| Preface | p. xi |
| Acknowledgements | p. xiv |
| Matrices and vectors | p. 1 |
| Matrices | p. 1 |
| Exercises | p. 11 |
| Vectors in R[superscript 2] | p. 14 |
| Exercises | p. 20 |
| Vectors in R[superscript 3] | p. 21 |
| Lines | p. 24 |
| Planes | p. 27 |
| Exercises | p. 35 |
| Vectors in R[superscript n] | p. 38 |
| Flats | p. 40 |
| Exercises | p. 43 |
| Applications (optional) | p. 45 |
| Commodity bundles | p. 45 |
| Linear production models | p. 46 |
| Price vectors | p. 46 |
| Linear programming | p. 47 |
| Dual problem | p. 48 |
| Game theory | p. 49 |
| Functions of one variable | p. 51 |
| Intervals | p. 51 |
| Real valued functions of one real variable | p. 51 |
| Some elementary functions | p. 52 |
| Power functions | p. 52 |
| Exponential functions | p. 53 |
| Trigonometric functions | p. 53 |
| Combinations of functions | p. 55 |
| Inverse functions | p. 57 |
| Inverses of the elementary functions | p. 59 |
| Root functions | p. 59 |
| Exponential and logarithmic functions | p. 60 |
| Derivatives | p. 61 |
| Existence of derivatives | p. 63 |
| Derivatives of inverse functions | p. 64 |
| Calculation of derivatives | p. 65 |
| Derivatives of elementary functions and their inverses | p. 65 |
| Derivatives of combinations of functions | p. 66 |
| Exercises | p. 69 |
| Higher order derivatives | p. 71 |
| Taylor series for functions of one variable | p. 72 |
| Conic sections | p. 76 |
| Exercises | p. 80 |
| Functions of several variables | p. 87 |
| Real valued functions of two variables | p. 87 |
| Linear and affine functions | p. 88 |
| Quadric surfaces | p. 89 |
| Partial derivatives | p. 92 |
| Tangent plane | p. 96 |
| Gradient | p. 98 |
| Derivative | p. 100 |
| Directional derivatives | p. 100 |
| Exercises | p. 106 |
| Functions of more than two variables | p. 110 |
| Tangent hyperplanes | p. 111 |
| Directional derivatives | p. 114 |
| Exercises | p. 115 |
| Applications (optional) | p. 120 |
| Indifference curves | p. 120 |
| Profit maximisation | p. 122 |
| Contract curve | p. 122 |
| Stationary points | p. 125 |
| Stationary points for functions of one variable | p. 125 |
| Optimisation | p. 128 |
| Constrained optimisation | p. 130 |
| The use of computer systems | p. 132 |
| Exercises | p. 135 |
| Stationary points for functions of two variables | p. 139 |
| Gradient and stationary points | p. 142 |
| Stationary points for functions of more than two variables | p. 143 |
| Exercises | p. 145 |
| Vector functions | p. 149 |
| Vector valued functions | p. 149 |
| Affine functions and flats | p. 153 |
| Derivatives of vector functions | p. 155 |
| Manipulation of vector derivatives | p. 162 |
| Chain rule | p. 163 |
| Second derivatives | p. 168 |
| Taylor series for scalar valued functions of n variables | p. 170 |
| Exercises | p. 174 |
| Optimisation of scalar valued functions | p. 179 |
| Change of basis in quadratic forms | p. 179 |
| Positive and negative definite | p. 185 |
| Maxima and minima | p. 188 |
| Convex and concave functions | p. 193 |
| Exercises | p. 200 |
| Constrained optimisation | p. 203 |
| Constraints and gradients | p. 208 |
| Lagrange's method--optimisation with one constraint | p. 210 |
| Lagrange's method--general case | p. 217 |
| Constrained optimisation--analytic criteria | p. 219 |
| Exercises | p. 220 |
| Applications (optional) | p. 223 |
| The Nash bargaining problem | p. 223 |
| Inventory control | p. 224 |
| Least squares analysis | p. 227 |
| Kuhn--Tucker conditions | p. 229 |
| Linear programming | p. 231 |
| Saddle points | p. 233 |
| Inverse functions | p. 235 |
| Local inverses of scalar valued functions | p. 235 |
| Differentiability of local inverse functions | p. 237 |
| Inverse trigonometric functions | p. 238 |
| Local inverses of vector valued functions | p. 241 |
| Coordinate systems | p. 247 |
| Polar coordinates | p. 254 |
