ENGINEERING MATHEMATICS WEB
CONTENTS PART ONE MODELING, COMPUTERS, AND ERROR ANALYSIS 1 1 MOTIVATION 1 2 PART ORGANIZATION CHAPTER MATHEMATICAL MODELING, NUMERICAL METHODS, AND PROBLEM SOLVING A SIMPLE MATHEMATICAL MODEL CONSERVATION LAWS IN ENGINEERING AND SCIENCE NUMERICAL METHODS COVERED IN THIS BOOK CASE STUDY: IT’S A REAL DRAG PROBLEMS MATLAB FUNDAMENTALS THE MATLAB ENVIRONMENT ASSIGNMENT MATHEMATICAL OPERATIONS USE OF BUILT-IN FUNCTIONS GRAPHICS OTHER RESOURCES CASE STUDY: EXPLORATORY DATA ANALYSIS PROGRAMMING WITH MATLAB M-FILES INPUT-OUTPUT STRUCTURED PROGRAMMING NESTING AND INDENTATION PASSING FUNCTIONS TO M-FILES CASE STUDY: BUNGEE JUMPER VELOCITY ROUNDOFF AND TRUNCATION ERRORS ERRORS ROUNDOFF ERRORS TRUNCATION ERRORS TOTAL NUMERICAL ERROR BLUNDERS, MODEL ERRORS, AND DATA UNCERTAINTY PART TWO ROOTS AND OPTIMIZATION 2 1 OVERVIEW 2 2 PART ORGANIZATION ROOTS: BRACKETING METHODS ROOTS IN ENGINEERING AND SCIENCE GRAPHICAL METHODS BRACKETING METHODS AND INITIAL GUESSES BISECTION FALSE POSITION CASE STUDY: GREENHOUSE GASES AND RAINWATER ROOTS: OPEN METHODS SIMPLE FIXED-POINT ITERATION NEWTON-RAPHSON SECANT METHODS BRENT’S METHOD MATLAB FUNCTION: FZERO POLYNOMIALS CASE STUDY: PIPE FRICTION OPTIMIZATION INTRODUCTION AND BACKGROUND ONE-DIMENSIONAL OPTIMIZATION MULTIDIMENSIONAL OPTIMIZATION CASE STUDY: EQUILIBRIUM AND MINIMUM POTENTIAL ENERGY PART THREE LINEAR SYSTEMS 3 1 OVERVIEW 3 2 PART ORGANIZATION LINEAR ALGEBRAIC EQUATIONS AND MATRICES MATRIX ALGEBRA OVERVIEW SOLVING LINEAR ALGEBRAIC EQUATIONS WITH MATLAB CASE STUDY: CURRENTS AND VOLTAGES IN CIRCUITS GAUSS ELIMINATION SOLVING SMALL NUMBERS OF EQUATIONS NAIVE GAUSS ELIMINATION PIVOTING TRIDIAGONAL SYSTEMS CASE STUDY: MODEL OF A HEATED ROD LU FACTORIZATION 10 1 OVERVIEW OF LU FACTORIZATION 10 2 GAUSS ELIMINATION AS LU FACTORIZATION 10 3 CHOLESKY FACTORIZATION 10 4 MATLAB LEFT DIVISION MATRIX INVERSE AND CONDITION 11 1 THE MATRIX INVERSE 11 2 ERROR ANALYSIS AND SYSTEM CONDITION 11 3 CASE STUDY: INDOOR AIR POLLUTION ITERATIVE METHODS 12 1 LINEAR SYSTEMS: GAUSS-SEIDEL 12 2 NONLINEAR SYSTEMS 12 3 CASE STUDY: CHEMICAL REACTIONS EIGENVALUES 13 1 MATHEMATICAL BACKGROUND 13 2 PHYSICAL BACKGROUND 13 3 THE POWER METHOD 13 4 MATLAB FUNCTION: EIG 13 5 CASE STUDY: EIGENVALUES AND EARTHQUAKES PART FOUR CURVE FITTING 4 1 OVERVIEW 4 2 PART ORGANIZATION LINEAR REGRESSION 14 1 STATISTICS REVIEW 14 2 RANDOM NUMBERS AND SIMULATION 14 3 LINEAR LEAST-SQUARES REGRESSION 14 4 LINEARIZATION OF NONLINEAR RELATIONSHIPS 