Thoroughly updated, Zill’s Advanced Engineering Mathematics, Third Edition is a compendium of many mathematical topics for students planning a career in engineering or the sciences. A key strength of this text is Zill’s emphasis on differential equations as mathematical models, discussing the constructs and pitfalls of each. The Third Edition is comprehensive, yet flexible, to meet the unique needs of various course offerings ranging from ordinary differential equations to vector calculus. Numerous new projects contributed by esteemed mathematicians have been added. Key Features o The entire text has been modernized to prepare engineers and scientists with the mathematical skills required to meet current technological challenges. o The new larger trim size and 2-color design make the text a pleasure to read and learn from. o Numerous NEW engineering and science projects contributed by top mathematicians have been added, and are tied to key mathematical topics in the text. o Divided into five major parts, the text’s flexibility allows instructors to customize the text to fit their needs. The first eight chapters are ideal for a complete short course in ordinary differential equations. o The Gram-Schmidt orthogonalization process has been added in Chapter 7 and is used in subsequent chapters. o All figures now have explanatory captions. Supplements o Complete Instructor’s Solutions: Includes all solutions to the exercises found in the text. PowerPoint Lecture Slides and additional instructor’s resources are available online. o Student Solutions to accompany Advanced Engineering Mathematics, Third Edition: This student supplement contains the answers to every third problem in the textbook, allowing students to assess their progress and review key ideas and concepts discussed throughout the text.

 Unit One: Ordinary Differential Equations – Part One Introduction – Unit One Chapter 1: First-Order Differential Equations Introduction – Chapter 1 Section 1.1 Introduction Section 1.2 Terminology Section 1.3 The Direction Field Section 1.4 Picard Iteration Section 1.5 Existence and Uniqueness for the Initial Value Problem Review Exercises – Chapter 1 Chapter 2: Models Containing ODEs Introduction – Chapter 2 Section 2.1 Exponential Growth and Decay Section 2.2 Logistic Models Section 2.3 Mixing Tank Problems – Constant and Variable Volumes Section 2.4 Newton’s Law of Cooling Review Exercises – Chapter 2 Chapter 3: Methods for Solving First-Order ODEs Introduction – Chapter 3 Section 3.1 Separation of Variables Section 3.2 Equations with Homogeneous Coefficients Section 3.3 Exact Equations Section 3.4 Integrating Factors and the First-Order Linear Equation Section 3.5 Variation of Parameters and the First-Order Linear Equation Section 3.6 The Bernoulli Equation Review Exercises – Chapter 3 Chapter 4: Numeric Methods for Solving First-Order ODEs Introduction – Chapter 4 Section 4.1 Fixed-Step Methods – Order and Error Section 4.2 The Euler Method Section 4.3 Taylor Series Methods Section 4.4 Runge-Kutta Methods Section 4.5 Adams-Bashforth Multistep Methods Section 4.6 Adams-Moulton Predictor-Corrector Methods Section 4.7 Milne’s Method Section 4.8 rkf45, the Runge-Kutta-Fehlberg Method Review Exercises – Chapter 4 Chapter 5: Second-Order Differential Equations Introduction – Chapter 5 Section 5.1 Springs ‘n’ Things Section 5.2 The Initial Value Problem Section 5.3 Overview of the Solution Process Section 5.4 Linear Dependence and Independence Section 5.5 Free Undamped Motion Section 5.6 Free Damped Motion Section 5.7 Reduction of Order and Higher-Order Equations Section 5.8 The Bobbing Cylinder Section 5.9 Forced Motion and Variation of Parameters Section 5.10 