Thinking About Equations A Practical Guide for Developing Mathematical Intuition in the Physical Sciences and Engineering

Matt A. Bernstein, PhD, is Professor of Radiologic Physics at the Mayo Clinic, where he holds appointments in the Departments of Radiology and Biomedical Engineering. A Fellow of the International Society for Magnetic Resonance in Medicine (ISMRM) and Editorial Board member of Magnetic Resonance in Medicine, Dr. Bernstein has published over sixty journal articles, mainly in the field of MRI physics.

William A. Friedman, PhD, is Emeritus Professor at the University of Wisconsin and Affiliate Professor at the University of Washington. A Fellow of the American Physical Society, Dr. Friedman has over forty years of academic experience and has authored more than one hundred journal articles in the field of nuclear physics.

Acknowledgments.

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.

Exercises.

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.

Exercises.

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.

Exercises.

4 Diagrams, Graphs, and Symmetry.

4.1 Introduction.

4.2 Diagrams for Equations.

4.3 Graphical Solutions.

4.4 Symmetry to Simplify Equations.

Exercises.

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.

Exercises.

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.

Exercises.

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.

Exercises.

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.

Exercises.

Index.

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