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### Summary

### Summary

This paperback edition is not available in the U.S. and Canada. This book offers a comprehensive presentation of the mathematics required to tackle problems in economic analysis. To give a better understanding of the mathematical concepts, the text follows the logic of the development of mathematics rather than that of an economics course. After a review of the fundamentals of sets, numbers, and functions, the book covers limits and continuity, the calculus of functions of one variable, linear algebra, multivariate calculus, and dynamics. To develop the student's problem-solving skills, the book works through a large number of examples and economic applications. The second edition includes simple game theory, l'Hôpital's rule, Leibniz's rule, and a more intuitive development of the Hamiltonian. An instructor's manual is available.

### Table of Contents

Preface | p. xiii |

I Introduction and Fundamentals | |

1 Introduction | p. 3 |

1.1 What Is an Economic Model? | p. 3 |

1.2 How to Use This Book | p. 8 |

1.3 Conclusion | p. 9 |

2 Review of Fundamentals | p. 11 |

2.1 Sets and Subsets | p. 11 |

2.2 Numbers | p. 23 |

2.3 Some Properties of Point Sets in Rn | p. 33 |

2.4 Functions | p. 43 |

2.5 Proof, Necessary and Sufficient Conditions * | p. 60 |

3 Sequences, Series, and Limits | p. 67 |

3.1 Definition of a Sequence | p. 67 |

3.2 Limit of a Sequence | p. 70 |

3.3 Present-Value Calculations | p. 75 |

3.4 Properties of Sequences | p. 84 |

3.5 Series | p. 89 |

II Univariate Calculus and Optimization | |

4 Continuity of Functions | p. 115 |

4.1 Continuity of a Function of One Variable | p. 115 |

4.2 Economic Applications of Continuous and Discontinuous Functions | p. 125 |

4.3 Intermediate-Value Theorem | p. 143 |

5 The Derivative and Differential for Functions of One Variable | p. 155 |

5.1 Definition of a Tangent Line | p. 155 |

5.2 Definition of the Derivative and the Differential | p. 162 |

5.3 Conditions for Differentiability | p. 169 |

5.4 Rules of Differentiation | p. 175 |

5.5 Higher-Order Derivatives: Concavity and Convexity of a Function | p. 208 |

5.6 Taylor Series Formula and the Mean-Value Theorem | p. 218 |

6 Optimization of Functions of One Variable | p. 227 |

6.1 Necessary Conditions for Unconstrained Maxima and Minima | p. 227 |

6.2 Second-Order Conditions | p. 253 |

6.3 Optimization over an Interval | p. 265 |

III Linear Algebra | |

7 Systems of Linear Equations | p. 279 |

7.1 Solving Systems of Linear Equations | p. 279 |

7.2 Linear Systems in n-Variables | p. 293 |

8 Matrices | p. 317 |

8.1 General Notation | p. 317 |

8.2 Basic Matrix Operations | p. 324 |

8.3 Matrix Transposition | p. 340 |

8.4 Some Special Matrices | p. 345 |

9 Determinants and the Inverse Matrix | p. 353 |

9.1 Defining the Inverse | p. 353 |

9.2 Obtaining the Determinant and Inverse of a 3 x 3 Matrix | p. 370 |

9.3 The Inverse of an n x n Matrix and Its Properties | p. 376 |

9.4 Cramer's Rule | p. 386 |

10 Some Advanced Topics in Linear Algebra * | p. 405 |

10.1 Vector Spaces | p. 405 |

10.2 The Eigenvalue Problem | p. 421 |

10.3 Quadratic Forms | p. 436 |

IV Multivariate Calculus | |

11 Calculus for Functions of n-Variables | p. 455 |

11.1 Partial Differentiation | p. 455 |

11.2 Second-Order Partial Derivatives | p. 469 |

11.3 The First-Order Total Differential | p. 477 |

11.4 Curvature Properties: Concavity and Convexity | p. 498 |

11.5 More Properties of Functions with Economic Applications | p. 513 |

11.6 Taylor Series Expansion * | p. 534 |

12 Optimization of Functions of n-Variables | p. 545 |

12.1 First-Order Conditions | p. 545 |

12.2 Second-Order Conditions | p. 560 |

12.3 Direct Restrictions on Variables | p. 569 |

13 Constrained Optimization | p. 585 |

13.1 Constrained Problems and Approaches to Solutions | p. 585 |

13.2 Second-Order Conditions for Constrained Optimization | p. 616 |

13.3 Existence, Uniqueness, and Characterization of Solutions | p. 622 |

14 Comparative Statics | p. 631 |

14.1 Introduction to Comparative Statics | p. 631 |

14.2 General Comparative-Statics Analysis | p. 643 |

14.3 The Envelope Theorem | p. 660 |

15 Concave Programming and the Kuhn-Tucker Conditions | p. 677 |

15.1 The Concave-Programming Problem | p. 677 |

15.2 Many Variables and Constraints | p. 686 |

V Integration and Dynamic Methods | |

16 Integration | p. 701 |

16.1 The Indefinite Integral | p. 701 |

16.2 The Riemann (Definite) Integral | p. 709 |

16.3 Properties of Integrals | p. 721 |

16.4 Improper Integrals | p. 733 |

16.5 Techniques of Integration | p. 742 |

17 An Introduction to Mathematics for Economic Dynamics | p. 753 |

17.1 Modeling Time | p. 754 |

18 Linear, First-Order Difference Equations | p. 763 |

18.1 Linear, First-Order, Autonomous Difference Equations | p. 763 |

18.2 The General, Linear, First-Order Difference Equation | p. 780 |

19 Nonlinear, First-Order Difference Equations | p. 789 |

19.1 The Phase Diagram and Qualitative Analysis | p. 789 |

19.2 Cycles and Chaos | p. 799 |

20 Linear, Second-Order Difference Equations | p. 811 |

20.1 The Linear, Autonomous, Second-Order Difference Equation | p. 811 |

20.2 The Linear, Second-Order Difference Equation with a Variable Term | p. 838 |

21 Linear, First-Order Differential Equations | p. 849 |

21.1 Autonomous Equations | p. 849 |

21.2 Nonautonomous Equations | p. 870 |

22 Nonlinear, First-Order Differential Equations | p. 879 |

22.1 Autonomous Equations and Qualitative Analysis | p. 879 |

22.2 Two Special Forms of Nonlinear, First-Order Differential Equations | p. 888 |

23 Linear, Second-Order Differential Equations | p. 897 |

23.1 The Linear, Autonomous, Second-Order Differential Equation | p. 897 |

23.2 The Linear, Second-Order Differential Equation with a Variable Term | p. 919 |

24 Simultaneous Systems of Differential and Difference Equations | p. 929 |

24.1 Linear Differential Equation Systems | p. 929 |

24.2 Stability Analysis and Linear Phase Diagrams | p. 951 |

24.3 Systems of Linear Difference Equations | p. 976 |

25 Optimal Control Theory | p. 999 |

25.1 The Maximum Principle | p. 1002 |

25.2 Optimization Problems Involving Discounting | p. 1014 |

25.3 Alternative Boundary Conditions on x(T) | p. 1026 |

25.4 Infinite-Time Horizon Problems | p. 1040 |

25.5 Constraints on the Control Variable | p. 1053 |

25.6 Free-Terminal-Time Problems (T Free) | p. 1063 |

Appendix: Complex Numbers and Circular Functions | p. 1081 |

Answers | p. 1091 |

Index | p. 1123 |