Efnisyfirlit

• Cover
• Half Title
• Dedications
• Title Page
• Copyright
• Contents
• Preface
• Acknowledgements
• 1. Matrices and vectors
• 1.1 Matrices◊
• 1.2 Exercises
• 1.3 Vectors in R2
• 1.4 Exercises
• 1.5 Vectors in R3
• 1.6 Lines
• 1.7 Planes
• 1.8 Exercises
• 1.9 Vectors in Rn
• 1.10 Flats
• 1.11 Exercises
• 1.12 Applications (optional)
• 1.12.1 Commodity bundles
• 1.12.2 Linear production models
• 1.12.3 Price vectors
• 1.12.4 Linear programming
• 1.12.5 Dual problem
• 1.12.6 Game theory
• 2. Functions of one variable
• 2.1 Intervals◊
• 2.2 Real valued functions of one real variable◊
• 2.3 Some elementary functions◊
• 2.3.1 Power functions
• 2.3.2 Exponential functions
• 2.3.3 Trigonometric functions
• 2.4 Combinations of functions◊
• 2.5 Inverse functions◊
• 2.6 Inverses of the elementary functions◊
• 2.6.1 Root functions
• 2.6.2 Exponential and logarithmic functions
• 2.7 Derivatives◊
• 2.8 Existence of derivatives◊
• 2.9 Derivatives of inverse functions◊
• 2.10 Calculation of derivatives
• 2.10.1 Derivatives of elementary functions and their inverses
• 2.10.2 Derivatives of combinations of functions◊
• 2.11 Exercises
• 2.12 Higher order derivatives
• 2.13 Taylor series for functions of one variable
• 2.14 Conic sections
• 2.15 Exercises
• 3. Functions of several variables
• 3.1 Real valued functions of two variables
• 3.1.1 Linear and affine functions
• 3.1.2 Quadric surfaces
• 3.2 Partial derivatives
• 3.3 Tangent plane
• 3.4 Gradient
• 3.5 Derivative
• 3.6 Directional derivatives
• 3.7 Exercises
• 3.8 Functions of more than two variables
• 3.8.1 Tangent hyperplanes
• 3.8.2 Directional derivatives
• 3.9 Exercises
• 3.10 Applications (optional)
• 3.10.1 Indifference curves
• 3.10.2 Profit maximisation
• 3.10.3 Contract curve
• 4. Stationary points
• 4.1 Stationary points for functions of one variable◊
• 4.2 Optimisation
• 4.3 Constrained optimisation
• 4.4 The use of computer systems
• 4.5 Exercises
• 4.6 Stationary points for functions of two variables
• 4.7 Gradient and stationary points
• 4.8 Stationary points for functions of more than two variables
• 4.9 Exercises
• 5. Vector functions
• 5.1 Vector valued functions
• 5.2 Affine functions and flats
• 5.3 Derivatives of vector functions
• 5.4 Manipulation of vector derivatives
• 5.5 Chain rule
• 5.6 Second derivatives
• 5.7 Taylor series for scalar valued functions of n variables
• 5.8 Exercises
• 6. Optimisation of scalar valued functions
• 6.1 Change of basis in quadratic forms◊
• 6.2 Positive and negative definite
• 6.3 Maxima and minima
• 6.4 Convex and concave functions
• 6.5 Exercises
• 6.6 Constrained optimisation
• 6.7 Constraints and gradients
• 6.8 Lagrange’s method – optimisation with one constraint
• 6.9 Lagrange’s method – general case♣
• 6.10 Constrained optimisation – analytic criteria♣
• 6.11 Exercises
• 6.12 Applications (optional)
• 6.12.1 The Nash bargaining problem
• 6.12.2 Inventory control
• 6.12.3 Least squares analysis
• 6.12.4 Kuhn–Tucker conditions
• 6.12.5 Linear programming
• 6.12.6 Saddle points
• 7. Inverse functions
• 7.1 Local inverses of scalar valued functions
• 7.1.1 Differentiability of local inverse functions
• 7.1.2 Inverse trigonometric functions
• 7.2 Local inverses of vector valued functions
• 7.3 Coordinate systems
