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E-grāmata: Mathematical Methods in Engineering

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(University of Notre Dame, Indiana), (University of Notre Dame, Indiana)
  • Formāts: PDF+DRM
  • Izdošanas datums: 26-Jan-2015
  • Izdevniecība: Cambridge University Press
  • Valoda: eng
  • ISBN-13: 9781316190920
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Mathematical Methods in EngineeringPalielināt
  • Formāts: PDF+DRM
  • Izdošanas datums: 26-Jan-2015
  • Izdevniecība: Cambridge University Press
  • Valoda: eng
  • ISBN-13: 9781316190920
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This book is designed for engineering graduate students. It connects mathematics to a variety of methods used for engineering problems by walking the reader stepwise through examples that have been worked in detail followed by numerous homework problems to reinforce learning and connect the subject matter to engineering applications.

This text focuses on a variety of topics in mathematics in common usage in graduate engineering programs including vector calculus, linear and nonlinear ordinary differential equations, approximation methods, vector spaces, linear algebra, integral equations and dynamical systems. The book is designed for engineering graduate students who wonder how much of their basic mathematics will be of use in practice. Following development of the underlying analysis, the book takes students step-by-step through a large number of examples that have been worked in detail. Students can choose to go through each step or to skip ahead if they so desire. After seeing all the intermediate steps, they will be in a better position to know what is expected of them when solving homework assignments, examination problems, and when they are on the job. Each chapter concludes with numerous exercises for the student that reinforce the chapter content and help connect the subject matter to a variety of engineering problems. Students today have grown up with computer-based tools including numerical calculations and computer graphics; the worked-out examples as well as the end-of-chapter exercises often use computers for numerical and symbolic computations and for graphical display of the results.

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Designed for engineering graduate students, this book connects basic mathematics to a variety of methods used in engineering problems.
Preface xiii
1 Multivariable Calculus
1(63)
1.1 Implicit Functions
1(10)
1.1.1 One Independent Variable
1(4)
1.1.2 Many Independent Variables
5(1)
1.1.3 Many Dependent Variables
6(5)
1.2 Inverse Function Theorem
11(2)
1.3 Functional Dependence
13(5)
1.4 Leibniz Rule
18(2)
1.5 Optimization
20(11)
1.5.1 Unconstrained Optimization
21(1)
1.5.2 Calculus of Variations
22(6)
1.5.3 Constrained Optimization: Lagrange Multipliers
28(3)
1.6 Non-Cartesian Coordinate Transformations
31(33)
1.6.1 Jacobian Matrices and Metric Tensors
34(9)
1.6.2 Co variance and Contravariance
43(6)
