Description
Mathematical Methods for Engineering and Science by Merle C. Potter, ISBN-13: 978-3031261503
[PDF eBook eTextbook]
- Publisher: Springer; 2nd ed. 2023 edition (10 March 2023)
- Language: English
- 514 pages
- ISBN-10: 303126150X
- ISBN-13: 978-3031261503
This book introduces undergraduate students of engineering and science to applied mathematics essential to the study of many problems. Topics are differential equations, power series, Laplace transforms, matrices and determinants, vector analysis, partial differential equations, complex variables, and numerical methods. Approximately, 160 examples and 1000 homework problems aid students in their study. This book presents mathematical topics using derivations rather than theorems and proofs. This textbook is uniquely qualified to apply mathematics to physical applications (spring-mass systems, electrical circuits, conduction, diffusion, etc.), in a manner that is efficient and understandable.
This book is written to support a mathematics course after differential equations, to permit several topics to be covered in one semester, and to make the material comprehensible to undergraduates. An Instructor Solutions Manual, and also a Student Solutions Manual that provides solutions to select problems, is available.
Table of Contents:
Contents
Preface
1 Ordinary Differential Equations
1.1 INTRODUCTION
1.2 DEFINITIONS
1.3 DIFFERENTIAL EQUATIONS OF FIRST ORDER
1.3.1 Separable Equations
1.3.2 Exact Equations
1.3.3 Integrating Factors
1.4 PHYSICAL APPLICATIONS
1.4.1 Simple Electrical Circuits
1.4.2 The Rate Equation
1.4.3 Fluid Flow
1.4.4 Dynamics
1.5 LINEAR DIFFERENTIAL EQUATIONS
1.6 HOMOGENEOUS SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS
1.7 SPRING–MASS SYSTEM—FREE MOTION
1.7.1 Undamped Motion
1.7.2 Damped Motion
1.7.3 The Electrical Circuit Analog
1.8 NONHOMOGENEOUS SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS
1.9 SPRING–MASS SYSTEM—FORCED MOTION
1.9.1 Resonance
1.9.2 Near Resonance
1.9.3 Forced Oscillations with Damping
1.10 PERIODIC INPUT FUNCTIONS—FOURIER SERIES
1.10.1 Even and Odd Functions
1.10.2 Half-Range Expansions
1.10.3 Forced Oscillations
1.11 THE CAUCHY EQUATION
1.12 VARIATION OF PARAMETERS
1.13 MISCELLANEOUS INFORMATION
PROBLEMS
2 Power-Series Methods
2.1 POWER SERIES
2.2 LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS
2.3 LEGENDRE’S EQUATION
2.4 THE METHOD OF FROBENIUS
2.4.1 Distinct Roots Not Differing by an Integer
2.4.2 Double Roots
2.4.3 Roots Differing by an Integer
2.5 BESSEL’S EQUATION
PROBLEMS
3 Laplace Transforms
3.1 INTRODUCTION
3.2 THE LAPLACE TRANSFORM
3.3 LAPLACE TRANSFORMS OF DERIVATIVES AND INTEGRALS
3.4 DERIVATIVES AND INTEGRALS OF LAPLACE TRANSFORMS
3.5 LAPLACE TRANSFORMS OF PERIODIC FUNCTIONS
3.6 INVERSE TRANSFORMS—PARTIAL FRACTIONS
3.6.1 Unrepeated Linear Factor (s – a)
3.6.2 Repeated Linear Factor (s − a)m
3.7 SOLUTION OF DIFFERENTIAL EQUATIONS
PROBLEMS
4 Matrices and Determinants
4.1 INTRODUCTION
4.2 MATRICES
4.3 ADDITION OF MATRICES
4.4 THE TRANSPOSE AND SOME SPECIAL MATRICES
4.5 MATRIX MULTIPLICATION—DEFINITION
4.6 MATRIX MULTIPLICATION—ADDITIONAL PROPERTIES