• 7.4 Polar coordinates
• 7.5 Differential operators♣
• 7.6 Exercises
• 7.7 Application (optional): contract curve
• 8. Implicit functions
• 8.1 Implicit differentiation
• 8.2 Implicit functions
• 8.3 Implicit function theorem
• 8.4 Exercises
• 8.5 Application (optional): shadow prices
• 9. Differentials
• 9.1 Matrix algebra and linear systems◊
• 9.2 Differentials
• 9.3 Stationary points
• 9.4 Small changes
• 9.5 Exercises
• 9.6 Application (optional): Slutsky equations
• 10. Sums and integrals
• 10.1 Sums◊
• 10.2 Integrals◊
• 10.3 Fundamental theorem of calculus◊
• 10.4 Notation◊
• 10.5 Standard integrals◊
• 10.6 Partial fractions◊
• 10.7 Completing the square◊
• 10.8 Change of variable◊
• 10.9 Integration by parts◊
• 10.10 Exercises
• 10.11 Infinite sums and integrals♣
• 10.12 Dominated convergence♣
• 10.13 Differentiating integrals♣
• 10.14 Power series♣
• 10.15 Exercises
• 10.16 Applications (optional)
• 10.16.1 Probability
• 10.16.2 Probability density functions
• 10.16.3 Binomial distribution
• 10.16.4 Poisson distribution
• 10.16.5 Mean
• 10.16.6 Variance
• 10.16.7 Standardised random variables
• 10.16.8 Normal distribution
• 10.16.9 Sums of random variables
• 10.16.10 Cauchy distribution
• 10.16.11 Auctions
• 11. Multiple integrals
• 11.1 Introduction
• 11.2 Repeated integrals
• 11.3 Change of variable in multiple integrals♣
• 11.4 Unbounded regions of integration♣
• 11.5 Multiple sums and series♣
• 11.6 Exercises
• 11.7 Applications (optional)
• 11.7.1 Joint probability distributions
• 11.7.2 Marginal probability distributions
• 11.7.3 Expectation, variance and covariance
• 11.7.4 Independent random variables
• 11.7.5 Generating functions
• 11.7.6 Multivariate normal distributions
• 12. Differential equations of order one
• 12.1 Differential equations
• 12.2 General solutions of ordinary equations
• 12.3 Boundary conditions
• 12.4 Separable equations
• 12.5 Exact equations
• 12.6 Linear equations of order one
• 12.7 Homogeneous equations
• 12.8 Change of variable
• 12.9 Identifying the type of first order equation
• 12.10 Partial differential equations
• 12.11 Exact equations and partial differential equations
• 12.12 Change of variable in partial differential equations
• 12.13 Exercises
• 13. Complex numbers
• 13.1 Quadratic equations
• 13.2 Complex numbers
• 13.3 Modulus and argument
• 13.4 Exercises
• 13.5 Complex roots
• 13.6 Polynomials
• 13.7 Elementary functions♣
• 13.8 Exercises
• 13.9 Applications (optional)
• 13.9.1 Characteristic functions
• 13.9.2 Central limit theorem
• 14. Linear differential and difference equations
• 14.1 The operator P(D)
• 14.2 Difference equations and the shift operator E
• 14.3 Linear operators♣
• 14.4 Homogeneous, linear, differential equations♣
• 14.5 Complex roots of the auxiliary equation
• 14.6 Homogeneous, linear, difference equations
• 14.7 Nonhomogeneous equations♣
• 14.7.1 Nonhomogeneous differential equations
• 14.7.2 Nonhomogeneous difference equations
• 14.8 Convergence and divergence♣
• 14.9 Systems of linear equations♣
• 14.10 Change of variable♣
• 14.11 Exercises
• 14.12 The difference operator (optional)♣
• 14.13 Exercises
• 14.14 Applications (optional)
• 14.14.1 Cobweb models
• 14.14.2 Gambler’s ruin
• Answers to starred exercises with some hints and solutions
• Appendix
• Index