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The Mathematical Mechanic: Using Physical Reasoning to Solve Problems
Everybody knows that mathematics is indispensable to physics--imagine where we'd be today if Einstein and Newton didn't have the math to back up their ideas. But how many people realize that physics can be used to produce many astonishing and strikingly elegant solutions in mathematics? Mark Levi shows how in this delightful book, treating readers to a host of entertaining problems and mind-bending puzzlers that will amuse and inspire their inner physicist.
Levi turns math and physics upside down, revealing how physics can simplify proofs and lead to quicker solutions and new theorems, and how physical solutions can illustrate why results are true in ways lengthy mathematical calculations never can. Did you know it's possible to derive the Pythagorean theorem by spinning a fish tank filled with water? Or that soap film holds the key to determining the cheapest container for a given volume? Or that the line of best fit for a data set can be found using a mechanical contraption made from a rod and springs? Levi demonstrates how to use physical intuition to solve these and other fascinating math problems. More than half the problems can be tackled by anyone with precalculus and basic geometry, while the more challenging problems require some calculus. This one-of-a-kind book explains physics and math concepts where needed, and includes an informative appendix of physical principles.
The Mathematical Mechanic will appeal to anyone interested in the little-known connections between mathematics and physics and how both endeavors relate to the world around us.
||Princeton University Press|
||July 06, 2009|
|Average Customer Rating:
|| based on 12 reviews|
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110 of 112 found the following review helpful:
Two streams of thought are unified at last!Aug 01, 2009
By Peter Haggstrom
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.
14 of 15 found the following review helpful:
What if Pythagoras had met Newton?Nov 10, 2009
By Joseph Horton
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.
10 of 10 found the following review helpful:
Very Original and Thought-ProvokingDec 24, 2009
By G. Poirier
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.
9 of 12 found the following review helpful:
A new way of looking at mathNov 12, 2009
By Lance C. Hibbeler
In "The Mathematical Mechanic," Levi explains his (mostly his, but apparently a few others) way of reasoning behind various mathematical proofs. Rather than pages of algebra or calculus, a lot of mathematical things can be proved by relatively simple physical arguments, such as force and moment equilibrium, or the behavior of linear circuit components like resistors and capacitors. Such "mathematical things" include the Pythagorean theorem, inequalities, minimization problems, differential geometry, and complex variables. I have a pretty good mechanical intuition thanks to my training as an engineer, so a lot of the proofs clicked instantly with me...the physical line of reasoning espoused by Levi makes the math easier to understand. Well no, the math is still the same, but the trip to the answer is done in terms of things that are easy for me to understand and visualize, rather than just equations on the page.
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...
4 of 5 found the following review helpful:
Great concept, wrong examplesJul 25, 2010
By A. Martinez
I really wanted to like this book. Its still a good book in terms of the idea is trying to convey. However the examples given in the book are not usefull at all for anyone with at least some calculus background. Also, applying this kind of physical thinking to simple problems if your math background is not that strong will make your life harder. For example, the proof of the sine rule seems like a contrived explanation; why not just use the height itself to prove it instead of the differences in pressure? Just focus on the geometry; I think that's still physics.
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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