Introduction to Applied Mathematics for Environmental Science

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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

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