| Differential operators | p. 256 |
| Exercises | p. 259 |
| Application (optional): contract curve | p. 263 |
| Implicit functions | p. 265 |
| Implicit differentiation | p. 265 |
| Implicit functions | p. 266 |
| Implicit function theorem | p. 268 |
| Exercises | p. 275 |
| Application (optional): shadow prices | p. 277 |
| Differentials | p. 281 |
| Matrix algebra and linear systems | p. 281 |
| Differentials | p. 282 |
| Stationary points | p. 287 |
| Small changes | p. 289 |
| Exercises | p. 290 |
| Application (optional): Slutsky equations | p. 292 |
| Sums and integrals | p. 295 |
| Sums | p. 295 |
| Integrals | p. 297 |
| Fundamental theorem of calculus | p. 298 |
| Notation | p. 300 |
| Standard integrals | p. 302 |
| Partial fractions | p. 304 |
| Completing the square | p. 308 |
| Change of variable | p. 309 |
| Integration by parts | p. 311 |
| Exercises | p. 313 |
| Infinite sums and integrals | p. 316 |
| Dominated convergence | p. 320 |
| Differentiating integrals | p. 323 |
| Power series | p. 325 |
| Exercises | p. 326 |
| Applications (optional) | p. 328 |
| Probability | p. 328 |
| Probability density functions | p. 329 |
| Binomial distribution | p. 330 |
| Poisson distribution | p. 331 |
| Mean | p. 333 |
| Variance | p. 334 |
| Standardised random variables | p. 335 |
| Normal distribution | p. 336 |
| Sums of random variables | p. 339 |
| Cauchy distribution | p. 341 |
| Auctions | p. 341 |
| Multiple integrals | p. 345 |
| Introduction | p. 345 |
| Repeated integrals | p. 347 |
| Change of variable in multiple integrals | p. 353 |
| Unbounded regions of integration | p. 360 |
| Multiple sums and series | p. 361 |
| Exercises | p. 363 |
| Applications (optional) | p. 366 |
| Joint probability distributions | p. 366 |
| Marginal probability distributions | p. 366 |
| Expectation, variance and covariance | p. 367 |
| Independent random variables | p. 368 |
| Generating functions | p. 369 |
| Multivariate normal distributions | p. 370 |
| Differential equations of order one | p. 373 |
| Differential equations | p. 373 |
| General solutions of ordinary equations | p. 374 |
| Boundary conditions | p. 375 |
| Separable equations | p. 375 |
| Exact equations | p. 382 |
| Linear equations of order one | p. 384 |
| Homogeneous equations | p. 387 |
| Change of variable | p. 389 |
| Identifying the type of first order equation | p. 392 |
| Partial differential equations | p. 393 |
| Exact equations and partial differential equations | p. 395 |
| Change of variable in partial differential equations | p. 397 |
| Exercises | p. 399 |
| Complex numbers | p. 405 |
| Quadratic equations | p. 405 |
| Complex numbers | p. 406 |
| Modulus and argument | p. 407 |
| Exercises | p. 409 |
| Complex roots | p. 410 |
| Polynomials | p. 412 |
| Elementary functions | p. 414 |
| Exercises | p. 416 |
| Applications (optional) | p. 417 |
| Characteristic functions | p. 417 |
| Central limit theorem | p. 419 |
| Linear differential and difference equations | p. 421 |
| The operator P (D) | p. 421 |
| Difference equations and the shift operator E | p. 423 |
| Linear operators | p. 425 |
| Homogeneous, linear, differential equations | p. 426 |
| Complex roots of the auxiliary equation | p. 429 |
| Homogeneous, linear, difference equations | p. 431 |
| Nonhomogeneous equations | p. 434 |
| Nonhomogeneous differential equations | p. 435 |
| Nonhomogeneous difference equations | p. 441 |
| Convergence and divergence | p. 444 |
| Systems of linear equations | p. 447 |
| Change of variable | p. 453 |
| Exercises | p. 454 |
| The difference operator (optional) | p. 458 |
| Exercises | p. 461 |
| Applications (optional) | p. 462 |
| Cobweb models | p. 462 |
| Gambler's ruin | p. 467 |
| Answers to starred exercises with some hints and solutions | p. 471 |
| Appendix | p. 549 |
| Index | p. 550 |
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