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/ **ISBN-13:** 9781111427412

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PART I:

1. FIRST-ORDER DIFFERENTIAL EQUATIONS.

Terminology and Separable Equations. Linear Equations. Exact Equations. Homogeneous, Bernoulli and Riccsti Equations. Additional Applications. Existence and Uniqueness Questions.

2. LINEAR SECOND-ORDER EQUATIONS.

The Linear Second-Order Equations. The Constant Coefficient Case. The Nonhomogeneous Equation. Spring Motion. Euler’s Differential Equation.

3. THE LAPLACE TRANSFORM

Definition and Notation. Solution of Initial Value Problems. Shifiting and the Heaviside Function. Convolution. Impulses and the Delta Function. Solution of Systems. Polynomial Coefficients. Appendix on Partial Fractions Decompositions.

4. SERIES SOLUTIONS.

Power Series Solutions. Frobenius Solutions.

5. APPROXIMATION OF SOLUTIONS

Direction Fields. Euler’s Method. Taylor and Modified Euler Methods.

PART II:

6. VECTORS AND VECTOR SPACES.

Vectors in the Plane and 3 – Space. The Dot Product. The Cross Product. The Vector Space Rn. Orthogonalization. Orthogonal Complements and Projections. The Function Space C[a,b].

7. MATRICES AND LINEAR SYSTEMS.

Matrices. Elementary Row Operations. Reduced Row Echelon Form. Row and Column Spaces. Homogeneous Systems. Nonhomogeneous Systems. Matrix Inverses. Least Squares Vectors and Data Fitting. LU – Factorization. Linear Transformations.

8. DETERMINANTS.

Definition of the Determinant. Evaluation of Determinants I. Evaluation of Determinants II. A Determinant Formula for A-1. Cramer’s Rule. The Matrix Tree Theorem.

9. EIGENVALUES, DIAGONALIZATION AND SPECIAL MATRICES

Eigenvalues and Eigenvectors. Diagonalization. Some Special Types of Matrices.

10. SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS

Linear Systems. Solution of X’=AX for Constant A. Solution of X’=AX+G. Exponential Matrix Solutions. Applications and Illustrations of Techniques. Phase Portaits.

PART III:

11. VECTOR DIFFERENTIAL CALCULUS.

Vector Functions of One Variable. Velocity and Curvature. Vector Fields and Streamlines. The Gradient Field. Divergence and Curl.

12. VECTOR INTEGRAL CALCULUS.

Line Integrals. Green’s Theorem. An Extension of Green’s Theorem. Independence of Path and Potential Theory. Surface Integrals. Applications of Surface Integrals. Lifting Green’s Theorem to R3. The Divergence Theorem of Gauss. Stokes’s Theorem. Curvilinear Coordinates.

PART IV:

13. FOURIER SERIES.

Why Fourier Series? The Fourier Series of a Function. Sine and Cosine Series. Integration and Differentiation of Fourier Series. Phase Angle Form. Complex Fourier Series. Filtering of Signals.

14. THE FOURIER INTEGRAL AND TRANSFORMS.

The Fourier Integral. Fourier Cosine and Sine Integrals. The Fourier Transform. Fourier Cosine and Sine Transforms. The Discrete Fourier Transform. Sampled Fourier Series. DFT Approximation of the Fourier Transform.

15. SPECIAL FUNCTIONS AND EIGENFUNCTION EXPANSIONS.

Eigenfunction Expansions. Legendre Polynomials. Bessel Functions.

PART V:

16. THE WAVE EQUATION.

Derivation of the Wave Equation. Wave Motion on an Interval. Wave Motion in an Infinite Medium. Wave Motion in a Semi-Infinite Medium. Laplace Transform Techniques. Characteristics and d’Alembert’s Solution. Vibrations in a Circular Membrane I. Vibrations in a Circular Membrane II. Vibrations in a Rectangular Membrane.

17. THE HEAT EQUATION.

Initial and Boundary Conditions. The Heat Equation on [0, L]. Solutions in an Infinite Medium. Laplace Transform Techniques. Heat Conduction in an Infinite Cylinder. Heat Conduction in a Rectangular Plate.

18. 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 Equation for a Sphere. The Neumann Problem.

PART VI:

19. COMPLEX NUMBERS AND FUNCTIONS.

Geometry and Arithmetic of Complex Numbers. Complex Functions. The Exponential and Trigonometric Functions. The Complex Logarithm. Powers.

20. COMPLEX INTEGRATION.

The Integral of a Complex Function. Cauchy’s Theorem. Consequences of Cauchy’s Theorem.

21. SERIES REPRESENTATIONS OF FUNCTIONS.

Power Series. The Laurent Expansion.

22. SINGULARITIES AND THE RESIDUE THEOREM.

Singularities. The Residue Theorem. Evaluation of Real Integrals. Residues and the Inverse Laplace Transform.

23. CONFORMAL MAPPINGS AND APPLICATIONS.

Conformal Mappings. Construction of Conformal Mappings. Conformal Mappings and Solutions of Dirichlet Problems. Models of Plane Fluid Flow.

APPENDIX: A MAPLE PRIMER.

ANSWERS TO SELECTED PROBLEMS.

ONLINE CONTENT:

ADDITIONAL CHAPTER: COUNTING AND PROBABILITY

ADDITIONAL CHAPTER: STATISTICS

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