Advanced Engineering Mathematics

This well-known text is written to provide students in engineering and the applied sciences with a sound presentation of post-calculus mathematics. It contains ample material for a two-year sequence in applied mathematics or for a number of short courses such as ordinary differential equations, partial differential equations and boundary-value problems, vector analysis, applied linear algebra, and complex variables. The courses are offered in either the mathematics department or in the engineering department, and sometimes in both. Advanced Engineering Mathematics is respected for its clarity, accuracy, and utility. It features a large number of carefully and completely worked examples, many involving engineering applications; numerous two-color illustrations that enhance the visual perception and quick identification of distinct entities in each figure; and over 5,000 exercises which range from routine practice problems to more difficult applications. Its broad coverage and detailed illustrative examples make this a valuable reference book for undergraduate students and graduate students alike.

Table of Contents:
1. ORDINARY DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. Variables and Functions. Classification of Differential Equations. Solutions of Differential Equations. Solution Curves and Integral Curves. Differential Equations with Prescribed Solutions. Existence and Uniqueness of Solutions. Exact First-Order Equations. Integrating Factors for First-Order Equations. Separable First-Order Equations. Homogeneous First-Order Equations. Linear First-Order Equations. Special First-Order Equations. Second-Order Equations of Reducible Order. Orthogonal Trajectories. Applications of First-Order Differential. Equations. 2. LINEAR DIFFERENTIAL EQUATIONS. A Fundamental Existence and Uniqueness Theorem. Families of Solutions. Solutions of Nonhomogeneous Equations. Variation of Parameters and Reduction of Order. Homogeneous Second-Order Equations with Constant Coefficients. Homogeneous Equations of Higher-Order. Nonhomogeneous Equations with Constant Coefficients. The Euler-Cauchy Differential Equation. Power Series Solutions. The Method of Frobenius. Applications of Linear Differential Equations with Constant. Coefficients. Green's Functions. 3. COMPLEX NUMBERS AND LINEAR ALGEBRA. Complex Numbers. The Algebra of Vectors. The Algebra of Matrices. Special Matrices. Determinants. Systems of Linear Algebraic Equations. Special Linear Systems, Inverses, Adjoints, and Cramer's Rule. Characteristic Value Problems. 4. SIMULTANEOUS LINEAR DIFFERENTIAL EQUATIONS. Solutions, Consistency, and Equivalence of Linear Differential Systems. The Reduction of a Differential System to an Equivalent System. Fundamental Concepts and Theorems Concerning First-Order Systems. Theorems for General Linear Differential Systems. Linear Differential Systems with Constant Coefficients. 5. NUMERICAL METHODS. Introduction. The Differences of a Function. Interpolation Formulas. Numerical Differentiation and Integration. The Method of Least Squares. The Numerical Solution of Ordinary Differential Equations. Difference Equations. Difference Equations and The Numerical Solution of Differential. Equations. Generating Functions and the G- and Z- Transformations. 6. THE DESCRIPTIVE THEORY OF NONLINEAR DIFFERENTIAL EQUATIONS. Introduction. The Phase Plane and Critical Points. Critical Points and the Trajectories of Linear Systems. Critical Points of Systems which are Approximately Linear. Systems which are Not Approximately Linear. Periodic Solutions and Limit Cycles. 7. MECHANICAL SYSTEMS AND ELECTRIC CIRCUITS. Introduction. Systems with One Degree of Freedom. The Translational Mechanical System. The Series Electric Circuit. Systems with Several Degrees of Freedom. Systems with Many Degrees of Freedom. Electromechanical Analogies. 8. FOURIER SERIES. Periodic Functions. The Euler Formulas. Alternative Formulas for the Fourier Coefficients. Half-Range Expansions. Alternative Forms of Fourier Series. Applications of Fourier Series. 9. FOURIER INTEGRALS AND FOURIER TRANSFORMS. The Fourier Integral as the Limit of a Fourier Series. Fourier Integral Approximations and the Gibbs Phenomenon. Properties of Fourier Transforms. Applications of Fourier Integrals. Singularity Functions and Their Fourier Transforms. From the Fourier Integral to the Laplace Transform. 10. THE LAPLACE TRANSFORMATION. Introduction. Definitions and Basic Theory. Uniform Convergence and Its Consequences. The General Method. The Transforms of Special Functions. The Shifting Theorems. The Differentiation and Integration of Transforms. Limit Theorems. The Heaviside Expansion Theorems. The Transforms of Periodic Functions. Convolution and the Duhamel Formulas. The L-, T-, and Z- Transformations Compared. 11. PARTIAL DIFFERENTIAL EQUATIONS. Introduction. The Derivation of Equations. The d'Alembert Solution of the Wave Equation. Characteristics and the Classification of Partial Differential. Equations. Separation of Variables. Orthogonal Functions and the General Expansion Problems. Further Applications. Laplace Transformation Methods. The Numerical Solution of Partial Differential Equations. 12. BESSEL FUNCTIONS AND LEGENDRE POLYNOMIALS. Introduction. The Series Solution of Bessel's Equation. Modified Bessel Functions. Equations Solvable in Terms of Bessel Functions. Identities for the Bessel Functions. The Generating Function for {Jn(x)}. The Orthogonality of the Bessel Functions. Applications of Bessel Functions. Legendre Polynomials. 13. VECTOR SPACES AND LINEAR TRANSFORMATIONS. Vector Spaces. Subspaces, Linear Dependence and Linear Independence. Bases and Dimension. Angles, Projections, and the Gram-Schmidt Orthogonalization Process. Linear Transformations. Sums, Products, and Inverses of Linear Transformations. Linear Operator Equations. 14. APPLICATIONS AND FURTHER PROPERTIES OF MATRICES. Miscellaneous Applications. The Discrete and Fast Fourier Transforms. Rank and the Equivalence of Matrices. The Existence of Green's Functions for General Linear Differential Systems. Quadratic Forms. Characterist