Forced Motion and Undetermined Coefficients Section 5.11 Resonance Section 5.12 The Euler Equation Section 5.13 The Green’s Function Technique for IVPs Review Exercises – Chapter 5 Chapter 6: The Laplace Transform Introduction – Chapter 6 Section 6.1 Definition and Examples Section 6.2 Transform of Derivatives Section 6.3 First Shifting Law Section 6.4 Operational Laws Section 6.5 Heaviside Functions and the Second Shifting Law Section 6.6 Pulses and the Third Shifting Law Section 6.7 Transforms of Periodic Functions Section 6.8 Convolution and the Convolution Theorem Section 6.9 Convolution Products by the Convolution Theorem Section 6.10 The Dirac Delta Function Section 6.11 Transfer Function, Fundamental Solution, and the Green’s Function Review Exercises – Chapter 6 Unit Two: Infinite Series Introduction – Unit Two Chapter 7: Sequences and Series of Numbers Introduction – Chapter 7 Section 7.1 Sequences Section 7.2 Infinite Series Section 7.3 Series with Positive Terms Section 7.4 Series with Both Negative and Positive Terms Review Exercises – Chapter 7 Chapter 8: Sequences and Series of Functions Introduction – Chapter 8 Section 8.1 Sequences of Functions Section 8.2 Pointwise Convergence Section 8.3 Uniform Convergence Section 8.4 Convergence in the Mean Section 8.5 Series of Functions Review Exercises – Chapter 8 Chapter 9: Power Series Introduction – Chapter 9 Section 9.1 Taylor Polynomials Section 9.2 Taylor Series Section 9.3 Termwise Operations on Taylor Series Review Exercises – Chapter 9 Chapter 10: Fourier Series Introduction – Chapter 10 Section 10.1 General Formalism Section 10.2 Termwise Integration and Differentiation Section 10.3 Odd and Even Functions and Their Fourier Series Section 10.4 Sine Series and Cosine Series Section 10.5 Periodically Driven Damped Oscillator Section 10.6 Optimizing Property of Fourier Series Section 10.7 Fourier-Legendre Series Review Exercises – Chapter 10 Chapter 11: Asymptotic Series Introduction – Chapter 11 Section 11.1 Computing with Divergent Series Section 11.2 Definitions Section 11.3 Operations with Asymptotic Series Review Exercises – Chapter 11 Unit Three: Ordinary Differential Equations – Part Two Introduction – Unit Three Chapter 12: Systems of First-Order ODEs Introduction – Chapter 12 Section 12.1 Mixing Tanks – Closed Systems Section 12.2 Mixing Tanks – Open Systems Section 12.3 Vector Structure of Solutions Section 12.4 Determinants and Cramer’s Rule Section 12.5 Solving Linear Algebraic Equations Section 12.6 Homogeneous Equations and the Null Space Section 12.7 Inverses Section 12.8 Vectors and the Laplace Transform Section 12.9 The Matrix Exponential Section 12.10 Eigenvalues and Eigenvectors Section 12.11 Solutions by Eigenvalues and Eigenvectors Section 12.12 Finding Eigenvalues and Eigenvectors Section 12.13 System versus Second-Order ODE Section 12.14 Complex Eigenvalues Section 12.15 The Deficient Case Section 12.16 Diagonalization and Uncoupling Section 12.17 A Coupled Linear Oscillator Section 12.18 Nonhomogeneous Systems and Variation of Parameters Section 12.19 Phase Portraits Section 12.20 Stability Section 12.21 Nonlinear Systems Section 12.22 Linearization Section 12.23 The Nonlinear Pendulum Review Exercises – Chapter 12 Chapter 13: Numerical Techniques: First-Order Systems and Second-Order ODEs Introduction – Chapter 13 Section 13.1 Runge-Kutta-Nystrom Section 13.2 rk4 for First-Order Systems Review Exercises – Chapter 13 Chapter 14: Series Solutions Introduction – Chapter 14 Section 14.1 Power Series Section 14.2 Asymptotic Solutions Section 14.3 Perturbation Solution of an Algebraic Equation Section 14.4 Poincare Perturbation Solution for Differential Equations Section 