The Mathematical Mechanic: Using Physical Reasoning to Solve Problems [Hardcover]

Mark Levi (Author)

Editorial Reviews

Review

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 )

Review

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" )

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Product Details

• Hardcover: 196 pages
• Publisher: Princeton University Press; 1 edition (July 6, 2009)
• Language: English
• ISBN-10: 0691140200
• ISBN-13: 978-0691140209
• Product Dimensions: 9.3 x 6.3 x 0.8 inches
• Shipping Weight: 1.1 pounds (View shipping rates and policies)
• Average Customer Review: 3.7 out of 5 stars  See all reviews (10 customer reviews)
• Amazon Bestsellers Rank: #68,758 in Books (See Top 100 in Books)

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  Books > Science > Physics > Mathematical Physics

Customer Reviews

Most Helpful Customer Reviews

101 of 103 people found the following review helpful:

5.0 out of 5 stars Two streams of thought are unified at last!, July 31, 2009

By	Peter Haggstrom (BONDI BEACH, NSW Australia) - See all my reviews (REAL NAME)

This review is from: The Mathematical Mechanic: Using Physical Reasoning to Solve Problems (Hardcover)

Mark Levi's book "The Mathematical Mechanic" is a wonderful attempt to integrate physical reasoning with mathematical reasoning. These two strands have historically run in parallel and only occasionally have they been united at least at a pedagogical level. There seems to be a trend among Russian mathematicians particularly in the area of differential equations whereby they use physical reasoning to illuminate the more abstract mathematical approaches that are taken. V I Arnold is an example someone who has been known to integrate the two approaches. Perhaps Levi's Russian roots explain some of the impetus for this book. As mathematics becomes more and more specialised I fear that fewer mathematicians have the time or even inclination to think about the interconnections between physical reasoning and their own area. Levi's book is an antidote to that trend and he is to be congratulated for his efforts.

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.

13 of 14 people found the following review helpful:

5.0 out of 5 stars What if Pythagoras had met Newton?, November 9, 2009

By	Joseph Horton (Hoover, AL USA) - See all my reviews (REAL NAME)

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This review is from: The Mathematical Mechanic: Using Physical Reasoning to Solve Problems (Hardcover)

I loved geometry, thought it was the greatest thing since forever. The way I proved theorems was to visualize the constructs in motion. It made the stuff come alive for me, and I saw relationships that, well, others didn't seem to appreciate.

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.

7 of 7 people found the following review helpful:

4.0 out of 5 stars Very Original and Thought-Provoking, December 24, 2009

By	G. Poirier (Orleans, ON, Canada) - See all my reviews (TOP 500 REVIEWER)    (REAL NAME)

This review is from: The Mathematical Mechanic: Using Physical Reasoning to Solve Problems (Hardcover)

In this unusual book, the author discusses mathematical formulas and theorems using purely physical arguments, thus eliminating the usual detailed mathematical approaches. Some of the mathematical subject areas that are discussed include geometry, conics, integration and complex variables. Some of the physical disciplines that are used are mechanics, electricity, fluid dynamics and statics and optics. I found the level of difficulty to vary throughout the book; much of the material is clear, simple and really quite fascinating, while some of it is rather complex, significantly more challenging and often quite difficult to follow, i.e., real head-scratchers. What didn't help in the latter category were the several editorial mistakes which became rather annoying in the long run. The writing style is friendly, authoritative and generally clear but undoubtedly assumes a certain level of mathematical sophistication on the part of the reader. In my view, this is a book better suited for careful study at one's own pace rather than be leisurely read as one would a popular science/math book or a novel. Consequently, serious math/science buffs could certainly enjoy perusing this book and learn a great deal from it; however, it could also be used by math/physics students as a supplementary reference in an advanced math or physics course (as suggested by the author).

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.