Enhanced descriptions from Syndetics:

Through four editions, Peter O'Neil has made rigorous engineering mathematics topics accessible to thousands of students by emphasizing visuals, numerous examples, and interesting mathematical models. ADVANCED ENGINEERING MATHEMATICS featuries a greater number of examples and problems and is fine-tuned throughout to improve the clear flow of ideas. The computer plays a more prominent role than ever in generating computer graphics used to display concepts. And problem sets incorporate the use of such leading software packages as MAPLE. Computational assistance, exercises and projects have been included to encourage students to make use of these computational tools. The content is organized into eight-parts and covers a wide spectrum of topics including Ordinary Differential Equations, Vectors and Linear Algebra, Systems of Differential Equations, Vector Analysis, Fourier Analysis, Orthogonal Expansions, and Wavelets, Special Functions, Partial Differential Equations, Complex Analysis, and Historical Notes.

Table of contents provided by Syndetics

• Part 1 Ordinary Differential Equations (p. 1)
• Chapter 1 First-Order Differential Equations (p. 3)
• 1.1 Preliminary Concepts (p. 3)
• 1.2 Separable Equations (p. 11)
• 1.3 Linear Differential Equations (p. 23)
• 1.4 Exact Differential Equations (p. 28)
• 1.5 Integrating Factors (p. 34)
• 1.6 Homogeneous, Bernoulli, and Riccati Equations (p. 40)
• 1.7 Applications to Mechanics, Electrical Circuits, and Orthogonal Trajectories (p. 48)
• 1.8 Existence and Uniqueness for Solutions of Initial Value Problems (p. 61)
• Chapter 2 Second-Order Differential Equations (p. 65)
• 2.1 Preliminary Concepts (p. 65)
• 2.2 Theory of Solutions y" + p(x)y' + q(x)y = f(x) (p. 66)
• 2.3 Reduction of Order (p. 74)
• 2.4 The Constant Coefficient Homogeneous Linear Equation (p. 77)
• 2.5 Euler's Equation (p. 82)
• 2.6 The Nonhomogeneous Equation y" + p(x)y' + q(x)y = f(x) (p. 86)
• 2.7 Application of Second-Order Differential Equations to a Mechanical System (p. 98)
• Chapter 3 The Laplace Transform (p. 113)
• 3.1 Definition and Basic Properties (p. 113)
• 3.2 Solution of Initial Value Problems Using the Laplace Transform (p. 122)
• 3.3 Shifting Theorems and the Heaviside Function (p. 127)
• 3.4 Convolution (p. 142)
• 3.5 Unit Impulses and the Dirac Delta Function (p. 147)
• 3.6 Laplace Transform Solution of Systems (p. 152)
• 3.7 Differential Equations with Polynomial Coefficients (p. 157)
• Chapter 4 Series Solutions (p. 163)
• 4.1 Power Series Solutions of Initial Value Problems (p. 164)
• 4.2 Power Series Solutions Using Recurrence Relations (p. 169)
• 4.3 Singular Points and the Method of Frobenius (p. 174)
• 4.4 Second Solutions and Logarithm Factors (p. 181)
• 4.5 Appendix on Power Series (p. 189)
• Part 2 Vectors and Linear Algebra (p. 199)
• Chapter 5 Vectors and Vector Spaces (p. 201)
• 5.1 The Algebra and Geometry of Vectors (p. 201)
• 5.2 The Dot Product (p. 209)
• 5.3 The Cross Product (p. 216)
• 5.4 The Vector Space R[superscript n] (p. 222)
• 5.5 Linear Independence, Spanning Sets, and Dimension in R[superscript n] (p. 228)
• 5.6 Abstract Vector Spaces (p. 235)
• Chapter 6 Matrices and Systems of Linear Equations (p. 241)
• 6.1 Matrices (p. 242)
• 6.2 Elementary Row Operations and Elementary Matrices (p. 256)
• 6.3 The Row Echelon Form of a Matrix (p. 263)
• 6.4 The Row and Column Spaces of a Matrix and Rank of a Matrix (p. 271)
• 6.5 Solution of Homogeneous Systems of Linear Equations (p. 278)
• 6.6 The Solution Space of AX = O (p. 287)
• 6.7 Nonhomogeneous Systems of Linear Equations (p. 290)
• 6.8 Summary for Linear Systems (p. 301)
• 6.9 Matrix Inverses (p. 304)
• Chapter 7 Determinants (p. 311)