14.5 The Nonlinear Spring and Lindstedt’s Method Section 14.6 The Method of Krylov and Bogoliubov Review Exercises – Chapter 14 Chapter 15: Boundary Value Problems Introduction – Chapter 15 Section 15.1 Analytic Solutions Section 15.2 Numeric Solutions Section 15.3 Least-Squares, Rayleigh-Ritz, Galerkin, and Collocation Techniques Section 15.4 Finite Elements Review Exercises – Chapter 15 Chapter 16: The Eigenvalue Problem Introduction – Chapter 16 Section 16.1 Regular Sturm-Liouville Problems Section 16.2 Bessel’s Equation Section 16.3 Legendre’s Equation Section 16.4 Solution by Finite Differences Review Exercises – Chapter 16 Unit Four: Vector Calculus Introduction – Unit Four Chapter 17: Space Curves Introduction – Chapter 17 Section 17.1 Curves and Their Tangent Vectors Section 17.2 Arc Length Section 17.3 Curvature Section 17.4 Principal Normal and Binormal Vectors Section 17.5 Resolution of R” into Tanential and Normal Components Section 17.6 Applications to Dynamics Review Exercises – Chapter 17 Chapter 18: The Gradient Vector Introduction – Chapter 18 Section 18.1 Visualizing Vector Fields and Their Flows Section 18.2 The Directional Derivative and Gradient Vector Section 18.3 Properties of the Gradient Vector Section 18.4 Lagrange Multipliers Section 18.5 Conservative Forces and the Scalar Potential Review Exercises – Chapter 18 Chapter 19: Line Integrals in the Plane Introduction – Chapter 19 Section 19.1 Work and Circulation Section 19.2 Flux through a Plane Curve Review Exercises – Chapter 19 Chapter 20: Additional Vector Differential Operators Introduction – Chapter 20 Section 20.1 Divergence and Its Meaning Section 20.2 Curl and Its Meaning Section 20.3 Products – One f and Two Operands Section 20.4 Products – Two f’s and One Operand Review Exercises – Chapter 20 Chapter 21: Integration Introduction – Chapter 21 Section 21.1 Surface Area Section 21.2 Surface Integrals and Surface Flux Section 21.3 The Divergence Theorem and the Theorems of Green and Stokes Section 21.4 Green’s Theorem Section 21.5 Conservative, Solenoidal, and Irrotational Fields Section 21.6 Integral Equivalents of div, grad, and curl Review Exercises – Chapter 21 Chapter 22: NonCartesian Coordinates Introduction – Chapter 22 Section 22.1 Mappings and Changes of Coordinates Section 22.2 Vector Operators in Polar Coordinates Section 22.3 Vector Operators in Cylindrical and Spherical Coordinates Review Exercises – Chapter 22 Chapter 23: Miscellaneous Results Introduction – Chapter 23 Section 23.1 Gauss’ Theorem Section 23.2 Surface Area for Parametrically Given Surfaces Section 23.3 The Equation of Continuity Section 23.4 Green’s Identities Review Exercises – Chapter 23 Unit Five: Boundary Value Problems for PDEs Introduction – Unit Five Chapter 24: Wave Equation Introduction – Chapter 24 Section 24.1 The Plucked String Section 24.2 The Struck String Section 24.3 D’Alembert’s Solution Section 24.4 Derivation of the Wave Equation Section 24.5 Longitudinal Vibrations in an Elastic Rod Section 24.6 Finite-Difference Solution of the One-Dimensional Wave Equation Review Exercises – Chapter 24 Chapter 25: Heat Equation Introduction – Chapter 25 Section 25.1 One-Dimensional Heat Diffusion Section 25.2 Derivation of the One-Dimensional Heat Equation Section 25.3 Heat Flow in a Rod with Insulated Ends Section 25.4 Finite-Difference Solution of the One-Dimensional Heat Equation Review Exercises – Chapter 25 Chapter 26: Laplace’s Equation in a Rectangle Introduction – Chapter 26 Section 26.1 Nonzero Temperature on the Bottom Edge Section 26.2 Nonzero Temperature on the Top Edge Section 26.3 Nonzero Temperature on the Left Edge Section 26.4 Finite-Difference Solution of Laplace’s