14 5 COMPUTER APPLICATIONS 14 6 CASE STUDY: ENZYME KINETICS GENERAL LINEAR LEAST-SQUARES AND NONLINEAR REGRESSION 15 1 POLYNOMIAL REGRESSION 15 2 MULTIPLE LINEAR REGRESSION 15 3 GENERAL LINEAR LEAST SQUARES 15 4 QR FACTORIZATION AND THE BACKSLASH OPERATOR 15 5 NONLINEAR REGRESSION 15 6 CASE STUDY: FITTING EXPERIMENTAL DATA FOURIER ANALYSIS 16 1 CURVE FITTING WITH SINUSOIDAL FUNCTIONS 16 2 CONTINUOUS FOURIER SERIES 16 3 FREQUENCY AND TIME DOMAINS 16 4 FOURIER INTEGRAL AND TRANSFORM 16 5 DISCRETE FOURIER TRANSFORM (DFT) 16 6 THE POWER SPECTRUM 16 7 CASE STUDY: SUNSPOTS POLYNOMIAL INTERPOLATION 17 1 INTRODUCTION TO INTERPOLATION 17 2 NEWTON INTERPOLATING POLYNOMIAL 17 3 LAGRANGE INTERPOLATING POLYNOMIAL 17 4 INVERSE INTERPOLATION 17 5 EXTRAPOLATION AND OSCILLATIONS SPLINES AND PIECEWISE INTERPOLATION 18 1 INTRODUCTION TO SPLINES 18 2 LINEAR SPLINES 18 3 QUADRATIC SPLINES 18 4 CUBIC SPLINES 18 5 PIECEWISE INTERPOLATION IN MATLAB 18 6 MULTIDIMENSIONAL INTERPOLATION 18 7 CASE STUDY: HEAT TRANSFER PART FIVE INTEGRATION AND DIFFERENTIATION 5 1 OVERVIEW 5 2 PART ORGANIZATION NUMERICAL INTEGRATION FORMULAS 19 1 INTRODUCTION AND BACKGROUND 19 2 NEWTON-COTES FORMULAS 19 3 THE TRAPEZOIDAL RULE 19 4 SIMPSON’S RULES 19 5 HIGHER-ORDER NEWTON-COTES FORMULAS 19 6 INTEGRATION WITH UNEQUAL SEGMENTS 19 7 OPEN METHODS 19 8 MULTIPLE INTEGRALS 19 9 CASE STUDY: COMPUTING WORK WITH NUMERICAL INTEGRATION NUMERICAL INTEGRATION OF FUNCTIONS 20 1 INTRODUCTION 20 2 ROMBERG INTEGRATION 20 3 GAUSS QUADRATURE 20 4 ADAPTIVE QUADRATURE 20 5 CASE STUDY: ROOT-MEAN-SQUARE CURRENT NUMERICAL DIFFERENTIATION 21 1 INTRODUCTION AND BACKGROUND 21 2 HIGH-ACCURACY DIFFERENTIATION FORMULAS 21 3 RICHARDSON EXTRAPOLATION 21 4 DERIVATIVES OF UNEQUALLY SPACED DATA 21 5 DERIVATIVES AND INTEGRALS FOR DATA WITH ERRORS 21 6 PARTIAL DERIVATIVES 21 7 NUMERICAL DIFFERENTIATION WITH MATLAB 21 8 CASE STUDY: VISUALIZING FIELDS PART SIX ORDINARY DIFFERENTIAL EQUATIONS 6 1 OVERVIEW 6 2 PART ORGANIZATION INITIAL-VALUE PROBLEMS 22 1 OVERVIEW 22 2 EULER’S METHOD 22 3 IMPROVEMENTS OF EULER’S METHOD 22 4 RUNGE-KUTTA METHODS 22 5 SYSTEMS OF EQUATIONS 22 6 CASE STUDY: PREDATOR-PREY MODELS AND CHAOS ADAPTIVE METHODS AND STIFF SYSTEMS 23 1 ADAPTIVE RUNGE-KUTTA METHODS 23 2 MULTISTEP METHODS 23 3 STIFFNESS 23 4 MATLAB APPLICATION: BUNGEE JUMPER WITH CORD 23 5 CASE STUDY: PLINY’S INTERMITTENT FOUNTAIN BOUNDARY-VALUE PROBLEMS 24 1 INTRODUCTION AND BACKGROUND 24 2 THE SHOOTING METHOD 24 3 FINITE-DIFFERENCE METHODS