Fundamental Math and Physics for Scientists and Engineers

Provides a concise overview of the core undergraduate physics and applied mathematics curriculum for students and practitioners of science and engineering

Fundamental Math and Physics for Scientists and Engineers summarizes college and university level physics together with the mathematics frequently encountered in engineering and physics calculations. The presentation provides straightforward, coherent explanations of underlying concepts emphasizing essential formulas, derivations, examples, and computer programs. Content that should be thoroughly mastered and memorized is clearly identified while unnecessary technical details are omitted. Fundamental Math and Physics for Scientists and Engineers is an ideal resource for undergraduate science and engineering students and practitioners, students reviewing for the GRE and graduate-level comprehensive exams, and general readers seeking to improve their comprehension of undergraduate physics.

• Covers topics frequently encountered in undergraduate physics, in particular those appearing in the Physics GRE subject examination

• Reviews relevant areas of undergraduate applied mathematics, with an overview chapter on scientific programming

• Provides simple, concise explanations and illustrations of underlying concepts

  Succinct yet comprehensive, Fundamental Math and Physics for Scientists and Engineers constitutes a reference for science and engineering students, practitioners and non-practitioners alike.

  Autorentext

David Yevick, P. Eng. (Ontario) is Professor of Physics at the University of Waterloo, Canada. He received his A.B. and Ph.D. degrees respectively from Harvard University in Physics (1973) and Princeton University in Particle Physics (1977). Dr. Yevick is a leading scientist in the numerical simulation of optical communication systems, in particular electric field propagation in guided-wave optics, optical processes in semiconductors, and communication system modeling. Dr. Yevick is a fellow of the APS, OSA, and IEEE. Hannah Yevick holds a Ph.D. in Biological Physics from the Curie Institute, France, as well as a M.A. from Columbia University, and a B.A. from the University of Pennsylvania in Physics. Her experience with the Physics GRE and graduate comprehensive exams has enhanced the text.

Inhalt
1 Introduction 1

2 Problem Solving 3

2.1 Analysis 3

2.2 Test-Taking Techniques 4

2.2.1 Dimensional Analysis 5

3 Scientific Programming 6

3.1 Language Fundamentals 6

3.1.1 Octave Programming 7

4 Elementary Mathematics 12

4.1 Algebra 12

4.1.1 Equation Manipulation 12

4.1.2 Linear Equation Systems 13

4.1.3 Factoring 14

4.1.4 Inequalities 15

4.1.5 Sum Formulas 16

4.1.6 Binomial Theorem 17

4.2 Geometry 17

4.2.1 Angles 18

4.2.2 Triangles 18

4.2.3 Right Triangles 19

4.2.4 Polygons 20

4.2.5 Circles 20

4.3 Exponential, Logarithmic Functions, and Trigonometry 21

4.3.1 Exponential Functions 21

4.3.2 Inverse Functions and Logarithms 21

4.3.3 Hyperbolic Functions 22

4.3.4 Complex Numbers and Harmonic Functions 23

4.3.5 Inverse Harmonic and Hyperbolic Functions 25

4.3.6 Trigonometric Identities 26

4.4 Analytic Geometry 28

4.4.1 Lines and Planes 28

4.4.2 Conic Sections 29

4.4.3 Areas, Volumes, and Solid Angles 31

5 Vectors and Matrices 32

5.1 Matrices and Matrix Products 32

5.2 Equation Systems 34

5.3 Traces and Determinants 35

5.4 Vectors and Inner Products 38

5.5 Cross and Outer Products 40

5.6 Vector Identities 41

5.7 Rotations and Orthogonal Matrices 42

5.8 Groups and Matrix Generators 43

5.9 Eigenvalues and Eigenvectors 45

5.10 Similarity Transformations 48

6 Calculus of a Single Variable 50

6.1 Derivatives 50

6.2 Integrals 54

6.3 Series 60

7 Calculus of Several Variables 62

7.1 Partial Derivatives 62

7.2 Multidimensional Taylor Series and Extrema 66

7.3 Multiple Integration 67

7.4 Volumes and Surfaces of Revolution 69

7.5 Change of Variables and Jacobians 70

8 Calculus of Vector Functions 72

8.1 Generalized Coordinates 72

8.2 Vector Differential Operators 77

8.3 Vector Differential Identities 81

8.4 Gauss's and Stokes' Laws and Green's Identities 82

8.5 Lagrange Multipliers 83

9 Probability Theory and Statistics 85

9.1 Random Variables, Probability Density, and Distributions 85

9.2 Mean, Variance, and Standard Deviation 86

9.3 Variable Transformations 86

9.4 Moments and Moment-Generating Function 86

9.5 Multivariate Probability Distributions, Covariance, and Correlation 87

9.6 Gaussian, Binomial, and Poisson Distributions 87

9.7 Least Squares Regression 91

9.8 Error Propagation 92

9.9 Numerical Models 93

10 Complex Analysis 94

10.1 Functions of a Complex Variable 94

10.2 Derivatives, Analyticity, and the CauchyRiemann Relations 95

10.3 Conformal Mapping 96

10.4 Cauchy's Theorem and Taylor and Laurent Series 97

10.5 Residue Theorem 101

10.6 Dispersion Relations 105

10.7 Method of Steepest Decent 106

11 Differential Equations 108

11.1 Linearity, Superposition, and Initial and Boundary Values 108

11.2 Numerical Solutions 109

11.3 First-Order Differential Equations 112

11.4 Wronskian 114

11.5 Factorization 115

11.6 Method of Undetermined Coefficients 115

11.7 Variation of Parameters 116

11.8 Reduction of Order 118

11.9 Series Solution and Method of Frobenius 118

11.10 Systems of Equations, Eigenvalues, and Eigenvectors 119

12 Transform Theory 122

12.1 Eigenfunctions and Eigenvectors 122 1...