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The Mathematics of Physics and Chemistry: A Comprehensive and Rigorous Book by Margenau and Murphy


The Mathematics of Physics and Chemistry by Margenau and Murphy




If you are looking for a comprehensive and rigorous book on the mathematics of physics and chemistry, you might want to check out The Mathematics of Physics and Chemistry by Henry Margenau and George M. Murphy. This book, originally published in 1956, is a classic in the field that covers a wide range of topics from basic to advanced mathematics, with applications to various branches of physical science. In this article, we will give you an overview of what the book is about, who are the authors, why is the book important, what are the topics covered in the book, and how you can get a copy of the book.




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Introduction




What is the book about?




The Mathematics of Physics and Chemistry is a two-volume book that aims to provide a comprehensive and systematic exposition of the mathematical methods used in physics and chemistry. The book is not just a collection of formulas and techniques, but also explains the underlying concepts, principles, and proofs behind them. The book also illustrates how the mathematical methods are applied to solve problems in various domains of physical science, such as mechanics, thermodynamics, electromagnetism, quantum theory, relativity, atomic physics, nuclear physics, solid state physics, and chemical physics.


Who are the authors?




The authors of The Mathematics of Physics and Chemistry are Henry Margenau and George M. Murphy, two distinguished physicists who have made significant contributions to their fields. Henry Margenau (1901-1997) was a professor of physics at Yale University for over 40 years. He was an expert in quantum mechanics, philosophy of science, foundations of physics, thermodynamics, statistical mechanics, optics, spectroscopy, molecular physics, biophysics, and parapsychology. He wrote over 300 papers and 15 books on these topics. He was also a recipient of many honors and awards, such as the Oersted Medal, the Wetherill Medal, the Max Planck Medal, and the Templeton Prize.


George M. Murphy (1905-1968) was a professor of physics at Ohio State University for over 30 years. He was an expert in mathematical physics, relativity theory, cosmology, differential geometry, group theory, tensor analysis, quantum mechanics, and nuclear physics. He wrote over 100 papers and 5 books on these topics. He was also a recipient of many honors and awards, such as the Guggenheim Fellowship, the Fulbright Fellowship, the Sigma Xi Award, and the Ohio Academy of Science Award.


Why is the book important?




The Mathematics of Physics and Chemistry is important for several reasons. First, it is one of the most comprehensive and rigorous books on the mathematics of physics and chemistry ever written. It covers a vast range of topics, from basic to advanced mathematics, with clear explanations, proofs, examples, and exercises. It also shows how the mathematical methods are relevant and useful for various branches of physical science, with applications to real-world problems. Second, it is a classic in the field that has influenced and inspired many generations of students, teachers, researchers, and practitioners of physics and chemistry. It is widely regarded as a masterpiece of mathematical physics and a reference work of lasting value. Third, it is still relevant and useful today, despite being over 60 years old. The mathematics of physics and chemistry has not changed much since the book was published, and many of the topics covered in the book are still important and active areas of research and development. The book can also serve as a bridge between the old and the new developments in the field, as it provides a solid foundation and a historical perspective on the mathematics of physics and chemistry.


Main Content




The structure of the book




The Mathematics of Physics and Chemistry consists of two volumes: Volume 1: Basic Mathematics and Volume 2: Advanced Mathematics. Each volume has its own table of contents, preface, introduction, bibliography, index, and appendix. The two volumes are complementary and can be read independently or together, depending on the level and interest of the reader.


Volume 1: Basic Mathematics




Volume 1 covers the basic mathematics that is essential for physics and chemistry. It consists of 13 chapters, which are:


  • Chapter 1: Numbers and Numerical Calculations



  • Chapter 2: Algebra



  • Chapter 3: Analytic Geometry



  • Chapter 4: Trigonometry



  • Chapter 5: Differential Calculus



  • Chapter 6: Integral Calculus



  • Chapter 7: Infinite Series



  • Chapter 8: Complex Numbers



  • Chapter 9: Functions of a Complex Variable



  • Chapter 10: Vector Analysis



  • Chapter 11: Matrices and Determinants



  • Chapter 12: Linear Equations



  • Chapter 13: Elementary Differential Equations



The chapters are organized in a logical and progressive order, starting from the simplest concepts and methods to the more advanced ones. Each chapter begins with an introduction that explains the purpose and scope of the chapter, followed by a series of sections that present the main topics, definitions, theorems, proofs, examples, and exercises. The exercises are divided into two types: drill exercises that test the basic skills and understanding of the material, and problem exercises that challenge the reader to apply the material to more complex situations. The answers to some of the exercises are given at the end of each chapter. The appendix contains some useful tables, such as logarithms, trigonometric functions, exponential functions, and binomial coefficients.


