Basic Mathematics for the Biological and Social Sciences

PrefaceAcknowledgments1. Basic Algebra 1.1. Symbols and Notation Factorials The Sigma Notation Functions 1.2. The Binomial Theorem An Application to Probability Theory 1.3. Partial Fractions Examples2. Graphical Methods 2.1. Introduction 2.2. The Graphical Presentation of Data 2.3. Special Types of Graph Changes of Scale The Histogram, or Block Diagram 2.4. Cartesian Coordinates in Two Dimensions 2.5. Features of Plane Curves Particular Points Symmetry Bounds and Impossible Areas Maxima, Minima and Inflexions Asymptotes 2.6. The Conic Sections 2.7. Polar Coordinates 2.8. Three-dimensional Problems 2.9. Three-dimensional Polar Coordinates Examples3. Trigonometrical or Circular Functions 3.1. Definitions 3.2. Properties of the Trigonometrical Functions 3.3. The Graphs of the Trigonometrical Functions 3.4. The Inverse Functions 3.5. Applications of Trigonometric Functions Examples4. Limits and Convergence 4.1. The Idea of a Limit 4.2. Definition of a Limit 4.3. Series and Convergence 4.4. Some Important Limits (i) A Theorem on Limits (ii) Rational Functions (iii) Rational Functions of x (iv) The Limits of (sin x)/x . . . and (1—cos x)/x 4.5. The Importance of Limits Examples5. Differentiation (1) 5.1. Introduction 5.2. The Derivative 5.3. Standard Derivatives 5.4. Rules for Differentiation 5.5. Higher Derivatives 5.6. Differentials 5.7. Maxima and Minima 5.8. Small Errors 5.9. Newton's Method of Approximation Examples6. The Exponential Function and Related Functions 6.1. Introduction 6.2. Definition and Properties of the Exponential Function 6.3. The Natural Logarithm 6.4. The Hyperbolic Functions 6.5. Growth Curves Examples7. Differentiation (2) 7.1. Taylor's and Maclaurin's Series 7.2. Functions of Several Variables 7.3. Partial Derivatives 7.4. Small Errors 7.5. Maxima and Minima 7.6. Taylor's Theorem 7.7. Partial Differential Equations 7.8. Change of Variables 7.9. Maximum Subject to Constraints 7.10. Implicit Functions Examples8. Integration 8.1. Introduction 8.2. Area and the Definite Integral 8.3. The Indefinite Integral 8.4. Improper Integrals 8.5. Techniques of Integration (i) Integration by Substitution (ii) Rational Functions and Partial Fractions (iii) Integration by Parts 8.6. Numerical Methods (i) The Trapezium Rule (ii) Simpson's Rule 8.7. Multiple Integrals 8.8. Miscellaneous Results (i) Differentiation of Integrals (ii) Stirling's Approximation (iii) Mean Values and Centers of Mass Examples9. Vectors 9.1. Scalar and Vector Quantities 9.2. A Digression on Mechanics 9.3. Vectors in One Dimension 9.4. Vectors and Their Components 9.5. Derivatives of Vectors; Gradients 9.6. The Product of Two Vectors 9.7. The Line-integral Examples10. Complex Numbers 10.1. Introduction 10.2. Elementary Manipulation of Complex Numbers Basic Rules Modulus and Argument The n-th Roots of a Complex Number Roots of Polynomials Complex Numbers and Vectors 10.3. Functions of Complex Variables The Exponential Function The Logarithmic Function Trigonometric and Hyperbolic Functions 10.4. Applications of Complex Algebra Differential Equations Alternating Current Electricity Examples11. Differential Equations 11.1. Introduction 11.2. Arbitrary Constants and Initial Conditions 11.3. A First-order Differential Equation 11.4. Linear Equations with Constant Coefficients 11.5. The Complementary Function 11.6. Damped Oscillations 11.7. Particular Integrals 11.8. Forced Oscillations and Resonance 11.9. Simple First-Order Equations (i) Variables Separable (ii) Exact Derivatives (iii) Linear Equations 11.10. Simultaneous Differential Equations 11.11. Numerical Methods ExamplesAppendix A. A Note on DefinitionsAppendix B. Infinite Series and ConvergenceAppendix C. Tables of the Exponential and Natural Logarithmic FunctionAnswers to ExamplesSuggestions for Further ReadingReferencesIndex