Equations play a central role in problem solving across variousfields of study. Understanding what an equation means is anessential step toward forming an effective strategy to solve it,and it also lays the foundation for a more successful andfulfilling work experience. Thinking About Equationsprovides an accessible guide to developing an intuitiveunderstanding of mathematical methods and, at the same time,presents a number of practical mathematical tools for successfullysolving problems that arise in engineering and the physicalsciences.
Equations form the basis for nearly all numerical solutions, andthe authors illustrate how a firm understanding of problem solvingcan lead to improved strategies for computational approaches. Eightsuccinct chapters provide thorough topical coverage, including:
- Approximation and estimation
- Isolating important variables
- Generalization and special cases
- Dimensional analysis and scaling
- Pictorial methods and graphical solutions
- Symmetry to simplify equations
Each chapter contains a general discussion that is integratedwith worked-out problems from various fields of study, includingphysics, engineering, applied mathematics, and physical chemistry.These examples illustrate the mathematical concepts and techniquesthat are frequently encountered when solving problems. Toaccelerate learning, the worked example problems are grouped by theequation-related concepts that they illustrate as opposed tosubfields within science and mathematics, as in conventionaltreatments. In addition, each problem is accompanied by acomprehensive solution, explanation, and commentary, and numerousexercises at the end of each chapter provide an opportunity to testcomprehension.
Requiring only a working knowledge of basic calculus andintroductory physics, Thinking About Equations is anexcellent supplement for courses in engineering and the physicalsciences at the upper-undergraduate and graduate levels. It is alsoa valuable reference for researchers, practitioners, and educatorsin all branches of engineering, physics, chemistry, biophysics, andother related fields who encounter mathematical problems in theirday-to-day work.
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About the Author
William A. Friedman, PhD, is Emeritus Professor at theUniversity of Wisconsin and Affiliate Professor at the Universityof Washington. A Fellow of the American Physical Society, Dr.Friedman has over forty years of academic experience and hasauthored more than one hundred journal articles in the field ofnuclear physics.
Table of ContentsPreface.
List of Worked-Out Example Problems.
1 Equations Representing Physical Quantities.
1.1 Systems of Units.
1.2 Conversion of Units.
1.3 Dimensional Checks and the Use of Symbolic Parameters.
1.4 Arguments of Transcendental Functions.
1.5 Dimensional Checks to Generalize Equations.
1.6 Other Types of Units.
1.7 Simplifying Intermediate Calculations.
2 A Few Pitfalls and a Few Useful Tricks.
2.1 A Few Instructive Pitfalls.
2.2 A Few Useful Tricks.
2.3 A Few “Advanced” Tricks.
3 Limiting and Special Cases.
3.1 Special Cases to Simplify and Check Algebra.
3.2 Special Cases and Heuristic Arguments.
3.3 Limiting Cases of a Differential Equation.
3.4 Transition Points.
4 Diagrams, Graphs, and Symmetry.
4.2 Diagrams for Equations.
4.3 Graphical Solutions.
4.4 Symmetry to Simplify Equations.
5 Estimation and Approximation.
5.1 Powers of Two for Estimation.
5.2 Fermi Questions.
5.3 Estimates Based on Simple Physics.
5.4 Approximating Definite Integrals.
5.5 Perturbation Analysis.
5.6 Isolating Important Variables.
6 Introduction to Dimensional Analysis and Scaling.
6.1 Dimensional Analysis: An Introduction.
6.2 Dimensional Analysis: A Systematic Approach.
6.3 Introduction to Scaling.
7 Generalizing Equations.
7.1 Binomial Expressions.
7.2 Motivating a General Expression.
7.3 Recurring Themes.
7.4 General yet Simple: Euler’s Identity.
7.5 When to Try to Generalize.
8 Several Instructive Examples.
8.1 Choice of Coordinate System.
8.2 Solution Has Unexpected Properties.
8.3 Solutions in Search of Problems.
8.4 Learning from Remarkable Results.