Guide to Essential Math: A Review for Physics, Chemistry and Engineering Students

By Sy M. Blinder

This book reminds students in junior, senior and graduate level courses in physics, chemistry and engineering of the math they may have forgotten (or learned imperfectly) that is needed to succeed in science courses. The focus is on math actually used in physics, chemistry, and engineering, and the approach to mathematics begins with 12 examples of increasing complexity, designed to hone the student's ability to think in mathematical terms and to apply quantitative methods to scientific problems. Detailed illustrations and links to reference material online help further comprehension. The second edition features new problems and illustrations and features expanded chapters on matrix algebra and differential equations.

• Use of proven pedagogical techniques developed during the author’s 40 years of teaching experience
• New practice problems and exercises to enhance comprehension
• Coverage of fairly advanced topics, including vector and matrix algebra, partial differential equations, special functions and complex variables

• Language: English
• Publisher: Elsevier Science
• Release date: Feb 14, 2013
• ISBN: 9780124071582

Sy M. Blinder

Professor Blinder is Professor Emeritus of Chemistry and Physics at the University of Michigan, Ann Arbor and a senior scientist with Wolfram Research Inc., Champaign, IL.. After receiving his A.B. in Physics and Chemistry from Cornell University, he went on to receive an A. M in Physics, and a Ph. D. in Chemical Physics from Harvard University under Professors W. E. Moffitt and J. H. Van Vleck. He has held positions at Johns Hopkins University, Carnegie-Mellon University, Harvard University, University College London, Centre de Méchanique Ondulatoire Appliquée in Paris, the Mathematical Institute in Oxford, and the University of Michigan. Prof Blinder has won multiple awards for his work, published 4 books, and over 100 journal articles. His research interests include Theoretical Chemistry, Mathematical Physics, applications of quantum mechanics to atomic and molecular structure, theory and applications of Coulomb Propagators, structure and self-energy of the electron, supersymmetric quantum field theory, connections between general relativity and quantum mechanics.

Book preview

2006

Preface to Second Edition

I was much gratified by the reception of students and teachers to the First Edition of this Math Study Guide. In response to your many suggestions, I have made several improvements and additions. A number of Problems have been added to help solidify the student’s understanding of some of the more complex concepts. I have also clarified the coverage of several topics and added a Chapter on Group Theory and a Section on Hypergeometric Functions. The figures in the Second Edition are now being rendered in full color. Finally, in the first example in Chapter 1, I have updated the format of the NCAA basketball tournament to reflect its expansion to 68 teams.

A (translated) quotation attributed to Leibniz states that It is unworthy of excellent men to lose hours like slaves in the labor of calculation … With modern computer software, it is now possible to perform, with remarkabke facility, not only numerical but also symbolic calculations involving algebra and calculus. I am an enthusiastic user of Mathematica™ as an indispensible aid to my mathematical and scientific work. Other symbolic mathematical programs which will provide many of the same befefits are Maple™ and Mathcad™.

A useful adjunct to this book is the Wolfram Demonstration Project. This has been available on the Web at http://demonstrations.wolfram.com since 2007 and contains a growing collection, approaching 10,000 Demonstrations, mostly on scientific and mathematical topics. This should prove very instructive to the same audience to which this book is addressed. I feel privileged to have been associated with this project. I would also like to acknowledge the assistance of my colleagues at Wolfram Research for their unfailing assistance and encouragement.

Since this little book is simply an introduction to several useful mathematical concepts, the reader will undoubtedly need to seek other sources for more exhaustive coverage of specific topics. There exist, of course, thousands of excellent textbooks and references on mathematics, but I have found it very useful to refer to two online sites as a starting point. One is Wolfram MathWorld at http://mathworld.wolfram.com. The second is the continually expanding online encyclopedia: Wikipedia, at http://en.wikipedia.org.

My sincere thanks also to my Elsevier editors, Dr. Erin Hill-Parks and Ms. Tracey Miller, for their unfailing cooperation in getting this Second Edition into production.

SMB

Ann Arbor

Setember 2012

Chapter 1

Mathematical Thinking

Mathematics is both the queen and the handmaiden of all sciences—Eric Temple Bell.

