This innovative text was written for the one or two-semester, sophomore/junior level advanced maths course for engineers. It was built from the ground up using a Computer Algebra System, offering the student opportunities to visualize and experience the maths at every turn. The text has been designed to accommodate a variety of teaching styles, and varying levels on technology integration. It has a logical arrangement with many short self-contained sections, and many real-world applications of interest to engineering students. Chapter Introductions and Chapter Summaries help to make the material more accessible, and Chapter Review Exercises provides constant checks along the way. *A CD-ROM is included in the back of every book, which contains Maple worksheets. The Maple worksheets are fully integrated with the books content, and provide a great resource for students when working on exercise sections. The CD-ROM allows the instructor and the student to take full advantage of what the text has to offer. *Logical arrangement with many short self-contained sections. *Exercises are divided into two sections: those designed to be computed by hand (A exercises), and those to be computed with a Computer Algebra System program (B exercises). *Examples throughout which illustrate to students how concepts are conceived. *Multi-stage exercises appear in every section, which include exercises that demand interpretation and analysis. *Unit and Chapter Introductions summarize important concepts that students will learn in the upcoming chapter. *End of Chapter Review Exercises continually check students knowledge. *Full Glossary. Includes terms that have been partially defined, informally defined, and formally defined. *Sample syllabi. Included in the preface, to help instructors mold their course.

Table of Contents:
Preface. UNIT I. ORDINARY DIFFERENTIAL EQUATIONS - PART 1. 1. First-Order Differential Equations. Introduction. Terminology. The Direction Field. Picard Iteration. Existence and Uniqueness for the Initial Value Problem. 2. Models Containing ODEs. Exponential Growth and Decay. Logistic Models. Mixing Tank Problems—Constant and Variable Volumes. Newton's Law of Cooling. 3. Methods for Solving First-Order ODEs. Separation of Variables. Equations with Homogeneous Coefficients. Exact Equations. Integrating Factors and the First-Order Equations. Variation of Parameters and the First-Order Linear Equation. The Bernoulli Equation. 4. Numeric Methods for Solving First-Order ODEs. Fixed-Step Methods—Order and Error. The Euler Method. Taylor Series Methods. Runge-Kutta Methods. Adams-Bashforth Multistep Methods. Adams-Moulton Predictor-Corrector Methods. Milne's Method. RKF45, the Runge-Kutta-Fehlberg Method. 5. Second-Order Differential Equations. Springs `n' Things. The Initial Value Problem. Overview of the Solution Process. Linear Dependence and Independence. Free Undamped Motion. Free Damped Motion. Reduction of Order and Higher-Order Equations. The Bobbing Cylinder. Forced Motion and Variation Of Parameters. Forced Motion and Undetermined Coefficients. Resonance. The Euler Equation. The Green's Function Technique for IVPS. 6. The Laplace Transform. Definition and Examples. Transform of Derivatives. First Shifting Law. Operational Laws. Heaviside Functions and the Second Shifting Law. Pulses, and the Third Shifting Law. Transforms of Periodic Functions. Convolution and the Convolution Theorem. Convolution Products by the Convolution Theorem. The Dirac Delta Function. Transfer Function, Fundamental Solution, and The Green's Function. UNIT II. INFINITE SERIES. 7. Sequences and Series of Numbers. Sequences. Infinite Series. Series with Positive Terms. Series with Both Negative and Positive Terms. 8. Sequences and Series of Functions. Sequences of Functions. Pointwise Convergence. Uniform Convergence. Convergence in the Mean. Series of Functions. 9. Power Series. Taylor Polynomials. Taylor Series. Termwise Operations on Taylor Series. 