Engineering Mathematics

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 triple scalar product.- 14.12 The triple vector product.- 14.13 Differentiation of vectors.- 14.14 Workshop.- 14.15 Practical.- Summary.- Assignment.- Further exercises.- 15 Integration 1.- 15.1 The concept of integration.- 15.2 Rules for integration.- 15.3 Workshop.- 15.4 Integration by inspection.- 15.5 Integration by partial fractions.- 15.6 Integration by parts.- 15.7 Workshop.- 15.8 Practical.- Summary.- Assignment.- Further exercises.- 16 Integration 2.- 16.1 Integration by substitution.- 16.2 Workshop.- 16.3 Special substitutions.- 16.4 Integration using standard forms.- 16.5 Reduction formulas.- 16.6 Workshop.- 16.7 Practical.- Summary.- Assignment.- Further exercises.- 17 Integration 3.- 17.1 Definite integration.- 17.2 Improper integrals.- 17.3 Area under the curve.- 17.4 Volume of revolution.- Workshop.- 17.5 Length of a curve.- 17.6 Centres of mass.- 17.7 The theorems of Pappus.- 17.8 Moments of inertia.- 17.9 The perpendicular axis theorem.- 17.10 The parallel axis theorem.- 17.11 Average values.- 17.12 Radius of gyration.- 17.13 Practical.- Summary.- Assignment.- Further exercises.- 18 Numerical techniques.- 18.1 The solution of the equation f(x) = 0.- 18.2 Graphical methods.- 18.3 Iterative methods.- 18.4 The bisection method.- 18.5 The regula falsi method.- 18.6 The secant method.- 18.7 Newton's method.- 18.8 Workshop.- 18.9 Approximations to derivatives.- 18.10 Workshop.- 18.11 Numerical integration.- 18.12 The trapezoidal rule.- 18.13 Simpson's rule.- 18.14 Workshop.- 18.15 Practical.- Summary.- Assignment.- Further exercises.- 19 First-order differential equations.- 19.1 Terminology.- 19.2 Variables separable equations.- 19.3 Workshop.- 19.4 Linear equations.- 19.5 Workshop.- 19.6 Bernoulli's equation.- 19.7 Homogeneous equations.- 19.8 Workshop.- 19.9 Reducible equations.- 19.10 Boundary conditions.- 19.11 Practical.- Summary.- Assignment.- Further exercises.- 20 Second-order differential equations.- 20.1 Linear differential equations.- 20.2 The homogeneous case.- 20.3 Workshop.- 20.4 The non-homogeneous case.- 20.5 The particular solution.- 20.6 Workshop.- 20.7 The breakdown case.- 20.8 Workshop.- 20.9 Higher-order equations.- 20.10 Damping.- 20.11 Resonance.- 20.12 Transient and steady state.- 20.13 Practical.- Summary.- Assignment.- Further exercises.- 21 Difference equations.- 21.1 Variables.- 21.2 The homogeneous case.- 21.3 The auxiliary equation.- 21.4 Workshop.- 21.5 The non-homogeneous case.- 21.6 The particular solution.- 21.7 The breakdown case.- 21.8 Workshop.- 21.9 Practical.- Summary.- Assignment.- Further exercises.- 22 linear operators.- 22.1 Linear functions.- 22.2 Vector spaces.- 22.3 The addition rules.- 22.4 The scalar multiplication rules.- 22.5 Linear combinations.- 22.6 Linear dependence.- 22.7 Linear independence.- 22.8 Bases.- 22.9 The basis theorem.- 22.10 Transformations.- 22.11 The zero and identity operators.- 22.12 Linear operators.- 22.13 Algebraic rules for operators.- 22.14 Inverse operators.- 22.15 The operational method.- 22.16 Workshop.- 22.17 Practical.- Summary.- Assignment.- Further exercises.- 23 Descriptive statistics.- 23.1 Terminology.- 23.2 Random samples.- 23.3 Population statistics.- 23.4 Data.- 23.5 Pictorial representations.- 23.6 Pie charts.- 23.7 Bar charts.- 23.8 Histograms.- 23.9 Grouped data.- 23.10 Cumulative frequency diagrams.- 23.11 Measures of location and measures of spread.- 23.12 The mean.- 23.13 The mode.- 23.14 The median.- 23.15 The range.- 23.16 The mean absolute deviation.- 23.17 The standard deviation.- 23.18 Workshop.- 23.19 Practical.- Summary.- Assignment.- Further exercises.- 24 Probability.- 24.1 Concepts.- 24.2 The rules of probability.- 24.3 The sum rule.- 24.4 Mutually exclusive events.- 24.5 Conditional probability.- 24.6 The product rule.- 24.7 Independent events.- 24.8 Complementation rule.- 24.9 A random variable.- 24.10 The analytical method.- 24.11 The relative frequency method.- 24.12 The mathematical method.- 24.13 The binomial distribution.- 24.14 The mean of a probability distribution.- 24.15 The variance of a probability distribution.- 24.16 The Poisson distribution.- 24.17 Approximation for the binomial distribution.- 24.18 Continuous distributions.- 24.19 Mean and variance.- 24.20 The normal distribution.- 24.21 Discrete approximations.- 24.22 Continuity correction.- 24.23 Normal probability paper.- 24.24 Workshop.- 24.25 Practical.- Summary.- Assignment.- Further exercises.- Hints and solutions.- Symbols index.