The Mathematical Mechanic: Using Physical Reasoning to Solve Problems Hardcover – 26 Jul 2009
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One of Amazon.com science editors' Top 10 list for Science, Best for 2009
One of Choice's Outstanding Academic Titles for 2009
"The Mathematical Mechanic documents novel ways of viewing physics as a method of understanding mathematics. Levi uses physical arguments as tools to conjecture about mathematical concepts before providing rigorous proofs. . . . The Mathematical Mechanic is an excellent display of creative, interdisciplinary problem-solving strategies. The author has explained complex concepts with simplicity, yet the mathematics is accurate."--Mathematics Teacher
"A most interesting book. . . . Many of the ideas in it could be used as motivational or illustrative examples to support the teaching of non-specialists, especially physicists and engineers. In conclusion--a thoroughly enjoyable and thought-provoking read."--Nigel Steele, London Mathematical Society Newsletter
"The Mathematical Mechanic reverses the usual interaction of mathematics and physics. . . . Careful study of Levi's book may train readers to think of physical companions to mathematical problems. . . . Mathematicians will find The Mathematical Mechanic provides exercise in new ways of thinking. Instructors will find it contains material to supplement mathematics courses, helping physically-minded students approach mathematics and helping mathematically-minded students appreciate physics."--John D. Cook, MAA Reviews
"Mark Levi reverses the old stereotype that math is merely a tool to aid physicists by showing that many questions in mathematics can be easily solved by interpreting them as physical problems. . . . Some sections of the book require readers to brush up on their calculus but Levi's clear explanations, witty footnotes, and fascinating insights make the extra effort painless."--SEED Magazine
"The book is chock-full of these seemingly magical physical thought experiments involving bicycle wheels, pistons, springs, soap films, pendulums, and electric circuits, with applications to geometry, maximization and minimization problems, inequalities, optics, integrals, and complex functions. . . . I highly recommend it to anyone who is (even slightly) interested in physics, and appreciates mathematical elegance and cleverness. It would make a great gift for almost anyone, whether a high school student or university professor, armchair physicist or professional mathematician."--Boris Yorgey, The Math Less Traveled
"The Mathematical Mechanic is a pleasant surprise."--E. Kincanon, Choice
"This is a delightful and unusual book that is a welcome addition to the literature. Certainly, any calculus teacher and many others of us as well will want to have it on the shelf for ready reference. It not only will enhance our teaching experience but will also teach us (the instructors) something in the process."--Steven G. Krantz, UMAP Journal
From the Back Cover
"What a fun book! Mark Levi's physical arguments are so clever and surprising that they made me laugh with pleasure, again and again. The Mathematical Mechanic is downright magical--a real treat for anyone who loves intuition."--Steven Strogatz, author of Sync: How Order Emerges from Chaos in the Universe, Nature, and Daily Life
"This is an absolutely delightful book, full of surprises--even for mathematicians like myself--and beautifully written. It can be enjoyed by anyone, from someone just learning calculus to professional mathematicians and physicists."--Louis Nirenberg, recipient of the National Medal of Science
"This is an extraordinary book that only Mark Levi could have written. No one interested in mathematics or physics can fail to be amazed and delighted. It is witty and charming as well as deep, and accessible with very little background required--a tour de force!"--Nancy Kopell, Boston University, MacArthur Fellow
"The most imaginative and charming book on mechanics and geometry in the last fifty years--for lighting up tea times, for thrilling classrooms, as a present for a special friend, as company on a desert island."--Tadashi Tokieda, University of Cambridge
"This book shows how many mathematical theorems can be proved by looking at them in mechanical or geometrical terms. I found it to be very interesting and fun to read. I recommend it most enthusiastically."--Joseph Keller, recipient of the National Medal of Science
"The Mathematical Mechanic jazzes up the old married couple, math and physics. The book breathes fresh air into the (sometimes stale) relationship and invites us to rethink familiar topics in unfamiliar ways. It disorients us in the most delightful manner. Mark Levi's razor-edge writing and gentle humor permeate every page. I will turn to this book again and again for inspiration on teaching math to high school students."--Gregory Somers, State College Area High School, recipient of the Edyth May Sliffe Award for Distinguished Mathematics Teaching
"This book is a fresh, insightful, and highly original presentation of mathematical physics that will appeal to a broad spectrum of readers. I have not seen anything like it before. It is a book that a physicist or engineer would be proud to have written, and the fact that it has been written by a mathematician only adds to the book's authority. A definite winner."--Paul J. Nahin, author of Digital Dice
"I know of no other book quite like this, or even similar to it. After a couple of sentences of the introduction, I was hooked. The general theme--to show how physical reasoning can illuminate mathematical ideas and simplify proofs--is very attractive. This book will appeal to math enthusiasts at all levels, from high-school students on up."--Philip Holmes, coauthor of Celestial Encounters
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The book therefore has some value, but should not be used as an isolated text.