10. Fourier Series. General Formalism. Termwise Integration and Differentiation. Odd and Even Functions and Their Fourier Series. Sine Series and Cosine Series. Periodically Driven Damped Oscillator. Optimizing Property of Fourier Series. Fourier-Legendre Series. 11. Asymptotic Series. Computing with Divergent Series. Definitions. Operations with Asymptotic Series. UNIT III. ORDINARY DIFFERENTIAL EQUATIONS - PART 2. 12. Systems of First-Order ODEs. Mixing Tanks—Closed Systems. Mixing Tanks—Open Systems. Vector Structure of Solutions. Determinants and Cramer's Rule. Solving Linear Algebraic Equations. Homogeneous Equations and the Null Space. Inverses. Vectors and the Laplace Transform. The Matrix Exponential. Eigenvalues and Eigenvectors. Solution by Eigenvalues and Eigenvectors. Finding Eigenvalues and Eigenvectors. System Versus Second-Order ODE. Complex Eigenvalues. The Deficient Case. Diagonalization and Uncoupling. A Coupled Linear Oscillator. Nonhomogeneous Systems and Variation of Parameters. Phase Portraits. Stability. Nonlinear Systems. Linearization. The Nonlinear Pendulum. 13. Numerical Techniques: First-Order Systems and Second-Order ODEs. Runge-Kutta-Nystrom. RK4 for First-Order Systems. 14. Series Solutions. Power Series. Asymptotic Solutions. Perturbation Solution of an Algebraic Equation. Poincaré Perturbation Solution for Differential Equations. The Nonlinear Spring and Lindstedt's Method. The Method Of Krylov and Bogoliubov. 15. Boundary Value Problems. Analytic Solutions. Numeric Solutions. Least-Squares, Rayleigh-Ritz, Galerkin, And Collocation Techniques. Finite Elements. 16. The Eigenvalue Problem. Regular Sturm-Liouville Problems. Bessel's Equation. Legendre's Equation. Solution by Finite Differences. UNIT IV. VECTOR CALCULUS. 17. Space Curves. Curves and Their Tangent Vectors. Arc Length. Curvature. Principal Normal and Binormal Vectors. Resolution of R” into Tangential and Normal Components. Applications to Dynamics. 18. The Gradient Vector. Visualizing Vector Fields and Their Flows. The Directional Derivative and Gradient Vector. Properties of the Gradient Vector. Lagrange Multipliers. Conservative Forces and the Scalar Potential. 19. Line Integrals in the Plane. Work and Circulation. Flux Through a Plane Curve. 20. Additional Vector Differential Operators. Divergence and Its Meaning. Curl and Its Meaning. Products—One õ and Two Operands. Products—Two õ'S and One Operand. 21. Integration. Surface Area. Surface Integrals and Surface Flux. The Divergence Theorem and the Theorems Of Green and Stokes. Green's Theorem. Conservative, Solenoidal, and Irrotational Fields. Integral Equivalents of div, grad, and curl. 22. Non Cartesian Coordinates. Mappings and Changes of Coordinates. Vector Operators in Polar Coordinates. Vector Operators in Cylindrical and Spherical Coordinates. 23. Miscellaneous Results. Gauss' Theorem. Surface Area for Parametrically Given Surfaces. The Equation of Continuity. Green's Identities. UNIT V. BOUNDARY VALUE PROBLEMS FOR PDEs. 24. Wave Equation. The Plucked String. The Struck String. D'Alembert's Solution. Derivation of the Wave Equation. Longitudinal Vibrations in an Elastic Rod. Finite-Difference Solution of the One-Dimensional Wave Equation. 25. Heat Equation. One-Dimensional Heat Diffusion. Derivation of the One-Dimensional Heat Equation. Heat Flow in a Rod with Insulated Ends. Finite-Difference Solution of the One-Dimensional Heat Equation. 26. Laplace's Equation in a Rectangle. Nonzero Temperature on the Bottom Edge. Nonzero Temperature on the Top Edge. Nonzero Temperature on the Left Edge. Finite-Difference Solution Of Laplace's Equation in a Rectangle. 27. Nonhomogeneous Boundary Value Problems. One-Dimensional Heat Equation with Different Endpoint Temperatures. One-Dimensional Heat Equation with Time-Varying Endpoint Temperatures. 28.Time-Dependent Problems in Two Spatial Dimensions. Oscillations of a Rectangular Membrane. Time-Varying Temperatures in a Rectangular Plate. 29. Separation of Variables in Non Cartesian Coordinates. Laplace's Equation in a Disk. Laplace's Equation in a Cylinder. The Circular Drumhead. Laplace's Equation in a Sphere. The Spherical Dielectric. 