Synopses & Reviews
This book is intended to provide students with an efficient introduction and accessibility to ordinary and partial differential equations, linear algebra, vector analysis, Fourier analysis, and special functions and eigenfunction expansions, for their use as tools of inquiry and analysis in modeling and problem solving. It should also serve as preparation for further reading where this suits individual needs and interests. Although much of this material appears in Advanced Engineering Mathematics, 6th edition, ELEMENTS OF ADVANCED ENGINEERING MATHEMATICS has been completely rewritten to provide a natural flow of the material in this shorter format. Many types of computations, such as construction of direction fields, or the manipulation Bessel functions and Legendre polynomials in writing eigenfunction expansions, require the use of software packages. A short MAPLE primer is included as Appendix B. This is designed to enable the student to quickly master the use of MAPLE for such computations. Other software packages can also be used.
About the Author
Served on the faculty at the University of Minnesota, The College of William and Mary in Virginia, where he was chairman of mathematics, and the University of Alabama at Birmingham, where he was chairman of mathematics, dean of natural sciences and mathematics, and university provost. Primary research interests are in graph theory, combinatorial analysis and applications of mathematics to problems in the physical and biological sciences and engineering.
Table of Contents
Part I: ORDINARY DIFFERENTIAL EQUATIONS 1. First-Order Differential Equations Terminology and Separable Equations. Linear Equations. Exact Equations. Additional Applications. Existence and Uniqueness Questions. Direction Fields. Numerical Approximation of Solutions. 2. Linear Second-Order Equations Theory of the Linear Second-Order Equation. The Constant Coefficient Homogeneous Equation. Solutions of the Nonhomogeneous Equation. Spring Motion. 3. The Laplace Transform Definition and Notation. Solution of Initial Value Problems. Shifting and the Heaviside Function. Convolution. Impulses and the Dirac Delta Function. Appendix on Partial Fractions Decompositions. 4. Series Solutions Power Series Solutions. Frobenius Solutions. Part II: VECTORS, LINEAR ALGEBRA, AND SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS. 5. Algebra and Geometry of Vectors Vectors in the Plane and 3-Space. The Dot Product. The Cross Product. The Vector Space Rn. 6. Matrices and Systems of Linear Equations Matrices. Linear Homogeneous Systems. Nonhomogeneous Systems of Linear Equations. Matrix Inverses. 7. Determinants Definition of the Determinant. Evaluation of Determinants I. Evaluation of Determinants II. A Determinant Formula for A?1. Cramer's Rule. 8. Eigenvalues and Diagonalization Eigenvalues and Eigenvectors. Diagonalization. Some Special Matrices. 9. Systems of Linear Differential Equations Systems of Linear Differential Equations. Solution of X_ = AX when A Is Constant. Solution of X_ = AX + G. Part III: VECTOR ANALYSIS 10. Vector Differential Calculus Vector Functions of One Variable. Velocity and Curvature. Vector Fields and Streamlines. The Gradient Field. Divergence and Curl. 11. Vector Integral Calculus Line Integrals. Green's Theorem. An Extension of Green's Theorem. Potential Theory. Surface Integrals. Applications of Surface Integrals. The Divergence Theorem of Gauss. Stokes's Theorem. Part IV: FOURIER ANALYSIS AND EIGENFUNCTION EXPANSIONS 12. Fourier Series The Fourier Series of a Function. Sine and Cosine Series. Derivatives and Integrals of Fourier Series. Complex Fourier Series. 13. The Fourier Integral and Transforms The Fourier Integral. Fourier Cosine and Sine Integrals. The Fourier Transform. Fourier Cosine and Sine Transforms. 14 Eigenfunction Expansions General Eigenfunction Expansions. Fourier-Legendre Expansions. Fourier-Bessel Expansions. Part V: PARTIAL DIFFERENTIAL EQUATIONS 15. The Wave Equation Derivation of the Equation. Wave Motion on an Interval. Wave Motion in an Infinite Medium. Wave Motion in a Semi-Infinite Medium. d'Alembert's Solution. Vibrations in a Circular Membrane. Vibrations in a Rectangular Membrane. 16. The Heat Equation Initial and Boundary Conditions. The Heat Equation on [0, L]. Solutions in an Infinite Medium. Heat Conduction in an Infinite Cylinder. Heat Conduction in a Rectangular Plate. 17. The Potential Equation Laplace's Equation. Dirichlet Problem for a Rectangle. Dirichlet Problem for a Disk. Poisson's Integral Formula. Dirichlet Problem for Unbounded Regions. A Dirichlet Problem for a Cube. Steady-State Heat Equation for a Sphere. APPENDIX A Guide to Notation APPENDIX B A MAPLE Primer Answers to Selected Problems Index