Mathematics for physics/
Michael M. Woolfson, Malcolm S. Woolfson
- New York: Oxford University Press, 2007.
- xx, 783 p. : ill. ; 25 cm.
1. Useful formulae and relationships -- 1.1. Relationships for triangles -- 1.2. Trigonometric relationships -- 1.3. The binomial expansion (theorem) -- 1.4. The exponential e -- 1.5. Natural logarithms -- 1.6. Two-dimensional coordinate systems -- Problems -- 2. Dimensions and dimensional analysis -- 2.1. Basic units and dimensions -- 2.2. Dimensional homogeneity -- 2.3. Dimensional analysis -- 2.4. Electrical and magnetic units -- Problems -- 3. Sequences and series -- 3.1. Arithmetic series -- 3.2. Geometric series -- 3.3. Harmonic series -- 3.4. Tests for convergence -- 3.5. Power series -- Problems -- 4. Differentiation -- 4.1. The basic idea of a derivative -- 4.2. Chain rule -- 4.3. Product rule -- 4.4. Quotient rule -- 4.5. Maxima, minima, and higher-order derivatives -- 4.6. Expressing ex as a power series in x -- 4.7. Taylor's theorem -- Problems -- 5. Integration -- 5.1. Indefinite and definite integrals -- 5.2. Techniques of evaluating integrals -- 5.3. Substitution method -- 5.4. Partial fractions -- 5.5. Integration by parts -- 5.6. Integrating powers of cos x and sin x -- 5.7. The definite integral : area under the curve -- Problems -- 6. Complex numbers -- 6.1. Definition of a complex number -- 6.2. Argand diagram -- 6.3. Ways of describing a complex number -- 6.4. De Moivre's theorem -- 6.5. Complex conjugate -- 6.6. Division and reduction to real-plus-imaginary form -- 6.7. Modulus-argument form as an aid to integration -- 6.8. Circuits with alternating currents and voltages -- Problems. 7. Ordinary differential equations -- 7.1. Types of ordinary differential equation -- 7.2. Separation of variables -- 7.3. Homogeneous equations -- 7.4. The integrating factor -- 7.5. Linear constant-coefficient equations -- 7.6. Simple harmonic motion -- 7.7. Damped simple harmonic motion -- 7.8. Forced vibrations -- 7.9. An LCR circuit -- Problems -- 8. Matrices I and determinants -- 8.1. Definition of a matrix -- 8.2. Operations of matrix algebra -- 8.3. Types of matrix -- 8.4. Applications to lens systems -- 8.5. Application to special relativity -- 8.6. Determinants -- 8.7. Types of determinant -- 8.8. Inverse matrix -- 8.9. Linear equations -- Problems -- 9. Vector algebra -- 9.1. Scalar and vector quantities -- 9.2. Products of vectors -- 9.3. Vector representations of some rotational quantities -- 9.4. Linear dependence and independence -- 9.5. A straight line in vector form -- 9.6. A plane in vector form -- 9.7. Distance of a point from a plane -- 9.8. Relationships between lines and planes -- 9.9. Differentiation of vectors -- 9.10. Motion under a central force -- Problems -- 10. Conic sections and orbits -- 10.1. Kepler and Newton -- 10.2. Conic sections and the cone -- 10.3. The circle and the ellipse -- 10.4. The parabola -- 10.5. The hyperbola -- 10.6. The orbits of planets and Kepler's laws -- 10.7. The dynamics of orbits -- 10.8. Alpha-particle scattering -- Problems -- 11. Partial differentiation -- 11.1. What is partial differentiation? -- 11.2. Higher partial derivatives -- 11.3. The total derivative -- 11.4 Partial differentiation and thermodynamics -- 11.5. Taylor series for a function of two variables -- 11.6. Maxima and minima in a multidimensional space -- Problems. 12. Probability and statistics -- 12.1. What is probability? -- 12.2. Combining probabilities -- 12.3. Making selections -- 12.4. The birthday problem -- 12.5. Bayes' theorem -- 12.6. Too much information? -- 12.7. Mean ; variance and standard deviation ; median -- 12.8. Combining different estimates -- Problems -- 13. Coordinate systems and multiple integration -- 13.1. Two-dimensional coordinate systems -- 13.2. Integration in a rectangular Cartesian system -- 13.3. Integration with polar coordinates -- 13.4. Changing coordinate systems -- 13.5. Three-dimensional coordinate systems -- 13.6. Integration in three dimensions -- 13.7. Moments of inertia -- 13.8. Parallel-axis theorem -- 13.9. Perpendicular-axis theorem -- Problems -- 14. Distributions -- 14.1. Kinds of distribution -- 14.2. Firing at a target -- 14.3. Normal