Mathematical Methods Using Python : Applications in Physics and Engineering

This advanced undergraduate textbook presents a new approach to teaching mathematical methods for scientists and engineers. It provides a practical, pedagogical introduction to utilizing Python in Mathematical and Computational Methods courses. Both analytical and computational examples are integrated from its start. Each chapter concludes with a set of problems designed to help students hone their skills in mathematical techniques, computer programming, and numerical analysis. The book places less emphasis on mathematical proofs, and more emphasis on how to use computers for both symbolic and numerical calculations. It contains 182 extensively documented coding examples, based on topics that students will encounter in their advanced courses in Mechanics, Electronics, Optics, Electromagnetism, Quantum Mechanics etc.An introductory chapter gives students a crash course in Python programming and the most often used libraries (SymPy, NumPy, SciPy, Matplotlib). This is followed by chapters dedicated to differentiation, integration, vectors and multiple integration techniques. The next group of chapters covers complex numbers, matrices, vector analysis and vector spaces. Extensive chapters cover ordinary and partial differential equations, followed by chapters on nonlinear systems and on the analysis of experimental data using linear and nonlinear regression techniques, Fourier transforms, binomial and Gaussian distributions. The book is accompanied by a dedicated GitHub website, which contains all codes from the book in the form of ready to run Jupyter notebooks. A detailed solutions manual is also available for instructors using the textbook in their courses.Key Features: A unique teaching approach which merges mathematical methods and the Python programming skills which physicists and engineering students need in their courses Uses examples and models from physical and engineering systems, to motivate the mathematics being taught Students learn to solve scientific problems in three different ways: traditional pen-and-paper methods, using scientific numerical techniques with NumPy and SciPy, and using Symbolic Python (SymPy). Cover Half Title Title Page Copyright Page Dedication Contents Preface Chapter 1: Introduction to Python 1.1. Data Types and Variables in Python 1.2. Sequences in Python 1.2.1. Lists 1.2.2. Range Sequences and List Comprehensions 1.2.3. Tuple Sequences 1.2.4. Functions on Sequences 1.3. Functions, for Loops and Conditional Statements 1.4. Importing Python Libraries and Packages 1.5. The NumPy Library 1.5.1. Creating NumPy Arrays 1.5.2. Array Functions, Attributes and Methods 1.5.3. Arithmetic Operations with NumPy Arrays 1.5.4. Indexing and Slicing of NumPy Arrays 1.6. The Matplotlib Module 1.6.1. 2D Plots Using Matplotlib 1.6.2. 3D Plots Using Matplotlib 1.7. Symbolic Computation with SymPy 1.8. The lambdify() Function in Python 1.9. End of Chapter Problems Chapter 2: Differentiation 2.1. Derivatives of Single-Variable Functions 2.1.1. Rules for Differentiation 2.2. Differentiating Analytical Functions in Python 2.3. A Detailed Example: Derivation of Wien’s Displacement Law 2.4. Derivatives of Multivariable Functions 2.4.1. Introduction to Partial Differentiation 2.4.2. Total Differentials 2.4.3. Total Derivative of a Function - The Chain Rule Revisited 2.4.4. Maximum and Minimum Problems 2.5. Power Series Approximations of Functions 2.6. Numerical Evaluation of Derivatives 2.7. End of Chapter Problems Chapter 3: Integration 3.1. Integrals 3.2. Review of Elementary Integrals 3.3. Overview of Integration Methods in Python 3.4. Integration by Parts 3.5. Parametric Integration for Definite Integrals 3.6. Integrating Analytical Functions in Python 3.7. Fourier Series 3.8. Improper Integrals and Integrals of Special Functions 3.9. Integrating Functions Defined by NumPy Arrays 3.9.1. Simpson’s Rule 3.10. End of Chapter Problems Chapter 4: Vectors 4.1. Vector Basics 4.2. Scalar and Vector Fields in Python 4.3. Vector Multiplication 4.3.1. The Dot Product 4.3.2. The Cross Product 4.4. Triple Products 4.4.1. Triple Scalar Product 4.4.2. Triple Vector Product 4.5. Non-Cartesian Coordinates 4.5.1. Polar Coordinates 4.5.2. Cylindrical Coordinates 4.5.3. Spherical Coordinates 4.6. Differentiation of Vectors 4.7. Parametric Equations of Lines and Planes 4.8. End of Chapter Problems Chapter 5: Multiple Integrals 5.1. Multiple Integrals 5.2. Evaluation of Double Integrals 5.2.1. The Evaluation of Double Integrals Over a Rectangular Domain 5.2.2. The Evaluation of Double Integrals Over a Non-Rectangular Domain 5.3. Evaluation of Triple Integrals 5.4. Change of Variables in Multiple Integrals 5.4.1. Using Polar Coordinates 5.4.2. Using Cylindrical Coordinates 5.4.3. Using Spherical Coordinates 5.5. Application of Multiple Integrals: Moment of Inertia Tensor 5.5.1. Numerical Evaluation of Multiple Integrals 5.6. End of Chapter Problems Chapter 6: Complex Numbers 6.1. The Complex Plane 6.2. Trigonometric Functions and Complex Exponentials 