Mathematical Methods for Scientists and Engineers

DONALD McQUARRIE is Professor Emeritus from the Department of Chemistry at the University of California, Davis, USA. From his classic text on Statistical Mechanics to his recent quantum-first tour de force on Physical Chemistry, McQuarrie's best selling textbooks are highly acclaimed by the chemistry community.

'McQuarrie's Mathematical Methods for Scientists and Engineers is a well-written, carefully conceived panorama of an extensive mathematical landscape. From asymptotic analysis to linear algebra to partial differential equations and complex variables, McQuarrie provides relevant background, physical and mathematical intuition and motivation, and just the right dose of mathematical rigor to get the ideas across effectively. The large collection of examples and exercises will prove indispensable for teaching and learning the material.' - Dennis DeTurck, University of Pennsylvania, USA 'McQuarrie has produced a masterpiece that anyone would want to have handy when confronted with the need to perform serious mathematical analyses. It's destined to become a classic reference text...beautifully illustrated with two-color graphical side bars that emphasize the principles presented in the text. It includes vignettes of the lives of various mathematicians that almost make them seem human...as appealing as any reference work you might imagine on this topic...The text is authoritative and comprehensive...what every undergraduate student should master to become mathematically adept.' - Richard N Zare, Stanford University, USA, Chemical and Engineering News

