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

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