. As you go on, you will find that mathematics can provide indispensable short-cuts for solving complicated problems. Sometimes, mathematics is actually an alternative to thinking! At the very least, mathematics is one of the greatest labor-saving inventions ever created by Mankind.

Instead of outlining some sort of 12-Step Program for developing your mathematical skills, we will give 12 examples which will hopefully stimulate your imagination on how to think cleverly along mathematical lines. It will be worthwhile for you to study each example until you understand it completely and have internalized the line of reasoning. You might have to put off doing some of Examples 7–12 until you review some relevant calculus background in the later chapters of this book.

1.1 The NCAA March Madness Problem

. But there’s a much more elegant way to solve the problem. Since this is a single-elimination tournament, 67 of the 68 teams have to eventually lose a game. Therefore we must play exactly 67 games!

Problem 1.1.1

The College World Series is an annual tournament (every June in Omaha, NE) to determine the college baseball champion. Eight teams are split into two, four-team, double-elimination brackets (a team must lose two games to be eliminated), with the winner of each bracket playing in a best-of-three championship series. Calculate the total number of games played—there are actually four possibilities.

1.2 Gauss and the Arithmetic Series

A tale in mathematical mythology—it’s hard to know how close it is to the actual truth—tells of Carl Friedrich Gauss as a 10-year-old student. By one account, Gauss’ math teacher wanted to take a break so he assigned a problem he thought would keep the class busy for an hour or so. The problem was to add up all the integers from 1 to 100. Supposedly, Gauss almost immediately wrote down the correct answer 5050 and sat with his hands folded for the rest of the hour. The rest of his classmates got incorrect answers! Here’s how he did it. Gauss noted that the 100 integers could be arranged into 50 pairs:

. The general result for the sum of an arithmetic progression with n terms is:

(1.1)

where a are the first and last terms, respectively. This holds true whatever the difference between successive terms.

Problem 1.2.1

.

1.3 The Pythagorean Theorem

The most famous theorem in Euclidean geometry is usually credited to Pythagoras (ca. 500 BC). However, Babylonian tablets suggest that the result was known more than a thousand years earlier. The theorem states that the square of the hypotenuse c of a right triangle is equal to the sum of the squares of the lengths a and b of the other two sides:

(1.2)

The geometrical significance of the Pythagorean theorem is shown in Figure 1.1: the sum of the areas of the red and the blue squares equals the area of the purple square.

Figure 1.1 Geometrical interpretation of Pythagoras’ theorem. The purple area equals the sum of the blue and red areas. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this book.)

Well over 350 different proofs of the theorem have been published. Figure 1.2 shows a pictorial proof which requires neither words nor formulas.

Figure 1.2 Pictorial proof of Pythagorean theorem.

1.4 Torus Area and Volume

A torus or anchor ring, drawn in Figure 1.3, is the approximate shape of a donut or bagel. The radii R and r refer, respectively, to the circle through the center of the torus and the circle made by a cross-sectional cut. Generally, to determine the area and volume of a surface of revolution, it is necessary to evaluate double or triple integrals. However, long before calculus was invented, Pappus of Alexandria (ca. 3rd Century AD) proposed two theorems which can give the same results much more directly.

Figure 1.3 Torus. The surface of revolution with radius R .

The first theorem of Pappus states that the area A generated by the revolution of a curve about an external axis is equal to the product of the arc length of the generating curve and the distance traveled by the curve’s centroid. For a torus the generating curve is a small circle of radius r. Therefore the surface area of a torus is

(1.3)

. Therefore the volume of a torus is

(1.4)

For less symmetrical figures, finding the centroid will usually require doing an integration.

An incidental factoid. You probably know about the four-color theorem: on a plane or spherical surface, four colors suffice to draw a map in such a way that regions sharing a common boundary have different colors. On the surface of a torus it takes seven colors.

Problem 1.4.1

Using the theorems of Pappus, calculate the volume and surface area of a flat cylindrical disk of width w. Check the results using the volume and area formulas for cylinders.

1.5 Einstein’s Velocity Addition Law

(or 186,000 miles/s). Expressed mathematically, we can