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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 | |