differential equations
16.7 Dynamic forms of demand and supply functions
16.8 The general theory of consumers' choice
Examples XVI—Differential equations ; economic applications of differential equations

XVII. Expansions, Taylor's Series and Higher Order Differentials -
17.1 Limits and infinite series
17.2 The expansion of a function of one variable (Taylor's series)
17.3 Examples of the expansion of functions
17.4 The expansion of a function of two or more variables
17.5 A complete criterion for maximum and minimum values -
17.6 Second and higher order differentials
17.7 Differentials of a function of two independent variables
17.8 Differentials of a function of two dependent variables
Examples XVII—Infinite series ; expansions ; higher order differentials

XVIII. Determinants, Linear Equations and Quadratic Forms
18.1 The general notion of a determinant
18.2 The definition of determinants of various orders
18.3 Properties of determinants
18.4 Minors and co-factors of determinants
18.5 Linear and homogeneous functions of several variables
18.6 The solution of linear equations
18.7 Quadratic forms in two and three variables
18.8 Examples of quadratic forms
18.9 Two general results for quadratic forms
Examples XVIII—Determinants ; linear equations ; quadratic forms

XIX. Further Problems of Maximum and Minimum Values
19.1 Maximum and minimum values of a function of several variables
19.2 Relative maximum and minimum values -
19.3 Examples of maxirhum and minimum values -
19.4 The stability of demand for factors of production
19.5 Partial elasticities of substitution . . .
19.6 Variation of demand for factors of production .
19.7 The demand for consumers' goods (integrability case)
19.8 Demands for three consumers' goods (general case)
Examples XIX—General maximum and minimum problems ; economic maximum and minimum problems

XX. Some Problems m the Calculus of Variations
20.1 The general theory of functionals
20.2 The calculus of variations
20.3 The method of the calculus of variations
20.4 Solution of the simplest problem
20.5 Special cases of Euler's equation
20.6 Examples of solution by Euler's equation
20.7 A dynamic problem of monopoly
20.8 Other problems in the calculus of variations
Examples XX—Problems in the calculus of variations

Index :
Mathematical Methods
Economic Applications
Authors