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Mathematics for Physicists

Book Description

Mathematics for Physicists is a relatively short volume covering all the essential mathematics needed for a typical first degree in physics, from a starting point that is compatible with modern school mathematics syllabuses. Early chapters deliberately overlap with senior school mathematics, to a degree that will depend on the background of the individual reader, who may quickly skip over those topics with which he or she is already familiar. The rest of the book covers the mathematics that is usually compulsory for all students in their first two years of a typical university physics degree, plus a little more. There are worked examples throughout the text, and chapter-end problem sets.

Mathematics for Physicists features:

  • Interfaces with modern school mathematics syllabuses

  • All topics usually taught in the first two years of a physics degree

  • Worked examples throughout

  • Problems in every chapter, with answers to selected questions at the end of the book and full solutions on a website

  • This text will be an excellent resource for undergraduate students in physics and a quick reference guide for more advanced students, as well as being appropriate for students in other physical sciences, such as astronomy, chemistry and earth sciences.

    Table of Contents

    1. Editors' preface to the Manchester Physics Series
    2. Authors' preface
    3. Notes and website information
      1. ‘Starred’ material
      2. Website
      3. Examples, problems and solutions
    4. 1 Real numbers, variables and functions
      1. 1.1 Real numbers
      2. 1.2 Real variables
      3. 1.3 Functions, graphs and co-ordinates
      4. Problems 1
      5. Notes
    5. 2 Some basic functions and equations
      1. 2.1 Algebraic functions
      2. 2.2 Trigonometric functions
      3. 2.3 Logarithms and exponentials
      4. 2.4 Conic sections
      5. Problems 2
      6. Notes
    6. 3 Differential calculus
      1. 3.1 Limits and continuity
      2. 3.2 Differentiation
      3. 3.3 General methods
      4. 3.4 Higher derivatives and stationary points
      5. 3.5 Curve sketching
      6. Problems 3
      7. Notes
    7. 4 Integral calculus
      1. 4.1 Indefinite integrals
      2. 4.2 Definite integrals
      3. 4.3 Change of variables and substitutions
      4. 4.4 Integration by parts
      5. 4.5 Numerical integration
      6. 4.6 Improper integrals
      7. 4.7 Applications of integration
      8. Problems 4
      9. Notes
    8. 5 Series and expansions
      1. 5.1 Series
      2. 5.2 Convergence of infinite series
      3. 5.3 Taylor's theorem and its applications
      4. 5.4 Series expansions
      5. *5.5 Proof of d'Alembert's ratio test
      6. *5.6 Alternating and other series
      7. Problems 5
      8. Notes
    9. 6 Complex numbers and variables
      1. 6.1 Complex numbers
      2. 6.2 Complex plane: Argand diagrams
      3. 6.3 Complex variables and series
      4. 6.4 Euler's formula
      5. Problems 6
      6. Notes
    10. 7 Partial differentiation
      1. 7.1 Partial derivatives
      2. 7.2 Differentials
      3. 7.3 Change of variables
      4. 7.4 Taylor series
      5. 7.5 Stationary points
      6. *7.6 Lagrange multipliers
      7. *7.7 Differentiation of integrals
      8. Problems 7
      9. Notes
    11. 8 Vectors
      1. 8.1 Scalars and vectors
      2. 8.2 Products of vectors
      3. 8.3 Applications to geometry
      4. 8.4 Differentiation and integration
      5. Problems 8
      6. Notes
    12. 9 Determinants, Vectors and Matrices
      1. 9.1 Determinants
      2. 9.2 Vectors in <i xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg">n</i> Dimensions Dimensions
      3. 9.3 Matrices and linear transformations
      4. 9.4 Square Matrices
      5. Problems 9
      6. Notes
    13. 10 Eigenvalues and eigenvectors
      1. 10.1 The eigenvalue equation
      2. *10.2 Diagonalisation of matrices
      3. Problems 10
      4. Notes
    14. 11 Line and multiple integrals
      1. 11.1 Line integrals
      2. 11.2 Double integrals
      3. 11.3 Curvilinear co-ordinates in three dimensions
      4. 11.4 Triple or volume integrals
      5. Problems 11
    15. 12 Vector calculus
      1. 12.1 Scalar and vector fields
      2. 12.2 Line, surface, and volume integrals
      3. 12.3 The divergence theorem
      4. 12.4 Stokes' theorem
      5. Problems 12
      6. Notes
    16. 13 Fourier analysis
      1. 13.1 Fourier series
      2. 13.2 Complex Fourier series
      3. 13.3 Fourier transforms
      4. Problems 13
      5. Notes
    17. 14 Ordinary differential equations
      1. 14.1 First-order equations
      2. 14.2 Linear ODEs with constant coefficients
      3. *14.3 Euler's equation
      4. Problems 14
      5. Notes
    18. 15 Series solutions of ordinary differential equations
      1. 15.1 Series solutions
      2. 15.2 Eigenvalue equations
      3. 15.3 Legendre's equation
      4. 15.4 Bessel's equation
      5. Problems 15
      6. Notes
    19. 16 Partial differential equations
      1. 16.1 Some important PDEs in physics
      2. 16.2 Separation of variables: Cartesian co-ordinates
      3. 16.3 Separation of variables: polar co-ordinates
      4. *16.4 The wave equation: d'Alembert's solution
      5. *16.5 Euler Equations
      6. *16.6 Boundary conditions and uniqueness
      7. Problems 16
      8. Notes
    20. Answers to selected problems
      1. Problems 1
      2. Problems 2
      3. Problems 3
      4. Problems 4
      5. Problems 5
      6. Problems 6
      7. Problems 7
      8. Problems 8
      9. Problems 9
      10. Problems 10
      11. Problems 11
      12. Problems 12
      13. Problems 13
      14. Problems 14
      15. Problems 15
      16. Problems 16
    21. Index
    22. EULA