• 7.1 Permutations (p. 311)
• 7.2 Definition of the Determinant (p. 313)
• 7.3 Properties of Determinants (p. 315)
• 7.4 Evaluation of Determinants by Elementary Row and Column Operations (p. 319)
• 7.5 Cofactor Expansions (p. 324)
• 7.6 Determinants of Triangular Matrices (p. 328)
• 7.7 A Determinant Formula for a Matrix Inverse (p. 329)
• 7.8 Cramer's Rule (p. 332)
• 7.9 The Matrix Tree Theorem (p. 334)
• Chapter 8 Eigenvalues, Diagonalization, and Special Matrices (p. 337)
• 8.1 Eigenvalues and Eigenvectors (p. 337)
• 8.2 Diagonalization of Matrices (p. 345)
• 8.3 Orthogonal and Symmetric Matrices (p. 354)
• 8.4 Quadratic Forms (p. 363)
• 8.5 Unitary, Hermitian, and Skew-Hermitian Matrices (p. 368)
• Part 3 Systems of Differential Equations and Qualitative Methods (p. 375)
• Chapter 9 Systems of Linear Differential Equations (p. 377)
• 9.1 Theory of Systems of Linear First-Order Differential Equations (p. 377)
• 9.2 Solution of X' = AX When A Is Constant (p. 389)
• 9.3 Solution of X' = AX + G (p. 410)
• Chapter 10 Qualitative Methods and Systems of Nonlinear Differential Equations (p. 425)
• 10.1 Nonlinear Systems and Existence of Solutions (p. 425)
• 10.2 The Phase Plane, Phase Portraits, and Direction Fields (p. 428)
• 10.3 Phase Portraits of Linear Systems (p. 435)
• 10.4 Critical Points and Stability (p. 446)
• 10.5 Almost Linear Systems (p. 453)
• 10.6 Predator/Prey Population Models (p. 474)
• 10.7 Competing Species Models (p. 480)
• 10.8 Lyapunov's Stability Criteria (p. 489)
• 10.9 Limit Cycles and Periodic Solutions (p. 498)
• Part 4 Vector Analysis (p. 509)
• Chapter 11 Vector Differential Calculus (p. 511)
• 11.1 Vector Functions of One Variable (p. 511)
• 11.2 Velocity, Acceleration, Curvature, and Torsion (p. 517)
• 11.3 Vector Fields and Streamlines (p. 528)
• 11.4 The Gradient Field and Directional Derivatives (p. 535)
• 11.5 Divergence and Curl (p. 547)
• Chapter 12 Vector Integral Calculus (p. 553)
• 12.1 Line Integrals (p. 553)
• 12.2 Green's Theorem (p. 565)
• 12.3 Independence of Path and Potential Theory in the Plane (p. 572)
• 12.4 Surfaces in 3-Space and Surface Integrals (p. 583)
• 12.5 Applications of Surface Integrals (p. 596)
• 12.6 Preparation for the Integral Theorems of Gauss and Stokes (p. 602)
• 12.7 The Divergence Theorem of Gauss (p. 604)
• 12.8 The Integral Theorem of Stokes (p. 613)
• Part 5 Fourier Analysis, Orthogonal Expansions, and Wavelets (p. 623)
• Chapter 13 Fourier Series (p. 625)
• 13.1 Why Fourier Series? (p. 625)
• 13.2 The Fourier Series of a Function (p. 628)
• 13.3 Convergence of Fourier Series (p. 635)
• 13.4 Fourier Cosine and Sine Series (p. 651)
• 13.5 Integration and Differentiation of Fourier Series (p. 657)
• 13.6 The Phase Angle Form of a Fourier Series (p. 667)
• 13.7 Complex Fourier Series and the Frequency Spectrum (p. 673)
• Chapter 14 The Fourier Integral and Fourier Transforms (p. 681)
• 14.1 The Fourier Integral (p. 681)
• 14.2 Fourier Cosine and Sine Integrals (p. 685)
• 14.3 The Complex Fourier Integral and the Fourier Transform (p. 687)
• 14.4 Additional Properties and Applications of the Fourier Transform (p. 698)
• 14.5 The Fourier Cosine and Sine Transforms (p. 717)
• 14.6 The Finite Fourier Cosine and Sine Transforms (p. 719)
• 14.7 The Discrete Fourier Transform (p. 726)
• 14.8 Sampled Fourier Series (p. 733)
• 14.9 The Fast Fourier Transform (p. 745)
• Chapter 15 Special Functions, Orthogonal Expansions, and Wavelets (p. 765)
• 15.1 Legendre Polynomials (p. 765)
• 15.2 Bessel Functions (p. 783)
• 15.3 Sturm-Liouville Theory and Eigenfunction Expansions (p. 815)
• 15.4 Orthogonal Polynomials (p. 836)
• 15.5 Wavelets (p. 841)
• Part 6 Partial Differential Equations (p. 855)
• Chapter 16 The Wave Equation (p. 857)
