# Introduction to Applied Mathematics for Environmental Science

Springer Science & Business Media, 28 de juny 2006 - 317 pàgines
For many years, first as a student and later as a teacher, I have ob served graduate students in ecology and other environmental sci ences who had been required as undergraduates to take calculus courses. Those courses have often emphasized how to prove theo rems about the beautiful, logical structure of calculus, but have ne glected applications. Most of the time, the students have come out of such courses with little or no appreciation of how to apply calculus in their own work. Based on these observations, I developed a course de signed in part to re-teach calculus as an everyday tool in ecology and other environmental sciences. I emphasized derivations—working with story problems (sometimes quite complex ones)—in that course, and now in this book. The present textbook has developed out of my notes for that course. Its basic purpose is to describe various types of mathemati cal structures and how they can be apphed in environmental science. Thus, linear and non-linear algebraic equations, derivatives and in tegrals, and ordinary and partial differential equations are the basic kinds of structures, or types of mathematical models, discussed. For each, the discussion follows a pattern something like this: 1. An example of the type of structure, as apphed to environmental science, is given. 2. Next, a description of the structure is presented. 3. Usually, this is followed by other examples of how the structure arises in environmental science. 4. The analytic methods of solving and learning from the structure are discussed.

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### Continguts

 Introduction 1 12 PreCalculus Math Review 4 13 Trigonometry 5 15 Ratios and Percentages 7 16 Analysis Finding Symbolic Solutions versus Numerical Analysis 8 17 Notes on Significant Digits 10 18 Exercises 13 19 Questions and Answers 18
 Numerical Solution of Ordinary Differential Equations 147 72 The RungeKutta Method 151 73 Solving ODEs Numerically with MATLAB 159 74 Exercises 160 75 Questions and Answers 163 SecondOrder ODEs 166 81 Mass Transfer in Cartesian Coordinate Systems 167 82 Generalizations to Other Quantities 175

 Derivatives and Differentiation 19 22 Usefulness in Environmental Science 20 23 Taylor Series a Basis for Numerical Analysis 27 24 Numerical Differentiation 32 25 Checking Analytic Derivatives 36 26 Exercises 37 27 Questions and Answers 46 Integration 52 32 Usefulness in Environmental Science 53 33 Analytic Integration 57 34 Numerical Integration 60 35 DifferentiationIntegration Contrasts 65 36 Exercises 68 37 Questions and Answers 77 Ordinary Differential Equations 82 42 Solution of Simple FirstOrder ODEs 89 43 Checking Solutions of ODEs 94 44 Notes on Differential Equations 98 45 Analytic Solution of FirstOrder ODEs 102 46 Table of Solutions of Selected ODEs 106 47 Analytic Solution with MATLAB 109 48 Exercises 110 49 Questions and Answers 119 Further Topics in ODEs 123 52 Integrating Factors for Linear FirstOrder Differential Equations 126 53 Integrals for the integratingfactor method 131 54 Exercises 132 55 Questions and Answers 135 Systems of Ordinary Differential Equations 137 62 Exercises 141
 83 Conduction of Heat in a Solid 176 84 Coordinate Systems for Curved Geometry 178 85 Solving SecondOrder ODEs Analytically with MATLAB 182 86 Exercises 183 87 Questions and Answers 190 Linear Algebra 193 92 How Linear Systems Arise 194 An Introduction to Linear Algebra 197 94 Well and 111 Conditioned Linear Systems 208 95 Population modelling with Leslie matrices 212 96 Other Applications for Matrices 215 97 Exercises 216 98 Questions and Answers 222 NonLinear Equations 229 101 Roots of Nonlinear Equations 230 102 Repeated or Multiple Roots 239 103 Exercises 241 104 Questions and Answers 247 Partial Differential Equations 111 Partial Derivatives 250 112 Mass and Heat Transfer 253 113 Schmidts Graphical Method for Solving PDEs 259 114 Atmospheric Diffusion in Three Dimensions 262 115 Exercises 272 116 Questions and Answers 283 PreCalculus Math Review 285 Solutions to OddNumbered Exercises 292 List of Applications 307 Bibliography 310 Index 313 Copyright