1.6.3 Differentiation and Christoffel Symbols
49(4)
1.6.4 Summary of Identities
53(1)
1.6.5 Nonorthogonal Coordinates: Alternate Approach
54(3)
1.6.6 Orthogonal Curvilinear Coordinates
57(2)
Exercises
59(5)
2 Vectors and Tensors in Cartesian Coordinates
64(51)
2.1 Preliminaries
64(17)
2.1.1 Cartesian Index Notation
64(3)
2.1.2 Direction Cosines
67(5)
2.1.3 Scalars
72(1)
2.1.4 Vectors
72(1)
2.1.5 Tensors
73(8)
2.2 Algebra of Vectors
81(3)
2.2.1 Definitions and Properties
81(1)
2.2.2 Scalar Product
82(1)
2.2.3 Cross Product
82(1)
2.2.4 Scalar Triple Product
83(1)
2.2.5 Identities
83(1)
2.3 Calculus of Vectors
84(8)
2.3.1 Vector Functions
84(1)
2.3.2 Differential Geometry of Curves
84(8)
2.4 Line Integrals
92(2)
2.5 Surface Integrals
94(1)
2.6 Differential Operators
94(6)
2.6.1 Gradient
95(3)
2.6.2 Divergence
98(1)
2.6.3 Curl
98(1)
2.6.4 Laplacian
99(1)
2.6.5 Identities
99(1)
2.7 Curvature Revisited
100(4)
2.7.1 Trajectory
100(3)
2.7.2 Principal
103(1)
2.7.3 Gaussian
103(1)
2.7.4 Mean
104(1)
2.8 Special Theorems
104(11)
2.8.1 Green's Theorem
104(1)
2.8.2 Divergence Theorem
105(3)
2.8.3 Green's Identities
108(1)
2.8.4 Stokes' Theorem
108(2)
2.8.5 Extended Leibniz Rule
110(1)
Exercises
110(5)
3 First-Order Ordinary Differential Equations
115(31)
3.1 Paradigm Problem
115(2)
3.2 Separation of Variables
117(1)
3.3 Homogeneous Equations
118(2)
3.4 Exact Equations
120(2)
3.5 Integrating Factors
122(1)
3.6 General Linear Solution
123(2)
3.7 Bernoulli Equation
125(1)
3.8 Riccati Equation
126(3)
3.9 Reduction of Order
129(2)
3.9.1 Dependent Variable y Absent
129(1)
3.9.2 Independent Variable x Absent
129(2)
3.10 Factorable Equations
131(1)
3.11 Uniqueness and Singular Solutions
132(2)
3.12 Clairaut Equation
134(2)
3.13 Picard Iteration
136(3)
3.14 Solution by Taylor Series
139(1)
3.15 Delay Differential Equations
140(6)
Exercises
141(5)
4 Linear Ordinary Differential Equations
146(73)
4.1 Linearity and Linear Independence
146(3)
4.2 Complementary Functions
149(7)
4.2.1 Constant Coefficients
149(5)
4.2.2 Variable Coefficients
154(2)
4.3 Particular Solutions
156(13)
4.3.1 Undetermined Coefficients
156(2)
4.3.2 Variation of Parameters
158(2)
4.3.3 Green's Functions
160(6)
4.3.4 Operator D
166(3)
4.4 Sturm-Liouville Analysis
169(24)
4.4.1 General Formulation
170(1)
4.4.2 Adjoint of Differential Operators
171(4)
4.4.3 Linear Oscillator
175(4)
4.4.4 Legendre Differential Equation
179(3)
4.4.5 Chebyshev Equation
182(3)
4.4.6 Hermite Equation
185(3)
4.4.7 Laguerre Equation
188(1)
4.4.8 Bessel Differential Equation
189(4)
4.5 Fourier Series Representation
193(7)
4.6 Fredholm Alternative
200(1)
4.7 Discrete and Continuous Spectra
201(1)
4.8 Resonance
202(5)
4.9 Linear Difference Equations
207(12)
Exercises
211(8)
5 Approximation Methods
219(60)
5.1 Function Approximation
220(4)
5.1.1 Taylor Series
220(2)
5.1.2 Pade Approximants
222(2)
5.2 Power Series
224(14)
5.2.1 Functional Equations
224(2)
5.2.2 First-Order Differential Equations
226(4)
5.2.3 Second-Order Differential Equations
230(7)
5.2.4 Higher-Order Differential Equations
237(1)
5.3 Taylor Series Solution
238(2)