6.3. Arithmetic with Complex Numbers 6.4. Application of Complex Numbers in AC Circuits 6.5. Equations with Complex Numbers 6.6. Functions of a Complex Variable 6.6.1. Exponentials, Powers and Roots of Complex Numbers 6.6.2. Hyperbolic Functions 6.7. Complex Vectors 6.8. End of Chapter Problems Chapter 7: Matrices 7.1. The Structure of a Matrix 7.1.1. Defining Matrices in Python 7.1.2. Vectors as Matrices 7.2. Matrix Operations 7.2.1. Matrix Equivalence 7.2.2. Multiplication by a Scalar 7.2.3. Matrix Addition 7.2.4. Matrix Multiplication 7.2.5. Trace of a Matrix 7.2.6. Transpose and Hermitian Adjoint of a Matrix 7.3. The Determinant 7.3.1. Calculating Determinants 7.3.2. Determinants as Representations of Area and Volume 7.3.3. The Jacobian Determinant and Volume Transformations 7.4. Matrices and Systems of Linear Equations 7.4.1. Representing Systems of Linear Equations as a Matrix Equation 7.4.2. Representing Systems of Linear Equations as an Augmented Matrix 7.4.3. Homogeneous Equations 7.5. Matrices as Representations of Linear Operators 7.6. Eigenvalues and Eigenvectors 7.7. Diagonalization of a Matrix 7.8. End of Chapter Problems Chapter 8: Vector Analysis 8.1. Scalar and Vector Fields 8.2. The Gradient of a Scalar Field 8.2.1. The Gradient in Other Coordinate Systems 8.3. Properties of the Gradient 8.4. Divergence 8.5. Curl 8.6. Second Derivatives Using ∇ 8.7. Line Integrals 8.8. Conservative Fields 8.9. Area Integrals and Flux 8.10. Green’s Theorem in the Plane 8.11. The Divergence Theorem 8.12. Stokes’s Theorem 8.13. End of Chapter Problems Chapter 9: Vector Spaces 9.1. Definition of a Vector Space 9.2. Finite Dimensional Vector Spaces 9.2.1. Linear Independence 9.2.2. Basis Vectors 9.2.3. Beyond Cartesian Coordinates 9.2.4. The Gram-Schmidt Method 9.3. Vector Spaces of Functions 9.3.1. Definition 9.3.2. Functions and Linear Independence 9.3.3. Inner Products and Functions 9.3.4. Basis Functions 9.3.5. The Gram-Schmidt Method for Functions 9.4. Infinite Dimensional Vector Spaces 9.4.1. Quantum Mechanics and Hilbert Space 9.5. End of Chapter Problems Chapter 10: Ordinary Differential Equations 10.1. Definitions and Examples of Differential Equations 10.2. Separable Differential Equations 10.3. Symbolic Integration of Differential Equations Using SymPy 10.4. General Solution of First Order Linear ODEs: The Integrating Factor Method 10.5. ODEs for Oscillating Systems: General Considerations 10.6. The Simple Harmonic Oscillator 10.7. Numerical Integration of the ODE of a Simple Plane Pendulum 10.8. Damped Harmonic Oscillator 10.8.1. Case I: Overdamped Oscillations 10.8.2. Case II: Underdamped Oscillations 10.8.3. Case III: Critically Damped Oscillations 10.9. Forced Harmonic Oscillator 10.10. The Principle of Linear Superposition 10.11. Electrical Circuits 10.12. Phase Space 10.13. Systems of Differential Equations 10.14. The Legendre Equation 10.14.1. Associated Legendre Functions 10.15. The Bessel Equation 10.16. End of Chapter Problems Chapter 11: Partial Differential Equations 11.1. Separation of Variables 11.2. The Heat Equation 11.2.1. The Product Solution to the One-Dimensional Heat Equation 11.2.2. The Particular Solution of the Heat Equation: The Cooling Process in a Uniform Rod 11.3. The Laplace Equation 11.4. Application of the Laplace Equation in Electrostatics 11.5. The Wave Equation 11.6. Application of the Wave Equation for a Plucked String: Traveling Waves 11.7. The Laplace Equation in Cylindrical Coordinates 11.7.1. Application of the Laplace Equation: Electric Potential Inside a Cylinder 11.8. The Laplace Equation in Spherical Coordinates 11.8.1. Application of the Laplace Equation: The Electric Potential Due to a Charged Sphere 11.9. General Outline for Solving Linear PDEs 11.10. End of Chapter Problems Chapter 12: Analysis of Nonlinear Systems 12.1. The Difference Between Linear and Nonlinear Systems 12.2. Phase Portraits 12.3. Fixed Points and Equilibria 12.3.1. Finding Fixed Points 12.3.2. The Classification of Fixed Points 12.4. Bifurcations of Fixed Points 12.5. The Phase Portrait, Revisited 12.6. Nonlinear Oscillations and Limit Cycles 12.7. Chaos 12.7.1. The Lorenz Equations 12.7.2. Sensitivity to Initial Conditions 12.7.3. Lyapunov Exponents and the Horizon Time 12.8. How to Analyze a Nonlinear System 12.9. End of Chapter Problems Chapter 13: Analysis of Experimental Data 13.1. Estimating Uncertainties 13.2. Propagation of Uncertainties 13.3. Linear Regression and the Method of Least Squares 13.4. Data Transformations 13.4.1. Feature Engineering 13.4.2. Power Law Relationships and Log-Log Plots 13.4.3. Exponential Relationships and Log-Linear (Semi-Log) Plots 13.5. Multivariate Linear Regression 13.6. Nonlinear Regression 13.7. Fourier Transforms 13.8. Discrete Fourier transforms 13.9. Discrete and Continuous Random Variables 13.9.1. Discrete Random Variables 13.9.2. Continuous Random Variables 13.10. Useful Probability Functions 13.10.1. The Binomial Distribution 13.10.2. Normal Distribution 13.10.3. Poisson Distribution 13.11. End of Chapter Problems Further Reading and Additional Resources Index