distribution -- 14.4. Binomial distribution -- 14.5. Poisson distribution -- Problems -- 15. Hyperbolic functions -- 15.1. Definitions -- 15.2. Relationships linking hyperbolic functions -- 15.3. Differentiation of hyperbolic functions -- 15.4. Taylor expansions of sinh x and cosh x -- 15.5. Integration involving hyperbolic functions -- 15.6. Comments about analytical functions -- Problems -- 16. Vector analysis -- 16.1. Scalar and vector fields -- 16.2. Gradient (grad) and del operators -- 16.3. Conservative fields -- 16.4. Divergence (div) -- 16.5. Laplacian operator -- 16.6. Curl of a vector field -- 16.7. Maxwell's equations and the speed of light -- Problems. 17. Fourier analysis -- 17.1. Signals -- 17.2. The nature of signals -- 17.3. Amplitude-frequency diagrams -- 17.4. Fourier transform -- 17.5. The d-function, d(x) -- 17.6. Inverse Fourier transform -- 17.7. Several cosine signals -- 17.8. Parseval's theorem -- 17.9. Fourier series -- 17.10. Determination of the Fourier coefficients a₀, , and -- 17.11. Fourier or waveform synthesis -- 17.12. Power in periodic signals -- 17.13. Complex form for the Fourier series -- 17.14. Amplitude and phase spectrum -- 17.15. Alternative variables for Fourier analysis -- 17.16. Applications in physics -- 17.17. Summary -- Problems -- 18. Introduction to digital signal processing -- 18.1. More on sampling -- 18.2. Discrete Fourier transform (DFT) -- 18.3. Some concluding remarks -- Problems -- 19. Numerical methods for ordinary differential equations -- 19.1. The need for numerical methods -- 19.2. Euler methods -- 19.3. Runge-Kutta method -- 19.4. Numerov method -- Problems -- 20. Applications of partial differential equations -- 20.1. Types of partial differential equation -- 20.2. Finite differences -- 20.3. Diffusion -- 20.4. Explicit method -- 20.5. The Crank-Nicholson method -- 20.6. Poisson's and Laplace's equations -- 20.7. Numerical solution of a hot-plate problem -- 20.8. Boundary conditions for hot-plate problems -- 20.9. Wave equation -- 20.10. Finite-difference approach for a vibrating string -- 20.11. Two-dimensional vibrations -- Problems. 21. Quantum mechanics I : Schrödinger wave equation and observations -- 21.1. Transition from classical to modern physics : a brief history -- 21.2. Intuitive derivation of the Schrödinger wave equation -- 21.3. A particle in a one-dimensional box -- 21.4. Observations and operators -- 21.5. A square box and degeneracy -- 21.6. Probabilities of measurements -- 21.7. Simple harmonic oscillator -- 21.8. Three-dimensional simple harmonic oscillator -- 21.9. The free particle -- 21.10. Compatible and incompatible measurements -- 21.11. A potential barrier -- 21.12. Tunnelling -- 21.13. Other methods of solving the TISWE -- Problems -- 22. The Maxwell-Boltzmann distribution -- 22.1. Deriving the Maxwell-Boltzmann distribution -- 22.2. Retention of a planetary atmosphere -- 22.3. Nuclear fusion in stars -- Problems -- 23. The Monte Carlo method -- 23.1. Origin of the method -- 23.2. Random walk -- 23.3. A simple polymer model -- 23.4. Uniform distribution within a sphere and random directions -- 23.5. Generation of random numbers for non-uniform deviates -- 23.6. Equation of state of a liquid -- 23.7. Simulation of a fluid by the Monte Carlo method -- 23.8. Modelling a nuclear reactor -- 23.9. Description of a simple model reactor -- 23.10. A cautionary tale -- Problems -- 24. Matrices II -- 24.1. Population studies -- 24.2. Eigenvalues and eigenvectors -- 24.3. Diagonalization of a matrix -- 24.4. A vibrating system -- Problems. 