4.7 DETERMINANTS
4.8 THE ADJOINT AND THE INVERSE MATRICES
4.9 SOLUTION OF SIMULTANEOUS LINEAR ALGEBRAIC EQUATIONS
4.9.1 Nonhomogeneous Sets of Linear Algebraic Equations
4.9.2 Homogeneous Sets of Linear Algebraic Equations
4.9.3 Solutions to Sets of Linear Equations by MATLAB
4.10 LEAST-SQUARES FIT AND THE PSEUDO INVERSE
4.11 EIGENVALUES AND EIGENVECTORS
4.12 EIGENVALUE PROBLEMS IN ENGINEERING
4.12.1 Moments of Inertia
4.12.2 Stress
4.12.3 Linear Dynamic Systems and Stability
PROBLEMS
5 Vector Analysis
5.1 INTRODUCTION
5.2 VECTOR ALGEBRA
5.2.1 Definitions
5.2.2 Addition and Subtraction
5.2.3 Components of a Vector
5.2.4 Multiplication
5.3 VECTOR DIFFERENTIATION
5.3.1 Ordinary Differentiation
5.3.2 Partial Differentiation
5.4 THE GRADIENT
5.5 CYLINDRICAL AND SPHERICAL COORDINATES
5.5.1 Cylindrical Coordinates
5.5.2 Spherical Coordinates
5.6 INTEGRAL THEOREMS
5.6.1 The Divergence Theorem
5.6.2 Stokes’s Theorem
PROBLEMS
6 Partial Differential Equations
6.1 INTRODUCTION
6.2 WAVE MOTION
6.2.1 Vibration of a Stretched, Flexible String
6.2.2 The Vibrating Membrane
6.2.3 Longitudinal Vibrations of an Elastic Bar
6.2.4 Transmission-Line Equations
6.3 THE D’ALEMBERT SOLUTION OF THE WAVE EQUATION
6.4 SEPARATION OF VARIABLES
6.5 DIFFUSION
6.6 SOLUTION OF THE DIFFUSION EQUATION
6.6.1 A Long, Insulated Rod with Ends at Fixed Temperatures
6.6.2 A Long, Totally Insulated Rod
6.6.3 Two-Dimensional Heat Conduction in a Long, Rectangular Bar
6.7 ELECTRIC POTENTIAL ABOUT A SPHERICAL SURFACE
6.8 HEAT TRANSFER IN A CYLINDRICAL BODY
6.9 GRAVITATIONAL POTENTIAL
PROBLEMS
7 Complex Variables
7.1 INTRODUCTION
7.2 COMPLEX NUMBERS
7.3 ELEMENTARY FUNCTIONS
7.4 ANALYTIC FUNCTIONS
7.5 COMPLEX INTEGRATION
7.5.1 Green’s Theorem
7.5.2 Cauchy’s Integral Theorem
7.5.3 Cauchy’s Integral Formula
7.6 SERIES
7.7 RESIDUES
PROBLEMS
8 Numerical Methods
8.1 INTRODUCTION
8.2 FINITE-DIFFERENCE OPERATORS
8.3 THE DIFFERENTIAL OPERATOR RELATED TO THE DIFFERENCE OPERATORS
8.4 TRUNCATION ERROR
8.5 NUMERICAL INTEGRATION
8.6 NUMERICAL INTERPOLATION
8.7 ROOTS OF EQUATIONS
8.8 INITIAL-VALUE PROBLEMS—ORDINARY DIFFERENTIAL EQUATIONS
8.8.1 Taylor’s Method
8.8.2 Euler’s Method
8.8.3 Adams’ Method
8.8.4 Runge–Kutta Methods
8.8.5 Direct Method
8.9 HIGHER-ORDER EQUATIONS
8.10 BOUNDARY-VALUE PROBLEMS—ORDINARY DIFFERENTIAL EQUATIONS
8.10.1 Iterative Method
8.10.2 Superposition
8.10.3 Simultaneous Equations
8.11 NUMERICAL STABILITY
8.12 NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
8.12.1 The Diffusion Equation
8.12.2 The Wave Equation
8.12.3 Laplace’s Equation
PROBLEMS
Bibliography
Appendix A
Appendix B Introduction to MATLAB
B.1 INTRODUCTION
B.2 REAL AND COMPLEX NUMBERS
B.3 VECTORS AND MATRICES
B.4 FORMAT AND SCIENTIFIC NOTATION
B.5 PROGRAMMING LOOPS
B.6 PLOTTING
B.7 STRING ARRAYS
B.8 MATLAB FILES, INPUT, AND OUTPUT
B.8.1 Setting the Path
B.8.2 Script Files, or m-Files
B.8.3 Input Files and Output Files
B.8.4 Interactive Input and Output
B.9 FUNCTIONS
B.10 THE WORKSPACE BROWSER
B.11 FINAL REMARKS
Answers to Selected Problems
Index