Equation Review Exercises – Chapter 26 Chapter 27: Nonhomogeneous Boundary Value Problems Introduction – Chapter 27 Section 27.1 One-Dimensional Heat Equation with Different Endpoint Temperatures Section 27.2 One-Dimensional Heat Equation with Time-Varying Endpoint Temperatures Review Exercises – Chapter 27 Chapter 28: Time-Dependent Problems in Two Spatial Dimensions Introduction – Chapter 28 Section 28.1 Oscillations of a Rectangular Membrane Section 28.2 Time-Varying Temperatures in a Rectangular Plate Review Exercises – Chapter 28 Chapter 29: Separation of Variables in NonCartesian Coordinates Introduction – Chapter 29 Section 29.1 Laplace’s Equation in a Disk Section 29.2 Laplace’s Equation in a Cylinder Section 29.3 The Circular Drumhead Section 29.4 Laplace’s Equation in a Sphere Section 29.5 The Spherical Dielectric Review Exercises – Chapter 29 Chapter 30: Transform Techniques Introduction – Chapter 30 Section 30.1 Solution by Laplace Transform Section 30.2 The Fourier Integral Theorem Section 30.3 The Fourier Transform Section 30.4 Wave Equation on the Infinite String – Solution by Fourier Transform Section 30.5 Heat Equation on the Infinite Rod – Solution by Fourier Transform Section 30.6 Laplace’s Equation on the Infinite Strip – Solution by Fourier Transform Section 30.7 The Fourier Sine Transform Section 30.8 The Fourier Cosine Transform Review Exercises – Chapter 30 Unit Six: Matrix Algebra Introduction – Unit Six Chapter 31: Vectors as Arrows Introduction – Chapter 31 Section 31.1 The Algebra and Geometry of Vectors Section 31.2 Inner and Dot Products Section 31.3 The Cross-Product Review Exercises – Chapter 31 Chapter 32: Change of Coordinates Introduction – Chapter 32 Section 32.1 Change of Basis Section 32.2 Rotations and Orthogonal Matrices Section 32.3 Change of Coordinates Section 32.4 Reciprocal Bases and Gradient Vectors Section 32.5 Gradient Vectors and the Covariant Transformation Law Review Exercises – Chapter 32 Chapter 33: Matrix Computations Introduction – Chapter 33 Section 33.1 Summary Section 33.2 Projections Section 33.3 The Gram-Schmidt Orthogonalization Process Section 33.4 Quadratic Forms Section 33.5 Vector and Matrix Norms Section 33.6 Least Squares Review Exercises – Chapter 33 Chapter 34: Matrix Factorization Introduction – Chapter 34 Section 34.1 LU Decomposition Section 34.2 PJP-1 and Jordan Canonical Form Section 34.3 QR Decomposition Section 34.4 QR Algorithm for Finding Eigenvalues Section 34.5 SVD, The Singular Value Decomposition Section 34.6 Minimum-Length Least-Squares Solution, and the Pseudoinverse Review Exercises – Chapter 34 Unit Seven: Complex Variables Introduction – Unit Seven Chapter 35: Fundamentals Introduction – Chapter 35 Section 35.1 Complex Numbers Section 35.2 The Function w = f(z) = z2 Section 35.3 The Function w = f(z) = z3 Section 35.4 The Exponential Function Section 35.5 The Complex Logarithm Section 35.6 Complex Exponents Section 35.7 Trigonometric and Hyperbolic Functions Section 35.8 Inverses of Trigonometric and Hyperbolic Functions Section 35.9 Differentiation and the Cauchy-Riemann Equations Section 35.10 Analytic and Harmonic Functions Section 35.11 Integration Section 35.12 Series in Powers of z Section 35.13 The Calculus of Residues Review Exercises – Chapter 35 Chapter 36: Applications Introduction – Chapter 36 Section 36.1 Evaluation of Integrals Section 36.2 The Laplace Transform Section 36.3 Fourier Series and the Fourier Transform Section 36.4 The Root Locus Section 36.5 The Nyquist Stability Criterion Section 36.6 Conformal Mapping Section 36.7 The Joukowski Map Section 36.8 Solving the Dirichlet Problem by Conformal Mapping Section 36.9 Planar Fluid Flow Section 36.10 