Most Helpful Customer Reviews on Amazon.com (beta)
What Levy does is to take a large number of mathematical problems/theorems and show how physical reasoning using concepts such as conservation of energy, torque, resolution of forces, etc can be used to solve what are quite fundamental problems/theorems. In Chapter 2 he uses essentially torque concepts to prove the Pythagorean theorem be a thought experiment involving a right angled prism sitting in a water filled fish tank but attached to a spindle so it can rotate. The fact that it doesn't (ie there is zero net torque) leads directly to Pythagoras' Theorem.
Many of the problems turn upon one very basic physical principle and some careful reasoning about how that physical principle applies. For instance in working out why a triangle balances on the point of intersection of the medians the basic idea is a reductionist one and that is to conceptually slice a strip of the triangle. Since this strip balances and all the ones parallel to it will balance one can replicate the same argument for any other side and the point of balance will lie on the intersection of the medians. Levy spends a bit of time on geometrical optics and Fermat's principle and Snell's Law and gives a number of physical proofs for various formulas. There is that old favourite of saving a drowning victim by using Fermat's principle and this is explained in terms of Snell's law.
An interesting application of the general approach is to prove that the arithmetic mean is greater than the geometric mean by for throwing a switch. This all turns upon the concept of resistance along parallel paths and the result follows very quickly. Levy generalizes that approach to more complex arrangements. He covers Pappus Volume Theorem and applications of Ceva's Theorem. He also shows how you can compute the integral of sin x by using concepts of potential energy in the context of the movement of the pendulum. He touches on Hamiltonian mechanics and the Euler Lagrange equations and he even provides a hand waving proof of area preservation.
On page 125 there is a table of analogies between mechanics and analysis. For instance zero net work done is interpreted in an analytical sense in terms of preservation of the area. There is an interesting discussion of how an area preservation property can be viewed as a classical mechanical analog of the uncertainty principle in quantum mechanics. If an area preserving map squeezes some region about a point x we gain information about that point however because the map is area preserving it must stretch in the other direction (y) and this means that the range of values in the other direction is large so we lose information in that direction. If we think of the first variable x as signifying position and the second one being y which is identified with momentum, we then have the connection with the uncertainty principle.
I'm not aware of any other books that have systematically brought together this type of physical reasoning and its application to mathematical problems. In bringing together such a wide range of problems Levi has at the very least provided interested people with something to go on with in a more systematic fashion. The beauty of the book is that often a compelling physical reason for a particular mathematical equation can be much easier to remember and can actually illuminate the mathematical proof. One could even contemplate a little subculture of mathematics developing whereby people try to develop more and more inspired physical analogies for various mathematical theorems.
Levy does not assume a great level of mathematical sophistication however readers should have a reasonable grasp of basic concepts such as the resolution of forces, potential energy, kinetic energy and how the can be applied to a problem. There is no heavy-duty calculus or analysis involved and Levy has a very informal and chatty style.
I recommend this book without any reservation - it should have been written many years ago. I think students will find it enriches their understanding of the concepts.
Levi does this over and over again, but instead of merely making moving parts, he assigns the physical to what is otherwise purely mathematical. In addition to the stroll down the memory lane of my thought processes--and a reassurance that at least one other person the universe does this as well--it showed a few new ways of looking at commonplace things--like Pythagoras' theorem. He proves it using torques--torques?????--yeah, torques. Yet another proof involves concentric circles. Just read it--it's clever as anything. I grant you that I had to look at most of the analogies a couple times to get them, but get them I did.
It's a great way to spend a few hours. My bet is that this will be most useful to math and physics teachers. Is everything about physics and math intuitive? Certainly not, but enough is that having a strong sense of it is useful. It took my intuition to the next level.
As a final note, I disagree with the author's statement that this book "should appeal to ... many people who are not interested in mathematics because they find it dry or boring". Although I understand (and agree with) the author's implication that mathematics is very far from being dry and boring, I would expect that most of the people he refers to would have avoided mathematics in their lives and would thus be unwilling to read this book in the first place, or be unable to follow most of the discussions presented if they did try to read it.
