Instructors Manual to Accompany Linear Algebra and Ordinary Differential Equations

111 Pages·2018·2.873 MB·English

by  Jeffrey, Jeffrey

Instructor's Manual
to Accompany
Linear Algebra and
Ordinary
Differential Equations
ALAN JEFFREY
Blackwell Scientific Publications

Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
First published 1990 by CRC Press
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Reissued 2018 by CRC Press
© 1990 by Blackwell Scientific Publications, Inc.
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CONTENTS
Pan 1: Mathematical Prerequisites
1. Review of Topics from Analysis
1.2 Intervals and inequalities (pages 14,15), 1
1.3 Mathematical induction (pages 20-22), 1
1.4 Polynomials and partial fractions (page 28), 2
1.6 Complex numbers in Cartesian form (pages 43-45), 2
1.7 Complex numbers in polar form. Roots (pages 55—57), 2
1.8 Some properties of integrals (pages 72-78), 3
1.9 Linear difference equations (page 87), 5
Part 2: Vectors and Linear Algebra
2. Algebra of Vectors
3
2.3 Vectors - a geometrical approach in R (pages 104,105), 7
2.4 Vectors in component form (pages 113,114), 7
2.5 Scalar product (dot product) (pages 122,123), 8
2.6 Vector product (cross product) (pages 129,130), 9
2.7 Combinations of scalar and vector products (pages 134-136), 9
2.8 Geometrical applications of scalar and vector products (pages
145,146), 9
2.9 Vector spaces (pages 152-154), 10
3. Matrices
3.2 Addition of matrices, multiplication by a number and the transposition
operation (pages 162-167), 11
3.3 Matrix multiplication. Linear transformations. Differentiation (pages
180-185), 12
3.4 Systems of linear equations. Solution by elimination (pages
194-198), 13
3.5 Linear independence. Rank. Reduced echelon form (pages 208, 209), 14
3.7 Determinants (pages 225-228), 15
3.8 Determinants and rank. Cramer's rule (page 231), 15
3.9 Inverse matrices (pages 239-243), 16
3.10 Algebraic eigenvalue problems. Eigenvalues (pages 263—267), 17
3.11 Diagonalizability of matrices. The Cayley—Hamilton theorem (pages
277-280), 19
3.12 Quadratic forms (pages 291-293), 21
3.13 The LU and Cholesky factorization methods (page 305-307), 22
Part 3: Ordinary Differential Equations
4. First Order Ordinary Differential Equations
4.1 Differential equations and their origins (pages 317-319), 24
4.2 First order differential equations and isoclines (pages 324-327), 24
4.3 Separable equations (pages 336-338), 24
4.4 Exact differential equations and integrating factors (pages 345—347), 26
4.5 Linear first order differential equations (pages 356-361), 27
4.6 Orthogonal and isogonal trajectories (pages 370,371), 29
4.7 Existence, uniqueness and an iterative method of solution (pages
376-378), 29
4.8 Numerical solution of first order equations by the Runge-Kutta method
(pages 380,381), 31
5. Linear Higher Order Ordinary Differential Equations
5.1 Linear higher order ordinary differential equations (pages 394-397), 33
5.2 Second order constant coefficient equations — homogeneous case (pages
405,406), 33
5.3 Higher order constant coefficient equations — homogeneous case (page
411), 34
5.4 Differential operators (pages 414,415), 35
5.5 Nonhomogeneous linear differential equations (pages 428—431), 35
5.6 General reduction of the order of a linear differential equation. Integral
method (pages 435), 37
5.7 Oscillatory behavior (pages 462-466), 37
5.8 Reduction to the normal form u"-f(x)u=0 (pages 469), 39
5.9 The Green's function (pages 476,477), 39
6. Systems of Linear Differential Equations
6.1 First order linear homogeneous systems of differential equations (pages
499-506), 41
6.2 First order linear nonhomogeneous systems of differential equations
(pages 513-515), 44
6.3 Second order linear systems of differential equations (pages
540-545), 46
6.4 Qualitative theory: the phase plane and stability (pages 583-591), 48
6.5 Numerical solution of systems by the Runge-Kutta method (page 593),
51
7. Laplace Transform and z-transform
7.1 The Laplace transform — introductory ideas (pages 604,605), 53
7.2 Operational properties of the Laplace transform (pages 650-661), 53
7.3 Applications of the Laplace transform (pages 697—710), 59
7.4 The z-transform (pages 734-737), 63
7.5 Applications of the z—transform (pages 749,750), 66
8. Series Solution of Ordinary Differential Equations
8.1 Sequences, convergence and power series (pages 769-778), 68
8.2 Solving differential equations by Taylor series (pages 781-783), 70
8.3 Solution in the neighborhood of an ordinary point (pages 791,792), 70
8.4 Legendre's equation and Legendre polynomials (pages 799-803), 72
8.5 The gamma function T(x) (pages 807-810), 74
8.6 Frobenius' method and its extension (pages 830-S34), 75
8.7 Bessel functions (pages 846-850), 76
8.8 Asymptotic expansions (pages 874-878), 77
8.9 Numerical solution of second order equations by the Runge—Kutta
method (page 884), 79
9. Fourier Series, Sturm-Liouville Problems and Orthogonal Functions
9.1 Trigonometric series, periodic extension and convergence (pages
905-907), 81
9.2 The formal development of Fourier series (pages 946-957), 83
9.3 Convergence of Fourier series and related results (pages 969-971), 93
9.4 Integration and differentiation of Fourier series (pages 976-978), 93
9.7 Numerical harmonic analysis (page 988), 94
9.8 Representation of functions using orthogonal systems. Sturm-Liouville
problems (pages 997,998), 95
9.9 Expansions in terms of Bessel functions (pages 1004—1006), 96
9.10 Orthogonal polynomials (pages 1014-1016), 98