• Chapter 1: Functions of a Single Variable1-1 Functions1-2 Limits1-3 Continuity1-4 Differentiation1-5 Differentials1-6 Mean Value Theorems1-7 Integration1-8 Improper Integrals1-9 Uniform Convergence of IntegralsChapter 2: Infinite Series2-1 Infinite Sequences2-2 Convergence and Divergence of Infinite Series2-3 Tests for Convergence2-4 Alternating Series2-5 Uniform Convergence2-6 Power Series2-7 Taylor Series2-8 Applications of Taylor Series2-9 Asymptotic ExpansionsChapter 3: Functions Defined As Integrals3-1 The Gamma Function3-2 The Beta Function3-3 The Error Function3-4 The Exponential Integral3-5 Elliptic Integrals3-6 The Dirac Delta Function3-7 Bernoulli Numbers and Bernoulli PolynomialsChapter 4: Complex Numbers and Complex Functions4-1 Complex Numbers and the Complex Plane4-2 Functions of a Complex Variable4-3 Euler’s Formula and the Polar Form of Complex Numbers4-4 Trigonometric and Hyperbolic Functions4-5 The Logarithms of Complex Numbers4-6 Powers of Complex NumbersChapter 5: Vectors5-1 Vectors in Two Dimensions5-2 Vector Functions in Two Dimensions5-3 Vectors in Three Dimensions5-4 Vector Functions in Three Dimensions5-5 Lines and Planes in SpaceChapter 6: Functions of Several Variables6-1 Functions6-2 Limits and Continuity6-3 Partial Derivatives6-4 Chain Rules for Partial Differentiation6-5 Differentials and the Total Differential6-6 The Directional Derivative and the Gradient6-7 Taylor’s Formula in Several Variables6-8 Maxima and Minima6-9 The Method of Lagrange Multipliers6-10 Multiple IntegralsChapter 7: Vector Calculus7-1 Vector Fields7-2 Line Integrals7-3 Surface Integrals7-4 The Divergence Theorem7-5 Stokes’s TheoremChapter 8: Curvilinear Coordinates8-1 Plane Polar Coordinates8-2 Vectors in Plane Polar Coordinates8-3 Cylindrical Coordinates8-4 Spherical Coordinates8-5 Curvilinear Coordinates8-6 Some Other Coordinate SystemsChapter 9: Linear Algebra and Vector Spaces9-1 Determinants9-2 Gaussian Elimination9-3 Matrices9-4 Rank of a Matrix9-5 Vector Spaces9-6 Inner Product Spaces9-7 Complex Inner Product SpacesChapter 10: Matrices and Eigenvalue Problems10-1 Orthogonal and Unitary Transformations10-2 Eigenvalues and Eigenvectors10-3 Some Applied Eigenvalue Problems10-4 Change of Basis10-5 Diagonalization of Matrices10-6 Quadratic FormsChapter 11: Ordinary Differential Equations11-1 Differential Equations of First Order and First Degree11-2 Linear First-Order Differential Equations11-3 Homogeneous Linear Differential Equations with Constant Coefficients11-4 Nonhomogeneous Linear Differential Equations with Constant Coefficients11-5 Some Other Types of Higher-Order Differential Equations11-6 Systems of First-Order Differential Equations11-7 Two Invaluable Resources for Solutions to Differential EquationsChapter 12: Series Solutions of Differential Equations12-1 The Power Series Method12-2 Ordinary Points and Singular Points of Differential Equations12-3 Series Solutions Near an Ordinary Point: Legendre’s Equation12-4 Solutions Near Regular Singular Points12-5 Bessel’s Equation12-6 Bessel FunctionsChapter 13: Qualitative Methods for Nonlinear Differential Equations13-1 The Phase Plane13-2 Critical Points in the Phase Plane13-3 Stability of Critical Points13-4 Nonlinear Oscillators13-5 Population DynamicsChapter 14: Orthogonal Polynomials and Sturm–Liouville Problems14-1 Legendre Polynomials14-2 Orthogonal Polynomials14-3 Sturm–Liouville Theory14-4 Eigenfunction Expansions14-5 Green’s FunctionsChapter 15: Fourier Series15-1 Fourier Series as Eigenfunction Expansions15-2 Sine and Cosine Series15-3 Convergence of Fourier Series15-4 Fourier Series and Ordinary Differential EquationsChapter 16: Partial Differential Equations16-1 Some Examples of Partial Differential Equations16-2 Laplace’s Equation16-3 The One-Dimensional Wave Equation16-4 The Two-Dimensional Wave Equation16-5 The Heat Equation16-6 The Schrödinger Equationa. Particle in a Boxb. A Rigid Rotorc. The Electron in a Hydrogen Atom16-7 The Classification of Partial Differential EquationsChapter 17: Integral Transforms17-1 The Laplace Transform17-2 The Inversion of Laplace Transforms17-3 Laplace Transforms and Ordinary Differential Equations17-4 Laplace Transforms and Partial Differential Equations17-5 Fourier Transforms17-6 Fourier Transforms and Partial Differential Equations17-7 The Inversion Formula for Laplace TransformsChapter 18: Functions of a Complex Variable: Theory18-1 Functions, Limits, and Continuity18-2 Differentiation. The Cauchy–Riemann Equations18-3 Complex Integration. Cauchy’s Theorem18-4 Cauchy’s Integral Formula18-5 Taylor Series and Laurent Series18-6 Residues and the Residue TheoremChapter 19: Functions of a Complex Variable: Applications19-1 The Inversion Formula for Laplace Transforms19-2 Evaluation of Real, Definite Integrals19-3 Summation of Series19-4 Location of Zeros19-5 Conformal Mapping19-6 Conformal Mapping and Boundary Value Problems19-7 Conformal Mapping and Fluid FlowChapter 20: Calculus of Variations20-1 The Euler’s Equation20-2 Two Laws of Physics in Variational Form20-3 Variational Problems with Constraints20-4 Variational Formulation of Eigenvalue Problems20-5 Multidimensional Variational ProblemsChapter 21: Probability Theory and Stochastic Processes21-1 Discrete Random Variables21-2 Continuous Random Variables21-3 Characteristic Functions21-4 Stochastic Processes—General21-5 Stochastic Processes—Examplesa. Poisson Processb. The Shot EffectChapter 22: Mathematical Statistics22-1 Estimation of Parameters22-2 Three Key Distributions Used in Statistical Testsa. The Normal Distributionb. The Chi–Square Distributionc. Student t-Distribution22-3 Confidence Intervalsa. Confidence Intervals for the Mean of a Normal Distribution Whose Variance is Knownb. Confidence Intervals for the Mean of a Normal Distribution with Unknown Variancec. Confidence Intervals for the Variance of a Normal Distribution22-4 Goodness of Fit22-5 Regression and CorrelationIndex