5.4 Perturbation Methods
240(28)
5.4.1 Polynomial and Transcendental Equations
240(4)
5.4.2 Regular Perturbations
244(3)
5.4.3 Strained Coordinates
247(6)
5.4.4 Multiple Scales
253(3)
5.4.5 Boundary Layers
256(5)
5.4.6 Interior Layers
261(2)
5.4.7 WKBJ Method
263(3)
5.4.8 Solutions of the Type es(x)
266(1)
5.4.9 Repeated Substitution
267(1)
5.5 Asymptotic Methods for Integrals
268(11)
Exercises
271(8)
6 Linear Analysis
279(111)
6.1 Sets
279(1)
6.2 Integration
280(3)
6.3 Vector Spaces
283(21)
6.3.1 Normed
288(9)
6.3.2 Inner Product
297(7)
6.4 Gram-Schmidt Procedure
304(3)
6.5 Projection of Vectors onto New Bases
307(23)
6.5.1 Nonorthogonal
307(6)
6.5.2 Orthogonal
313(1)
6.5.3 Orthonormal
314(10)
6.5.4 Reciprocal
324(6)
6.6 Parseval's Equation, Convergence, and Completeness
330(1)
6.7 Operators
330(9)
6.7.1 Linear
332(2)
6.7.2 Adjoint
334(3)
6.7.3 Inverse
337(2)
6.8 Eigenvalues and Eigenvectors
339(11)
6.9 Rayleigh Quotient
350(4)
6.10 Linear Equations
354(5)
6.11 Method of Weighted Residuals
359(12)
6.12 Ritz and Rayleigh-Ritz Methods
371(2)
6.13 Uncertainty Quantification Via Polynomial Chaos
373(17)
Exercises
379(11)
7 Linear Algebra
390(90)
7.1 Paradigm Problem
390(1)
7.2 Matrix Fundamentals and Operations
391(8)
7.2.1 Determinant and Rank
391(1)
7.2.2 Matrix Addition
392(1)
7.2.3 Column, Row, and Left and Right Null Spaces
392(2)
7.2.4 Matrix Multiplication
394(2)
7.2.5 Definitions and Properties
396(3)
7.3 Systems of Equations
399(11)
7.3.1 Overconstrained
400(3)
7.3.2 Underconstrained
403(2)
7.3.3 Simultaneously Over- and Underconstrained
405(1)
7.3.4 Square
406(2)
7.3.5 Fredholm Alternative
408(2)
7.4 Eigenvalues and Eigenvectors
410(5)
7.4.1 Ordinary
410(4)
7.4.2 Generalized in the Second Sense
414(1)
7.5 Matrices as Linear Mappings
415(1)
7.6 Complex Matrices
416(3)
7.7 Orthogonal and Unitary Matrices
419(7)
7.7.1 Givens Rotation
422(1)
7.7.2 Householder Reflection
423(3)
7.8 Discrete Fourier Transforms
426(6)
7.9 Matrix Decompositions
432(24)
7.9.1 L · D · U
432(2)
7.9.2 Cholesky
434(1)
7.9.3 Row Echelon Form
435(4)
7.9.4 Q · U
439(2)
7.9.5 Diagonalization
441(6)
7.9.6 Jordan Canonical Form
447(2)
7.9.7 Schur
449(1)
7.9.8 Singular Value
450(3)
7.9.9 Polar
453(3)
7.9.10 Hessenberg
456(1)
7.10 Projection Matrix
456(2)
7.11 Least Squares
458(3)
7.11.1 Unweighted
459(1)
7.11.2 Weighted
460(1)
7.12 Neumann Series
461(1)
7.13 Matrix Exponential
462(2)
7.14 Quadratic Form
464(3)
7.15 Moore-Penrose Pseudoinverse
467(13)
Exercises
470(10)
8 Linear Integral Equations
480(17)
8.1 Definitions
480(1)
8.2 Homogeneous Fredholm Equations
481(6)
8.2.1 First Kind
481(1)
8.2.2 Second Kind
482(5)
8.3 Inhomogeneous Fredholm Equations
487(3)
8.3.1 First Kind
487(2)
8.3.2 Second Kind
489(1)
8.4 Fredholm Alternative
490(1)
8.5 Fourier Series Projection
490(7)
Exercises
495(2)
9 Dynamical Systems
497(88)
9.1 Iterated Maps
497(4)
9.2 Fractals
501(2)
9.2.1 Cantor Set
501(1)
9.2.2 Koch Curve
502(1)
9.2.3 Menger Sponge
502(1)
9.2.4 Weierstrass Function
503(1)
9.2.5 Mandelbrot and Julia Sets
503(1)
9.3 Introduction to Differential Systems