Volume 2: Advanced Mathematics




Volume 2 covers the advanced mathematics that is more specialized for physics and chemistry. It consists of 14 chapters, which are:


  • Chapter 14: Probability Theory



  • Chapter 15: Statistics



  • Chapter 16: Fourier Series



  • Chapter 17: Integral Transforms



  • Chapter 18: Special Functions



  • Chapter 19: Asymptotic Expansions



  • Chapter 20: Tensor Analysis



  • Chapter 21: Group Theory



  • Chapter 22: Quantum Mechanics I



  • Chapter 23: Quantum Mechanics II



  • Chapter 24: Relativity Theory I



  • Chapter 25: Relativity Theory II



  • Chapter 26: Thermodynamics I



  • Chapter 27: Thermodynamics II



The chapters are organized in a thematic order, grouping together related topics that have common applications in physics and chemistry. Each chapter begins with an introduction that explains the motivation and relevance of the chapter, followed by a series of sections that present the main topics, definitions, theorems, proofs, examples, and exercises. The exercises are divided into two types: drill exercises that test the basic skills and understanding of the material, and problem exercises that challenge the reader to apply the material to more complex situations. The answers to some of the exercises are given at the end of each chapter. The appendix contains some useful tables, such as Bessel functions, Legendre polynomials, Hermite polynomials, Laguerre polynomials, Chebyshev polynomials, and gamma function.


The topics covered in the book




The Mathematics of Physics and Chemistry covers a wide range of topics that are relevant and useful for physics and chemistry. Here are some of the main topics covered in the book:


Algebra and Analysis




The book covers the basic and advanced topics of algebra and analysis, such as numbers, numerical calculations, algebraic operations, equations, inequalities, functions, limits, continuity, differentiation, integration, series, complex numbers, complex functions, residues, contour integration, vector analysis, matrices, determinants, linear equations, eigenvalues and eigenvectors. These topics are essential for understanding and manipulating mathematical expressions and equations that arise in physics and chemistry.


Geometry and Topology




The book covers the basic and advanced topics of geometry and topology, such as analytic geometry, conic sections, curves, surfaces, solids, coordinate systems, transformations, tensors, differential geometry, curvature, geodesics, Riemannian geometry, topological spaces, continuity, compactness, connectedness, homotopy, and homology. These topics are important for describing and studying the shapes, properties, and relations of geometrical objects and spaces that appear in physics and chemistry.


Probability and Statistics




The book covers the basic and advanced topics of probability and statistics, such as probability theory, random variables, distributions, expectation, variance, covariance, correlation, central limit theorem, law of large numbers, sampling theory, estimation theory, hypothesis testing, regression analysis, analysis of variance, and design of experiments. These topics are useful for modeling and analyzing the uncertainty, variability, and randomness of physical and chemical phenomena and data.


Differential Equations and Integral Transforms




The book covers the basic and advanced topics of differential equations and integral transforms, such as ordinary differential equations, partial differential equations, linear differential equations, nonlinear differential equations, homogeneous differential equations, inhomogeneous differential equations, boundary value problems, initial value problems, separation of variables, method of characteristics, Green's functions, Fourier series, Fourier transform, Laplace transform, Hankel transform, Mellin transform, and Z-transform. These topics are important for solving the mathematical models that describe the dynamics and evolution of physical and chemical systems.


Special Functions and Series




The book covers the basic and advanced topics of special functions and series, such as power series, Taylor series, Maclaurin series, Laurent series, asymptotic expansions, Bessel functions, Legendre polynomials, Hermite polynomials, Laguerre polynomials, Chebyshev polynomials, hypergeometric functions, gamma function, beta function, error function, elliptic functions, theta functions, and modular functions. These topics are useful for representing and approximating the solutions of differential equations and integral transforms that arise in physics and chemistry.