Conformal Mapping of Elementary Flows Review Exercises – Chapter 36 Unit Eight: Numerical Methods Introduction – Unit Eight Chapter 37: Equations in One Variable – Preliminaries Introduction – Chapter 37 Section 37.1 Accuracy and Errors Section 37.2 Rate of Convergence Review Exercises – Chapter 37 Chapter 38: Equations in One Variable – Methods Introduction – Chapter 38 Section 38.1 Fixed-Point Iteration Section 38.2 The Bisection Method Section 38.3 Newton-Raphson Iteration Section 38.4 The Secant Method Section 38.5 Muller’s Method Review Exercises – Chapter 38 Chapter 39: Systems of Equations Introduction – Chapter 39 Section 39.1 Gaussian Arithmetic Section 39.2 Condition Numbers Section 39.3 Iterative Improvement Section 39.4 The Method of Jacobi Section 39.5 Gauss-Seidel Iteration Section 39.6 Relaxation and SOR Section 39.7 Iterative Methods for Nonlinear Systems Section 39.8 Newton’s Iteration for Nonlinear Systems Review Exercises – Chapter 39 Chapter 40: Interpolation Introduction – Chapter 40 Section 40.1 Lagrange Interpolation Section 40.2 Divided Differences Section 40.3 Chebyshev Interpolation Section 40.4 Spline Interpolation Section 40.5 Bezier Curves Review Exercises – Chapter 40 Chapter 41: Approximation of Continuous Functions Introduction – Chapter 41 Section 41.1 Least-Squares Approximation Section 41.2 Pade Approximations Section 41.3 Chebyshev Approximation Section 41.4 Chebyshev-Pade and Minimax Approximations Review Exercises – Chapter 41 Chapter 42: Numeric Differentiation Introduction – Chapter 42 Section 42.1 Basic Formulas Section 42.2 Richardson Extrapolation Review Exercises – Chapter 42 Chapter 43: Numeric Integration Introduction – Chapter 43 Section 43.1 Methods from Elementary Calculus Section 43.2 Recursive Trapezoid Rule and Romberg Integration Section 43.3 Gauss-Legendre Quadrature Section 43.4 Adaptive Quadrature Section 43.5 Iterated Integrals Review Exercises – Chapter 43 Chapter 44: Approximation of Discrete Data Introduction – Chapter 44 Section 44.1 Least-Squares Regression Line Section 44.2 The General Linear Model Section 44.3 The Role of Orthogonality Section 44.4 Nonlinear Least Squares Review Exercises – Chapter 44 Chapter 45: Numerical Calculation of Eigenvalues Introduction – Chapter 45 Section 45.1 Power Methods Section 45.2 Householder Reflections Section 45.3 QR Decomposition via Householder Reflections Section 45.4 Upper Hessenberg Form, Givens Rotations, and the Shifted QR-Algorithm Section 45.5 The Generalized Eigenvalue Problem Review Exercises – Chapter 45 Unit Nine: Calculus of Variations Introduction – Unit Nine Chapter 46: Basic Formalisms Introduction – Chapter 46 Section 46.1 Motivational Examples Section 46.2 Direct Methods Section 46.3 The Euler-Lagrange Equation Section 46.4 First Integrals Section 46.5 Derivation of the Euler-lagrange Equation Section 46.6 Transversality Conditions Section 46.7 Derivation of the Transversality Conditions Section 46.8 Three Generalizations Review Exercises – Chapter 46 Chapter 47: Constrained Optimization Introduction – Chapter 47 Section 47.1 Application of Lagrange Multipliers Section 47.2 Queen Dido’s Problem Section 47.3 Isoperimetric Problems Section 47.4 The Hanging Chain Section 47.5 A Variable-Endpoint Problem Section 47.6 Differential Constraints Review Exercises – Chapter 47 Chapter 48: Variational Mechanics Introduction – Chapter 48 Section 48.1 Hamilton’s Principle Section 48.2 The Simple Pendulum Section 48.3 A Compound Pendulum Section 48.4 The Spherical Pendulum Section 48.5 Pendulum with Oscillating Support Section 48.6 Legendre and Extended Legendre Transformations Section 48.7 Hamilton’s Canonical Equations Review Exercises – Chapter 48