Engineering Mathematics

The second edition differs from the first in three respects. First, the format is different. Wide margins are now provided so that readers can pencil in small individual notes and comments which may be of assistance to them later on. Second, each chapter has been provided with extra exercises. Generally these are of the more routine variety and have been incorporated hefore the assignment. All the exercises are supplied with answers which are located at the end of the book. Third, some marginal diagrams and ref erences have heen included to help illuminate the material and occasionally to indicate where a topic fits into the overall scheme. It is hoped that students will find in the new edition plenty to sustain the development of their mathematical knowledge and skills. The author thanks all those who have contrihuted to the production of this book. eWE Preface to the first edition Students reading for degrees and diplomas in Engineering and Applied Science arrive with a wide variety of mathematical backgrounds. Neverthe less by the end of the first year of study all of them must have achieved a minimum standard in mathematics and also have acquired sufficient skill to enable them to cope with the more advanced mathematical topics in the second year. Experience has shown that many students are unable to cope with the traditional mathematics textbooks because they find them remote and the concepts difficult to handle.

1 Numbers and logarithms.- 1.1 Numbers.- 1.2 Real numbers.- 1.3 Workshop.- 1.4 The real number system.- 1.5 Indices and logarithms.- 1.6 The binomial theorem.- 1.7 Workshop.- 1.8 Practical.- Summary.- Assignment.- Further exercises.- 2 Sets and functions.- 2.1 Set notation.- 2.2 Real intervals.- 2.3 Workshop.- 2.4 Functions.- 2.5 Workshop.- 2.6 Identities and equations.- 2.7 Polynomials and rational functions.- 2.8 Partial fractions.- 2.9 Workshop.- 2.10 Practical.- Summary.- Assignment.- Further exercises.- 3 Trigonometry and geometry.- 3.1 Coordinate systems.- 3.2 Circular functions.- 3.3 Trigonometrical identities.- 3.4 The form a cos ? + b sin ?.- 3.5 Solutions of equations.- 3.6 Workshop.- 3.7 Coordinate geometry.- 3.8 The straight line.- 3.9 The circle.- 3.10 The conic sections.- 3.11 Workshop.- 3.12 Practical.- Summary.- Assignment.- Further exercises.- 4 Limits, continuity and differentiation.- 4.1 Limits.- 4.2 The laws of limits.- 4.3 Workshop.- 4.4 Right and left limits.- 4.5 Continuity.- 4.6 Differentiability.- 4.7 Leibniz's theorem.- 4.8 Techniques of differentiation.- 4.9 Workshop.- 4.10 Logarithmic differentiation.- 4.11 Implicit differentiation.- 4.12 Parametric differentiation.- 4.13 Rates of change.- 4.14 Workshop.- 4.15 Practical.- Summary.- Assignment.- Further exercises.- 5 Hyperbolic functions.- 5.1 Definitions and identities.- 5.2 Differentiation of hyperbolic functions.- 5.3 Curve sketching.- 5.4 Workshop.- 5.5 Injective functions.- 5.6 Surjective functions.- 5.7 Bijective functions.- 5.8 Pseudo-inverse functions.- 5.9 Differentiation of inverse functions.- 5.10 The inverse circular functions.- 5.11 Workshop.- 5.12 The inverse hyperbolic functions.- 5.13 Practical.- Summary.- Assignment.- Further exercises.- 6 Further differentiation.- 6.1 Tangents and normals.- 6.2 Workshop.- 6.3 Intrinsic coordinates.- 6.4 The catenary.- 6.5 Curvature.- 6.6 Workshop.- 6.7 Practical.- Summary.- Assignment.- Further exercises.- 7 Partial differentiation.- 7.1 Functions.- 7.2 Continuity.- 7.3 Partial derivatives.- 7.4 Higher-order derivatives.- 7.5 Workshop.- 7.6 The formulas for a change of variables: the chain rule.- 7.7 The total differential.- 7.8 Workshop.- 7.9 Practical.- Summary.- Assignment.- Further exercises.- 8 Series expansions and their uses.- 8.1 The mean value property.- 8.2 Taylor's theorem.- 8.3 Workshop.- 8.4 L'Hospital's rule.- 8.5 Workshop.- 8.6 Maxima and minima.- 8.7 Workshop.- 8.8 Practical.- Summary.- Assignment.- Further exercises.- 9 Infinite series.- 9.1 Series.- 9.2 Convergence and divergence.- 9.3 Tests for convergence and divergence.- 9.4 Power series.- 9.5 Workshop.- 9.6 Practical.- Summary.- Assignment.- Further exercises.- 10 Complex numbers.- 10.1 Genesis.- 10.2 The complex plane: Argand diagram.- 10.3 Vectorial representation.- 10.4 Further properties of the conjugate.- 10.5 De Moivre's theorem.- 10.6 Workshop.- 10.7 The nth roots of a complex number.- 10.8 Power series.- 10.9 Workshop.- 10.10 Practical.- Summary.- Assignment.- Further exercises.- 11 Matrices.- 11.1 Notation.- 11.2 Matrix algebra.- 11.3 Workshop.- 11.4 Matrix equations.- 11.5 Zero, identity and inverse matrices.- 11.6 Algebraic rules.- 11.7 Practical.- Summary.- Assignment.- Further exercises.- 12 Determinants.- 12.1 Notation.- 12.2 Cramer's rule.- 12.3 Higher-order determinants.- 12.4 Rules for determinants.- 12.5 Workshop.- 12.6 Practical.- Summary.- Assignment.- Further exercises.- 13 Inverse matrices.- 13.1 The inverse of a square matrix.- 13.2 Row transformations.- 13.3 Obtaining inverses.- 13.4 Systematic elimination.- 13.5 Workshop.- 13.6 Pivoting.- 13.7 Practical.- 13.8 Concluding remarks.- Summary.- Assignment.- Further exercises.- 14 Vectors.- 14.1 Descriptions.- 14.2 Vector addition.- 14.3 Scalar multiplication.- 14.4 Components.- 14.5 The scalar product.- 14.6 Direction ratios and direction cosines.- 14.7 Applications.- 14.8 Algebraic properties.- 14.9 The vector product.- 14.10 Workshop.- 14.11 The tripl