Tensor Analysis and Group Theory




The book covers the basic and advanced topics of tensor analysis and group theory, such as tensors, covariant and contravariant tensors, mixed tensors, metric tensor, Christoffel symbols, covariant differentiation, Riemann-Christoffel tensor, Ricci tensor, Einstein tensor, groups, subgroups, cosets, normal subgroups, quotient groups, homomorphisms, isomorphisms, automorphisms, permutation groups, symmetry groups, point groups, space groups, representation theory, character theory, and irreducible representations. These topics are important for describing and studying the symmetry, invariance, and transformation properties of physical and chemical systems.


Quantum Mechanics and Relativity




The book covers the basic and advanced topics of quantum mechanics and relativity, such as wave mechanics, matrix mechanics, Schrödinger equation, Heisenberg uncertainty principle, commutation relations, operators, observables, eigenvalues and eigenfunctions, expectation values, Dirac notation, Hilbert space, linear algebra, Hermitian operators, unitary operators, projection operators, adjoint operators, spectral theorem, eigenfunction expansion, orthonormal basis, completeness relation, Dirac delta function, momentum representation, position representation, energy representation, angular momentum representation, spin representation, harmonic oscillator, hydrogen atom, perturbation theory, variational method, WKB approximation, scattering theory, Born approximation, partial wave analysis, phase shifts, cross sections, identical particles, Pauli exclusion principle, fermions and bosons, second quantization, creation and annihilation operators, Fock space, quantum statistics, Bose-Einstein statistics, Fermi-Dirac statistics, quantum electrodynamics, photons and electrons, gauge invariance, Lorentz transformations, special relativity, Minkowski space-time, relativistic kinematics and dynamics, relativistic mechanics, relativistic energy and momentum, relativistic mass and force, relativistic collisions and decays, four-vectors and tensors, electromagnetic field tensor, Maxwell equations in covariant form, Lorentz force law in covariant form, general relativity, equivalence principle, curved space-time, geodesics and parallel transport, Einstein field equations, Schwarzschild solution, gravitational redshift and time dilation, gravitational lensing and deflection of light, perihelion precession of Mercury's orbit, gravitational waves and radiation, and cosmology. These topics are essential for understanding and exploring the fundamental laws and phenomena of nature at the microscopic and macroscopic scales.


Thermodynamics and Statistical Mechanics




The book covers the basic and advanced topics of thermodynamics and statistical mechanics, such as thermodynamic systems, states, processes, variables, equilibrium, work, heat, energy, entropy, temperature, pressure, volume, internal energy, enthalpy, free energy, chemical potential, first law of thermodynamics, second law of thermodynamics, third law of thermodynamics, Carnot cycle and engine, efficiency and coefficient of performance, Clausius inequality and theorem, Kelvin-Planck statement and theorem, Clausius-Clapeyron equation, phase transitions, van der Waals equation of state, ideal gas law, real gas law, virial expansion, Maxwell relations, Gibbs-Duhem equation, Legendre transformations, Maxwell-Boltzmann statistics, Gibbs distribution, partition function, canonical ensemble, grand canonical ensemble, microcanonical ensemble, equipartition theorem, thermodynamic potentials, fluctuations and correlations, ensembles and ergodicity, Liouville theorem, Boltzmann equation, Boltzmann H-theorem and entropy, Boltzmann factor and distribution, Bose-Einstein statistics, Fermi-Dirac statistics, quantum gases, black-body radiation, Planck's law, Stefan-Boltzmann law, Wien's displacement law, Rayleigh-Jeans law, specific heat and heat capacity, Debye model and Einstein model of solids, lattice vibrations and phonons, electronic specific heat of metals, fermi energy and fermi level of metals, chemical equilibrium and reaction rate constants, mass action law and equilibrium constant. These topics are important for describing and studying the thermal behavior and statistical properties of physical and chemical systems.


