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| Preface | |
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| Topics in Linear Algebra | |
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| Review of Basic Linear Algebra | |
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| Linear Independence | |
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| The Rank of a Matrix | |
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| Main Results on Linear Systems | |
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| Eigenvalues | |
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| Diagonalization | |
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| Quadratic Forms | |
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| Quadratic Forms with Linear Constraints | |
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| Partitioned Matrices and Their Inverses | |
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| Multivariable Calculus | |
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| Gradients and Directional Derivatives | |
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| Convex Sets | |
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| Concave and Convex Functions I | |
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| Concave and Convex Functions II | |
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| Quasiconcave and Quasiconvex Functions | |
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| Taylor's Formula | |
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| Implicit and Inverse Function Theorems | |
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| Degrees of Freedom and Functional Dependence | |
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| Differentiability | |
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| Existence and Uniqueness of Solutions of Systems of Equations | |
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| Static Optimization | |
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| Extreme Points | |
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| Local Extreme Points | |
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| Equality Constraints: The Lagrange Problem | |
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| Local Second-Order Conditions | |
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| Inequality Constraints: Nonlinear Programming | |
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| Sufficient Conditions | |
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| Comparative Statics | |
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| Nonnegativity Constraints | |
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| Concave Programming | |
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| Precise Comparative Statics Results | |
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| Existence of Lagrange Multipliers | |
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| Topics in Integration | |
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| Review of One-Variable Integration | |
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| Leibniz's Formula | |
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| The Gamma Function | |
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| Multiple Integrals over Product Domains | |
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| Double Integrals over General Domains | |
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| The Multiple Riemann Integral | |
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| Change of Variables | |
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| Generalized Double Integrals | |
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| Differential Equations I: First-Order Equations in One Variable | |
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| Introduction | |
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| The Direction is Given: Find the Path! | |
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| Separable Equations | |
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| First-Order Linear Equations | |
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| Exact Equations and Integrating Factors | |
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| Transformation of Variables | |
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| Qualitative Theory and Stability | |
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| Existence and Uniqueness | |
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| Differential Equations II: Second-Order Equations and Systems in the Plane | |
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| Introduction | |
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| Linear Differential Equations | |
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| Constant Coefficients | |
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| Stability for Linear Equations | |
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| Simultaneous Equations in the Plane | |
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| Equilibrium Points for Linear Systems | |
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| Phase Plane Analysis | |
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| Stability for Nonlinear Systems | |
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| Saddle Points | |
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| Differential Equations III: Higher-Order Equations | |
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| Linear Differential Equations | |
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| The Constant Coefficients Case | |
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| Stability of Linear Differential Equations | |
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| Systems of Differential Equations | |
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| Stability for Nonlinear Systems | |
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| Qualitative Theory | |
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| A Glimpse at Partial Differential Equations | |
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| Calculus of Variations | |
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| The Simplest Problem | |
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| The Euler Equation | |
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| Why the Euler Equation is Necessary | |
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| Optimal Savings | |
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| More General Terminal Conditions | |
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| Control Theory: Basic Techniques | |
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| The Basic Problem | |
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| A Simple Case | |
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| Regularity Conditions | |
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| The Standard Problem | |
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| The Maximum Principle and the Calculus of Variations | |
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| Adjoint Variables as Shadow Prices | |
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| Sufficient Conditions | |
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| Variable Final Time | |
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| Current Value Formulations | |
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| Scrap Values | |
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| Infinite Horizon | |
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| Phase Diagrams | |
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| Control Theory with Many Variables | |
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| Several Control and State Variables | |
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| Some Examples | |
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| Infinite Horizon | |
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| Existence Theorems and Sensitivity | |
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| A Heuristic Proof of the Maximum Principle | |
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| Mixed Constraints | |
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| Pure State Constraints | |
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| Generalizations | |
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| Difference Equations | |
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| First-Order Difference Equations | |
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| Economic Applications | |
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| Second-Order Difference Equations | |
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| Linear Equations with Constant Coefficients | |
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| Higher-Order Equations | |
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| Systems of Difference Equations | |
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| Stability of Nonlinear Difference Equations | |
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| Discrete Time Optimization | |
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| Dynamic Programming | |
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| The Euler Equation | |
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| Infinite Horizon | |
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| The Maximum Principle | |
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| More Variables | |
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| Stochastic Optimization | |
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| Infinite Horizon Stationary Problems | |
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| Topology and Separation | |
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| Point Set Topology in R<sup>n</sup> | |
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| Topology and Convergence | |
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| Continuous Functions | |
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| Maximum Theorems | |
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| Convex Sets | |
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| Separation Theorems | |
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| Productive Economies and Frobenius's Theorem | |
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| Correspondences and Fixed Points | |
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| Correspondences | |
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| A General Maximum Theorem | |
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| Fixed Points for Contraction Mappings | |
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| Brouwer's and Kakutani's Fixed Point Theorems | |
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| Equilibrium in a Pure Exchange Economy | |
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| Sets, Completeness, and Covergence | |
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| Sets and Functions | |
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| Least Upper Bound Principle | |
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| Sequences of Real Numbers | |
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| Infimum and Supremum of Functions | |
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| Trigonometric Functions | |
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| Basic Definitions and Results | |
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| Differentiating Trigonometric Functions | |
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| Complex Numbers | |
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| Answers | |
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| References | |
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| Index | |