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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.
- This new text incorporates Maple by Maplesoft - a world leader in mathematical and analytical software
- Rigorous engineering mathematics topics are now made accessible with the emphasis of visuals, numerous examples, and interesting mathematical models
- Provides students with an efficient introduction to the required math necessary for engineering as well as the tools for inquiry and analysis used in modeling and problem solving.
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.
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
Cengage provides a range of supplements that are updated in coordination with the main title selection. For more information about these supplements, contact your Learning Consultant.
Instructor’s Solution Manual
This helpful Instructor’s Solution Manual saves you time as it provides detailed answers to nearly all of the book’s problems.
Instructor's Companion Website
Find everything you need for your course in one place. This password-protected Instructor’s website contains the Instructor’s Solutions Manual, Lecture Note PowerPoint® slides, two additional web chapters, and more than 20 supplementary web modules.