503(9)
9.3.1 Autonomous Example
504(4)
9.3.2 Nonautonomous Example
508(2)
9.3.3 General Approach
510(2)
9.4 High-Order Scalar Differential Equations
512(2)
9.5 Linear Systems
514(14)
9.5.1 Inhomogeneous with Variable Coefficients
514(1)
9.5.2 Homogeneous with Constant Coefficients
515(10)
9.5.3 Inhomogeneous with Constant Coefficients
525(3)
9.6 Nonlinear Systems
528(17)
9.6.1 Definitions
529(3)
9.6.2 Linear Stability
532(1)
9.6.3 Heteroclinic and Homoclinic Trajectories
533(6)
9.6.4 Nonlinear Forced Mass-Spring-Damper
539(2)
9.6.5 Lyapunov Functions
541(2)
9.6.6 Hamiltonian Systems
543(2)
9.7 Differential-Algebraic Systems
545(4)
9.7.1 Linear Homogeneous
545(3)
9.7.2 Nonlinear
548(1)
9.8 Fixed Points at Infinity
549(5)
9.8.1 Poincare Sphere
549(4)
9.8.2 Projective Space
553(1)
9.9 Bifurcations
554(5)
9.9.1 Pitchfork
555(1)
9.9.2 Transcritical
556(1)
9.9.3 Saddle-Node
557(1)
9.9.4 Hopf
558(1)
9.10 Projection of Partial Differential Equations
559(3)
9.11 Lorenz Equations
562(23)
9.11.1 Linear Stability
563(2)
9.11.2 Nonlinear Stability: Center Manifold Projection
565(4)
9.11.3 Transition to Chaos
569(4)
Exercises
573(12)
Appendix A
585(18)
A.1 Roots of Polynomial Equations
585(4)
A.1.1 First-Order
585(1)
A.1.2 Quadratic
585(1)
A.1.3 Cubic
586(1)
A.1.4 Quartic
587(2)
A.1.5 Quintic and Higher
589(1)
A.2 Cramer's Rule
589(1)
A.3 Gaussian Elimination
590(1)
A.4 Trapezoidal Rule
591(1)
A.5 Trigonometric Relations
591(2)
A.6 Hyperbolic Functions
593(1)
A.7 Special Functions
593(5)
A.7.1 Gamma
593(1)
A.7.2 Error
594(1)
A.7.3 Sine, Cosine, and Exponential Integral
594(1)
A.7.4 Hypergeometric
595(1)
A.7.5 Airy
596(1)
A.7.6 Dirac δ and Heaviside
596(2)
A.8 Complex Numbers
598(5)
A.8.1 Euler's Formula
598(1)
A.8.2 Polar and Cartesian Representations
599(1)
Exercises
600(3)
Bibliography 603(6)
Index 609
Joseph Powers joined the University of Notre Dame in 1989. His research has focused on the dynamics of high-speed reactive fluids and on computational science, especially as it applies to verification and validation of complex multi-scale systems. He has held positions at the NASA Lewis Research Center, the Los Alamos National Laboratory, the Air Force Research Laboratory, the Argonne National Laboratory, and the Chinese Academy of Sciences. He is a member of AIAA, APS, ASME, the Combustion Institute, and SIAM. He is the recipient of numerous teaching awards. Mihir Sen has been active in teaching and in research in thermal-fluids engineering especially in regard to problems relating to modeling, dynamics, and stability since obtaining his PhD from Massachusetts Institute of Technology. He has worked on reacting flows, natural and forced convection, flow in porous media, falling films, boiling, MEMS, heat exchangers, thermal control, and intelligent systems. He joined the University of Notre Dame in 1986 and received the Kaneb Teaching Award from the College of Engineering in 2001 and the Rev. Joyce University Award for Excellence in Undergraduate Teaching in 2009. He is a Fellow of ASME.
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