Merle C. Potter graduated with honors from MTU in 1958 with a B.S. degree in Mechanical Engineering and continued his education in Engineering Mechanics earning an M.S. degree from MTU in 1961. He continued his graduate education at the University of Michigan and earned a second M.S. degree in Aerospace Engineering in 1964 and a Ph.D. in Engineering Mechanics in 1965. Merle began his teaching career at Michigan Tech as an instructor in 1958. While pursuing his graduate degrees from the University of Michigan, he held the positions of the teaching fellow and instructor. Merle began an over thirty-year career with Michigan State University in 1965 as an assistant professor. He was promoted to an associate professor in 1969 and became a full professor in 1971. Merle retired and was given the title of the professor emeritus in 1994 and assisted the university as a visiting professor until 1998. For over forty years, Merle has had a significant influence on the engineering profession both as a professor and an author. He has written and co-authored many books on fluid mechanics, mathematical methods, thermodynamics, differential equations and review guides for the GRE, GMAT, fundamentals of engineering, and professional engineering exams. Merle has published nearly twenty archived journal papers, was the advisor for fourteen Ph.D. students, and was the recipient of several. NSF grants to support his research efforts. Merle was a member of Tau Beta Pi and ASME while a student at MTU. He has also participated in Phi Eta Sigma, Phi Kappa Phi, Pi Tau Sigma, Sigma Xi, ASEE, and AAM and was an active member of many major MSU committees. Merle has been honored with the Ford Faculty Scholarship in l961, the Teacher Scholar Award in 1969, the ASME Centennial Award in 1980, and the James Harry Potter Gold Medal in 2008.
Brian F. Feeny is a professor in the Department of Mechanical Engineering at Michigan State University. He obtained a B.S. with honors in Engineering Mechanics from the University of Wisconsin in 1984, an M.S. in Engineering Mechanics from Virginia Tech in 1986, and his Ph.D. in Theoretical and Applied Mechanics from Cornell University in 1990. He then held a postdoc in the Institute of Robotics at the Swiss Federal Institute of Technology (ETH) in Zürich and joined Michigan State University as an assistant professor in 1992. During his time at MSU, Brian has held visiting appointments at the National Institute of Standards and Technologies Manufacturing Engineering Lab and the Huazhong Agricultural University. He is a fellow of the American Society of Mechanical Engineers (ASME). He served as the chair of the ASME Technical Committee on Vibration and Sound and as an associate editor the ASME Journal of Vibration and Acoustics and the ASME Journal of Computational and Nonlinear Dynamics. He directs his department’s student exchange program between MSU and RWTH Aachen. His research is in vibrations, and he has published and supervised graduate students in areas including modal decomposition, parametric excitation, chaos, vibration with friction, wind-turbine blade vibration, pendulum vibration absorbers, and bio-locomotion.
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