30. Transform Techniques. Solution by Laplace Transform. The Fourier Integral Theorem. The Fourier Transform. Wave Equation on the Infinite String—Solution by Fourier Transform. Heat Equation on the Infinite Rod—Solution by Fourier Transform. Laplace's Equation on the Infinite Strip—Solution by Fourier Transform. The Fourier Sine Transform. The Fourier Cosine Transform. UNIT VI. MATRIX ALGEBRA. 31. Vectors As Arrows. The Algebra and Geometry of Vectors. Inner and Dot Products. The Cross Product. 32. Change of Coordinates. Change of Basis. Rotations and Orthogonal Matrices. Change of Coordinates. Reciprocal Bases and Gradient Vectors. Gradient Vectors and the Covariant Transformation Law. 33. Matrix Computations. Summary. Projections. The Gram-Schmidt Orthogonalization Process. Quadratic Forms. Vector and Matrix Norms. Least Squares. 34. Matrix Factorizations. LU Decomposition. PJP-1 and Jordan Canonical Form. QR Decomposition. QR Algorithm for Finding Eigenvalues. SVD, The Singular Value Decomposition. Minimum-Length Least-Squares Solution, and the Pseudoinverse. UNIT VII. COMPLEX VARIABLES. 35. Fundamentals. Complex Numbers. The Function w = f (z) = z2. The Function w = f (z) = z3. The Exponential Function. The Complex Logarithm. Complex Exponents. Trigonometric and Hyperbolic Functions. Inverses of Trigonometric and Hyperbolic Functions. Differentiation and the Cauchy-Riemann Equations. Analytic and Harmonic Functions. Integration. Series in Powers of z. The Calculus Of Residues. 36. Applications. Evaluation of Integrals. The Laplace Transform. Fourier Series and the Fourier Transform. The Root Locus. The Nyquist Stability Criterion. Conformal Mapping. The Joukowski Map. Solving the Dirichlet Problem by Conformal Mapping. Planar Fluid Flow. Conformal Mapping of Elementary Flows. UNIT VIII. NUMERICAL METHODS. 37. Equations in One Variable—Preliminaries. Accuracy and Errors. Rate of Convergence. 38. Equations in One Variable—Methods. Fixed-Point Iteration. The Bisection Method. Newton-Raphson Iteration. The Secant Method. Muller's Method. 39. Systems of Equations. Gaussian Arithmetic. Condition Numbers. Iterative Improvement. The Method of Jacobi. Gauss-Seidel Iteration. Relaxation and SOR. Iterative Methods for Nonlinear Systems. Newton's Iteration for Nonlinear Systems. 40. Interpolation. Lagrange Interpolation. Divided Differences. Chebyshev Interpolation. Spline Interpolation. Bezier Curves. 41. Approximation of Continuous Functions. Least-squares Approximation. Padé Approximations. Chebyshev Approximation. Chebyshev- Padé and Minimax Approximations. 42. Numeric Differentiation. Basic Formulas. Richardson Extrapolation. 43. Numeric Integration. Methods from Elementary Calculus. Recursive Trapezoid Rule and Romberg Integration. Gauss-Legendre Quadrature. Adaptive Quadrature. Iterated Integrals. 44. Approximation of Discrete Data. Least-Squares Regression Line. The General Linear Model. The Role of Orthogonality. Nonlinear Least Squares. 45. Numerical Calculation of Eigenvalues. Power Methods. Householder Reflections. QR Decomposition Via Householder Reflections. Upper Hessenberg Form, Givens Rotations, and the Shifted QR-Algorithm. The Generalized Eigenvalue Problem. UNIT IX. CALCULUS OF VARIATIONS. 46. Basic Formalisms. Motivational Examples. Direct Methods. The Euler-Lagrange Equation. First Integrals. Derivation of the Euler-Lagrange Equation. Transversality Conditions. Derivation of the Transversality Conditions. Three Generalizations. 47. Constrained Optimization. Applications of Lagrange Multipliers. Queen Dido's Problem. Isoperimetric Problems. The Hanging Chain. A Variable-Endpoint Problem. Differential Constraints. 48. Variational Mechanics. Hamilton's Principle. The Simple Pendulum. A Compound Pendulum. The Spherical Pendulum. Pendulum with Oscillating Support. Legendre and Extended Legendre Transformations. Hamilton's Canonical Equations. Answers to Selected Exercises. Bibliography. Index.