Electromagnetism and Optics




The book covers the basic and advanced topics of electromagnetism and optics, such as electric charge, electric field, electric potential, electric dipole, electric flux, Gauss's law for electricity, electric polarization, electric displacement vector, electric susceptibility and permittivity, capacitance and capacitors, electric current, electric resistance and resistivity, Ohm's law, Kirchhoff's laws for electric circuits, electromotive force and batteries, magnetic field, magnetic force, magnetic dipole, magnetic flux, Gauss's law for magnetism. magnetic polarization. magnetic induction vector. magnetic susceptibility and permeability. inductance and inductors. Faraday's law of electromagnetic induction. Lenz's law. self-induction and mutual induction. magnetic circuits. Ampere's circuital law. Biot-Savart law. Lorentz force law. Hall effect. electromagnetic waves. Maxwell equations in differential form. Maxwell equations in integral form. Poynting vector and theorem. electromagnetic energy density and momentum density. electromagnetic stress tensor. electromagnetic radiation pressure. electromagnetic boundary conditions. electromagnetic wave propagation in vacuum. electromagnetic wave propagation in media. electromagnetic wave polarization. electromagnetic wave reflection and refraction. Snell's law. Fresnel equations. Brewster's angle. total internal reflection. evanescent waves. electromagnetic wave interference. Young's double slit experiment. interference fringes. thin film interference. Newton's rings. Michelson interferometer. Fabry-Perot interferometer. electromagnetic wave diffraction. Huygens-Fresnel principle. Fraunhofer diffraction. Fresnel diffraction. diffraction grating. Rayleigh criterion. optical resolution. electromagnetic wave dispersion. group velocity and phase velocity. optical dispersion relation. refractive index and dispersion relation. Sellmeier equation. Cauchy equation. Abbe number. chromatic aberration. electromagnetic wave scattering. Rayleigh scattering. Mie scattering. Thomson scattering. Compton scattering. Raman scattering. electromagnetic wave absorption and attenuation. Beer-Lambert law and absorption coefficient.


Atomic and Molecular Physics




The book covers the basic and advanced topics of atomic and molecular physics, such as atomic structure and spectra, Bohr model of hydrogen atom, Rydberg formula and constant, Balmer series and other spectral series of hydrogen atom, Franck-Hertz experiment and atomic excitation, Zeeman effect and atomic magnetic moments, Stark effect and atomic electric moments, Pauli exclusion principle and atomic electronic configuration, Aufbau principle and Hund's rules, atomic orbital and quantum numbers, atomic term symbol and spectroscopic notation, selection rules and transition probabilities, fine structure and spin-orbit coupling, hyperfine structure and nuclear spin, Lamb shift and quantum electrodynamics, atomic clock and atomic frequency standard, laser and stimulated emission, maser and microwave amplification, optical pumping and population inversion, laser cooling and trapping of atoms, Bose-Einstein condensation of dilute atomic gases, molecular structure and spectra, molecular orbital and molecular electronic configuration, molecular term symbol and spectroscopic notation, molecular symmetry and point groups, molecular vibration and normal modes, molecular rotation and rotational spectra, vibrational-rotational spectra of diatomic molecules, electronic spectra of diatomic molecules, Franck-Condon principle and vibrational overlap integral, Born-Oppenheimer approximation Born-Oppenheimer approximation and electronic-nuclear separation, molecular dissociation and potential energy curves, molecular constants and spectroscopic parameters, molecular spectroscopy and spectroscopic techniques, infrared spectroscopy and vibrational spectra of polyatomic molecules, Raman spectroscopy and vibrational spectra of symmetric molecules, microwave spectroscopy and rotational spectra of polyatomic molecules, ultraviolet-visible spectroscopy and electronic spectra of polyatomic molecules, nuclear magnetic resonance spectroscopy and nuclear spin spectra of molecules, electron spin resonance spectroscopy and electron spin spectra of molecules, photoelectron spectroscopy and ionization spectra of molecules, mass spectrometry and mass spectra of molecules.


Nuclear and Particle Physics




The book covers the basic and advanced topics of nuclear and particle physics, such as nuclear structure and properties, nuclear size and shape, nuclear charge and mass, nuclear binding energy and mass defect, nuclear stability and decay modes, radioactive decay law and half-life, alpha decay and Geiger-Nuttall law, beta decay and Fermi theory, gamma decay and selection rules, nuclear reactions and Q-value, nuclear fission and chain reaction, nuclear fusion and thermonuc


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