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Foundation Mathematics for the Physical Sciences

Book Description

This tutorial-style textbook develops the basic mathematical tools needed by first and second year undergraduates to solve problems in the physical sciences. Students gain hands-on experience through hundreds of worked examples, self-test questions and homework problems. Each chapter includes a summary of the main results, definitions and formulae. Over 270 worked examples show how to put the tools into practice. Around 170 self-test questions in the footnotes and 300 end-of-section exercises give students an instant check of their understanding. More than 450 end-of-chapter problems allow students to put what they have just learned into practice. Hints and outline answers to the odd-numbered problems are given at the end of each chapter. Complete solutions to these problems can be found in the accompanying Student Solutions Manual. Fully-worked solutions to all problems, password-protected for instructors, are available at www.cambridge.org/foundation.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Preface
  7. 1. Arithmetic and geometry
    1. 1.1 Powers
    2. 1.2 Exponential and logarithmic functions
    3. 1.3 Physical dimensions
    4. 1.4 The binomial expansion
    5. 1.5 Trigonometric identities
    6. 1.6 Inequalities
    7. Summary
    8. Problems
    9. Hints and answers
  8. 2. Preliminary algebra
    1. 2.1 Polynomials and polynomial equations
    2. 2.2 Coordinate geometry
    3. 2.3 Partial fractions
    4. 2.4 Some particular methods of proof
    5. Summary
    6. Problems
    7. Hints and answers
  9. 3. Differential calculus
    1. 3.1 Differentiation
    2. 3.2 Leibnitz’s theorem
    3. 3.3 Special points of a function
    4. 3.4 Curvature of a function
    5. 3.5 Theorems of differentiation
    6. 3.6 Graphs
    7. Summary
    8. Problems
    9. Hints and answers
  10. 4. Integral calculus
    1. 4.1 Integration
    2. 4.2 Integration methods
    3. 4.3 Integration by parts
    4. 4.4 Reduction formulae
    5. 4.5 Infinite and improper integrals
    6. 4.6 Integration in plane polar coordinates
    7. 4.7 Integral inequalities
    8. 4.8 Applications of integration
    9. Summary
    10. Problems
    11. Hints and answers
  11. 5. Complex numbers and hyperbolic functions
    1. 5.1 The need for complex numbers
    2. 5.2 Manipulation of complex numbers
    3. 5.3 Polar representation of complex numbers
    4. 5.4 De Moivre’s theorem
    5. 5.5 Complex logarithms and complex powers
    6. 5.6 Applications to differentiation and integration
    7. 5.7 Hyperbolic functions
    8. Summary
    9. Problems
    10. Hints and answers
  12. 6. Series and limits
    1. 6.1 Series
    2. 6.2 Summation of series
    3. 6.3 Convergence of infinite series
    4. 6.4 Operations with series
    5. 6.5 Power series
    6. 6.6 Taylor series
    7. 6.7 Evaluation of limits
    8. Summary
    9. Problems
    10. Hints and answers
  13. 7. Partial differentiation
    1. 7.1 Definition of the partial derivative
    2. 7.2 The total differential and total derivative
    3. 7.3 Exact and inexact differentials
    4. 7.4 Useful theorems of partial differentiation
    5. 7.5 The chain rule
    6. 7.6 Change of variables
    7. 7.7 Taylor’s theorem for many-variable functions
    8. 7.8 Stationary values of two-variable functions
    9. 7.9 Stationary values under constraints
    10. 7.10 Envelopes
    11. 7.11 Thermodynamic relations
    12. 7.12 Differentiation of integrals
    13. Summary
    14. Problems
    15. Hints and answers
  14. 8. Multiple integrals
    1. 8.1 Double integrals
    2. 8.2 Applications of multiple integrals
    3. 8.3 Change of variables in multiple integrals
    4. Summary
    5. Problems
    6. Hints and answers
  15. 9. Vector algebra
    1. 9.1 Scalars and vectors
    2. 9.2 Addition, subtraction and multiplication of vectors
    3. 9.3 Basis vectors, components and magnitudes
    4. 9.4 Multiplication of two vectors
    5. 9.5 Triple products
    6. 9.6 Equations of lines, planes and spheres
    7. 9.7 Using vectors to find distances
    8. 9.8 Reciprocal vectors
    9. Summary
    10. Problems
    11. Hints and answers
  16. 10. Matrices and vector spaces
    1. 10.1 Vector spaces
    2. 10.2 Linear operators
    3. 10.3 Matrices
    4. 10.4 Basic matrix algebra
    5. 10.5 The transpose and conjugates of a matrix
    6. 10.6 The trace of a matrix
    7. 10.7 The determinant of a matrix
    8. 10.8 The inverse of a matrix
    9. 10.9 The rank of a matrix
    10. 10.10 Simultaneous linear equations
    11. 10.11 Special types of square matrix
    12. 10.12 Eigenvectors and eigenvalues
    13. 10.13 Determination of eigenvalues and eigenvectors
    14. 10.14 Change of basis and similarity transformations
    15. 10.15 Diagonalisation of matrices
    16. 10.16 Quadratic and Hermitian forms
    17. 10.17 The summation convention
    18. Summary
    19. Problems
    20. Hints and answers
  17. 11. Vector calculus
    1. 11.1 Differentiation of vectors
    2. 11.2 Integration of vectors
    3. 11.3 Vector functions of several arguments
    4. 11.4 Surfaces
    5. 11.5 Scalar and vector fields
    6. 11.6 Vector operators
    7. 11.7 Vector operator formulae
    8. 11.8 Cylindrical and spherical polar coordinates
    9. 11.9 General curvilinear coordinates
    10. Summary
    11. Problems
    12. Hints and answers
  18. 12. Line, surface and volume integrals
    1. 12.1 Line integrals
    2. 12.2 Connectivity of regions
    3. 12.3 Green’s theorem in a plane
    4. 12.4 Conservative fields and potentials
    5. 12.5 Surface integrals
    6. 12.6 Volume integrals
    7. 12.7 Integral forms for grad, div and curl
    8. 12.8 Divergence theorem and related theorems
    9. 12.9 Stokes’ theorem and related theorems
    10. Summary
    11. Problems
    12. Hints and answers
  19. 13. Laplace transforms
    1. 13.1 Laplace transforms
    2. 13.2 The Dirac δ-function and Heaviside step function
    3. 13.3 Laplace transforms of derivatives and integrals
    4. 13.4 Other properties of Laplace transforms
    5. Summary
    6. Problems
    7. Hints and answers
  20. 14. Ordinary differential equations
    1. 14.1 General form of solution
    2. 14.2 First-degree first-order equations
    3. 14.3 Higher degree first-order equations
    4. 14.4 Higher order linear ODEs
    5. 14.5 Linear equations with constant coefficients
    6. 14.6 Linear recurrence relations
    7. Summary
    8. Problems
    9. Hints and answers
  21. 15. Elementary probability
    1. 15.1 Venn diagrams
    2. 15.2 Probability
    3. 15.3 Permutations and combinations
    4. 15.4 Random variables and distributions
    5. 15.5 Properties of distributions
    6. 15.6 Functions of random variables
    7. 15.7 Important discrete distributions
    8. 15.8 Important continuous distributions
    9. 15.9 Joint distributions
    10. Summary
    11. Problems
    12. Hints and answers
  22. A: The base for natural logarithms
  23. B: Sinusoidal definitions
  24. C: Leibnitz’s theorem
  25. D: Summation convention
  26. E: Physical constants
  27. F: Footnote answers
  28. Index