• 16.1 The Wave Equation and Initial and Boundary Conditions (p. 857)
• 16.2 Fourier Series Solutions of the Wave Equation (p. 862)
• 16.3 Wave Motion Along Infinite and Semi-infinite Strings (p. 881)
• 16.4 Characteristics and d'Alembert's Solution (p. 895)
• 16.5 Normal Modes of Vibration of a Circular Elastic Membrane (p. 904)
• 16.6 Vibrations of a Circular Elastic Membrane, Revisited (p. 907)
• 16.7 Vibrations of a Rectangular Membrane (p. 910)
• Chapter 17 The Heat Equation (p. 915)
• 17.1 The Heat Equation and Initial and Boundary Conditions (p. 915)
• 17.2 Fourier Series Solutions of the Heat Equation (p. 918)
• 17.3 Heat Conduction in Infinite Media (p. 940)
• 17.4 Heat Conduction in an Infinite Cylinder (p. 949)
• 17.5 Heat Conduction in a Rectangular Plate (p. 953)
• Chapter 18 The Potential Equation (p. 955)
• 18.1 Harmonic Functions and the Dirichlet Problem (p. 955)
• 18.2 Dirichlet Problem for a Rectangle (p. 957)
• 18.3 Dirichlet Problem for a Disk (p. 959)
• 18.4 Poisson's Integral Formula for the Disk (p. 962)
• 18.5 Dirichlet Problems in Unbounded Regions (p. 964)
• 18.6 A Dirichlet Problem for a Cube (p. 972)
• 18.7 The Steady-State Heat Equation for a Solid Sphere (p. 974)
• 18.8 The Neumann Problem (p. 978)
• Chapter 19 Canonical Forms, Existence and Uniqueness of Solutions, and Well-Posed Problems (p. 987)
• 19.1 Canonical Forms (p. 987)
• 19.2 Existence and Uniqueness of Solutions (p. 996)
• 19.3 Well-Posed Problems (p. 998)
• Part 7 Complex Analysis (p. 1001)
• Chapter 20 Geometry and Arithmetic of Complex Numbers (p. 1003)
• 20.1 Complex Numbers (p. 1003)
• 20.2 Loci and Sets of Points in the Complex Plane (p. 1012)
• Chapter 21 Complex Functions (p. 1027)
• 21.1 Limits, Continuity, and Derivatives (p. 1027)
• 21.2 Power Series (p. 1040)
• 21.3 The Exponential and Trigonometric Functions (p. 1047)
• 21.4 The Complex Logarithm (p. 1056)
• 21.5 Powers (p. 1059)
• Chapter 22 Complex Integration (p. 1065)
• 22.1 Curves in the Plane (p. 1065)
• 22.2 The Integral of a Complex Function (p. 1070)
• 22.3 Cauchy's Theorem (p. 1081)
• 22.4 Consequences of Cauchy's Theorem (p. 1088)
• Chapter 23 Series Representations of Functions (p. 1101)
• 23.1 Power Series Representations (p. 1101)
• 23.2 The Laurent Expansion (p. 1113)
• Chapter 24 Singularities and the Residue Theorem (p. 1121)
• 24.1 Singularities (p. 1121)
• 24.2 The Residue Theorem (p. 1128)
• 24.3 Some Applications of the Residue Theorem (p. 1136)
• Chapter 25 Conformal Mappings (p. 1163)
• 25.1 Functions as Mappings (p. 1163)
• 25.2 Conformal Mappings (p. 1171)
• 25.3 Construction of Conformal Mappings Between Domains (p. 1182)
• 25.4 Harmonic Functions and the Dirichlet Problem (p. 1193)
• 25.5 Complex Function Models of Plane Fluid Flow (p. 1200)
• Part 8 Historical Notes (p. 1211)
• Chapter 26 Development of Areas of Mathematics (p. 1213)
• 26.1 Ordinary Differential Equations (p. 1213)
• 26.2 Matrices and Determinants (p. 1217)
• 26.3 Vector Analysis (p. 1218)
• 26.4 Fourier Analysis (p. 1220)
• 26.5 Partial Differential Equations (p. 1223)
• 26.6 Complex Function Theory (p. 1223)
• Chapter 27 Biographical Sketches (p. 1225)
• 27.1 Galileo Galilei (1564-1642) (p. 1225)
• 27.2 Isaac Newton (1642-1727) (p. 1227)
• 27.3 Gottfried Wilhelm Leibniz (1646-1716) (p. 1228)
• 27.4 The Bernoulli Family (p. 1229)
• 27.5 Leonhard Euler (1707-1783) (p. 1230)
• 27.6 Carl Friedrich Gauss (1777-1855) (p. 1230)
• 27.7 Joseph-Louis Lagrange (1736-1813) (p. 1231)
• 27.8 Pierre-Simon de Laplace (1749-1827) (p. 1232)
• 27.9 Augustin-Louis Cauchy (1789-1857) (p. 1233)
• 27.10 Joseph Fourier (1768-1830) (p. 1234)
• 27.11 Henri Poincare (1854-1912) (p. 1235)
• Answers and Solutions to Selected Odd-Numbered Problems (p. A1)
• Index (p. I1)