25. Quantum mechanics II : Angular momentum and spin -- 25.1. Measurement of angular momentum -- 25.2. The hydrogen atom -- 25.3. Electron spin -- 25.4. Many-electron systems -- Problems -- 26. Sampling theory -- 26.1. Samples -- 26.2. Sampling proportions -- 26.3. The significance of differences -- Problems -- 27. Straight-line relationships and the linear correlation coefficient -- 27.1. General considerations -- 27.2. Lines of regression -- 27.3. A numerical application -- 27.4. The linear correlation coefficient -- 27.5. A general least-squares straight line -- 27.6. Linearization of other forms of relationship -- Problems -- 28. Interpolation -- 28.1. Applications of interpolation -- 28.2. Linear interpolation -- 28.3. Parabolic interpolation -- 28.4. Gauss interpolation formula -- 28.5. Cubic spline interpolation -- 28.6. Multidimensional interpolation -- 28.7. Extrapolation -- Problems -- 29. Quadrature -- 29.1. Definite integrals -- 29.2. Trapezium method -- 29.3. Simpson's method (rule) -- 29.4. Romberg method -- 29.5. Gauss quadrature -- 29.6. Multidimensional quadrature -- 29.7. Monte Carlo integration -- Problems -- 30. Linear equations -- 30.1. Interpretation of linearly dependent and incompatible equations -- 30.2. Gauss elimination method -- 30.3. Conditioning of a set of equations -- 30.4. Gauss-Seidel method -- 30.5. Homogeneous equations -- 30.6. Least-squares solutions -- 30.7. Refinement procedures using least squares -- Problems -- 31. Numerical solution of equations -- 31.1. The nature of equations -- 31.2. Fixed-point iteration method -- 31.3. Newton-Raphson method -- Problems -- 32. Signals and noise -- 32.1. Introduction. 32. Signals, noise, and noisy signals -- 32.3. Mathematical and statistical description of noise -- 32.4. Auto- and cross-correlation functions -- 32.5. Detection of signals in noise -- 32.6. White noise -- 32.7. Concluding remarks -- Problems -- 33. Digital filters -- 33.1. Introduction -- 33.2. Fourier transform methods -- 33.3. Constant-coefficient digital filters -- 33.4. Other filter design methods -- 33.5. Summary of main results and concluding remarks -- Problems -- 34. Introduction to estimation theory -- 34.1. Introduction -- 34.2. Estimation of a constant -- 34.3. Taking into account the changes in the underlying model -- 34.4. Further methods -- 34.5. Concluding remarks -- Problems -- 35. Linear programming and optimization -- 35.1. Basic ideas of linear programming -- 35.2. Simplex method -- 35.3. Non-linear optimization ; gradient methods -- 35.4. Gradient method for two variables -- 35.5. A practical gradient method for any number of variables -- 35.6. Optimization with constraints, the Lagrange multiplier method -- Problems -- 36. Laplace transforms -- 36.1. Defining the Laplace transform -- 36.2. Inverse Laplace transforms -- 36.3. Solving differential equations with Laplace transforms -- 36.4. Laplace transforms and transfer functions -- Problems -- 37. Networks -- 37.1. Graphs and networks -- 37.2. Types of network -- 37.3. Finding cheapest paths -- 37.4. Critical path analysis -- Problems -- 38. Simulation with particles -- 38.1. Types of problem -- 38.2. Binary systems -- 38.3. An electron in a magnetic field -- 38.4. N-body problems -- 38.5. Molecular dynamics -- 38.6. Modelling plasmas -- 38.7. Collisionless particle-in-cell model -- Problems. 39. Chaos and physical calculations -- 39.1. The nature of chaos -- 39.2. An example from population studies -- 39.3. Other aspects of chaos -- Problem -- Appendices -- Appendix 1. Table of integrals -- Appendix 2. Inverse Fourier transform -- Appendix 3. Fourier transform of a sampled signal -- Appendix 4. Derivation of the discrete and inverse discrete Fourier transforms -- Appendix 5. Program OSCILLAT -- Appendix 6. Program EXPLICIT -- Appendix 7. Program HEATCRNI -- Appendix 8. Program SIMPLATE -- Appendix 9. Program STRING1 -- Appendix 10. Program DRUM -- Appendix 11. Program SHOOT -- Appendix 12. Program DRUNKARD -- Appendix 13. Program POLYMER -- Appendix 14. Program METROPOLIS -- Appendix 15. Program REACTOR -- Appendix 16. Program LESLIE -- Appendix 17. Eigenvalues and eigenvectors of Hermitian matrices -- Appendix 18. Distance of a point from a line -- Appendix 19. Program MULGAUSS -- Appendix 20. Program MCINT -- Appendix 21. Program GS -- Appendix 22. Second moments for uniform and Gaussian noise -- Appendix 23. Convolution theorem -- Appendix 24. Output from a filter when the input is a cosine -- Appendix 25. Program GRADMAX -- Appendix 26. Program NETWORK -- Appendix 27. Program GRAVBODY -- Appendix 28. Program ELECLENS -- Appendix 29. Program CLUSTER -- Appendix 30. Program FLUIDYN -- Appendix 31. Condition for collisionless PIC -- Appendix 32. Program PLASMA1 -- References and further reading -- Solutions to exercises and problems -- Index.