This short book is a relatively quick and easy read. Levi's style is very informal, almost to the point of conversational. You do need to be a little versed in mathematics to understand what's going on, i.e. why would you care about computing integrals if you don't know what an integral is?, but Levi doesn't explicitly calculus in his proofs. If you don't have the background, or it's been a while since you've used any of it, Levi has provided a nice summary in the appendix of the book. I would recommend this for mathematicians, physicists, and engineers. It certainly expands your line of thinking.
Aside: after reading the book, I'm left somewhat puzzled. In physics, we use math to describe physical principles. Physics is not the math we use to solve its problems, though the two subjects are undoubtedly and eternally married. The concept of "conservation of energy" is expressed in math-speak as "KE + PE = constant", or the concept of "forces must balance" as "sum of F = 0". We have the vector concept of a force (vector, as in a member of a normed vector space), where the magnitude is calculated from the 2-norm (n-component Pythagorean theorem) of its components. It doesn't surprise me that the Pythagorean falls out of an argument using forces in three-dimensional space, but somehow it seems like circular reasoning because our notion of forces, when discussed mathematically (not conceptually), are built upon the Pythagorean theorem. Similarly, to say Levi doesn't use calculus is disingenuous, as it is the foundation of the mathematical description of motion and forces- think high school physics...you can talk about motion, forces, work, and energy without explicitly using calculus. Maybe in thinking this I'm missing the point of the book...
And that 'conservation of difficulty law' he talks about is just wrong. For most problems there is one way of doing them with the least effort. That's what math is about: finding or inventing a 'framework' where the problem is the easiest to solve. Maybe he is referring to the math under the 'framework' up to first principles; in that case he might be phylosophically right, but a theorem in a 'framework' can be used without knowing much of the underlying math.
But there are some deep physical principles that can guide you in solving actual problems not talked about in this book: for example, the table on the first page of ch 3 that includes the following:
Calculus Physical interpretation
f(x) potential energy P(x)
f'(x) force F(x) = -P'(x)
So when the system is at equilibrium sum Fs = 0, you get the minimum of a function.
This is used in the chapter but for only trivial problems that anyone with some calculus can solve more easily.
There are lots of places in computer science and engineering where this principle from the book is used to find a differential equation that solves the problem and the equation is similar in form (or the same but with different constants) to an actual physical process:
For example in image segmentation, where a 'snake' and a 'ballon' shape with a certain differential equation is solved over 'time' until it wraps around a shape of interest.
A more concrete example is the Matlab program distmesh, where a good quality mesh is found by Delaunay triangulation (making a bad triangular mesh from points) and then the points are 'connected' by 'springs' and the springs are allowed to 'relax' until they touch the edges of the domain to be meshed and the system reaches 'force' equilibrium.
Also groups of robots can be made to perform tasks by attracting or repelling other robots in certain ways similar to the way particles interact in a gas or fluid. For example if the robots repel each other, they will fill any volume close to informly if distributed sensing is needed for example. The fact that they will fill the volume uniformly can be explained by thermodynamics.
These are examples where clearly a physical interpretation is more usefull than a mathematical one at solving (or at least inventing the solution to) an actual applied math problem.
This is especially true (up to a point) for solving actual physical problems. For example, using processors in a multicore supercomputer that solves for a certain fluid flow in a way that a processor's location maps to the actual position of the part of the flow that the core is actually simulating. There are other less trivial examples in computer science. Check out Dani Hillis' paper:
New Computer Architectures and Their Relationship to Physics or Why CS is No Good
Or his PhD thesis, that can be found online and it's quite readable.
He also refers to another physics gem: conservation of 'energy' implies the same triangle area after translating it. This is in the pythagorean theorem chapter. This has something to do with Noether's theorem (look it up in wikipedia, I can't post the link here). It could be nice if the author talked about this theorem explicitly and used it to 'transform' a math problem to a more physical looking one.
If you are getting this book for the complex analysis chapter, also buy the book: Visual Complex Analysis. It is way more informative and it has less contrived examples.
So overall the book is good because it tries to convey that physical reasoning is good for inventing solutions to some math problems. But take the examples with a grain of salt. I could also suggest not reading it if you don't have some basic calculus background since it might be counterproductive.
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