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.
Mark Levi introduces each physical problem, sometimes gives a hint or two, and then fully explains the solution. Here readers can test their critical-thinking skills against a whole assortment of puzzles and paradoxes involving floating and diving, sailing and gliding, gymnastics, bike riding, outer space, throwing a ball from a moving car, centrifugal force, gyroscopic motion, and, of course, falling cats.
Want to figure out how to open a wine bottle with a book? Or how to compute the square root of a number using a tennis shoe and a watch? Why Cats Land on Their Feet shows you how, and all that's required is a familiarity with basic high-school mathematics. This lively collection also features an appendix that explains all physical concepts used in the book, from Newton's laws to the fundamental theorem of calculus.
Some areas of particular interest are: an
extremely short derivation of the ellipticity of planetary orbits; a
statement and an explanation of the "tennis racket paradox"; a
heuristic explanation (and a rigorous treatment) of the gyroscopic
effect; a revealing equivalence between the dynamics of a particle and
statics of a spring; a short geometrical explanation of Pontryagin's
Maximum Principle, and more.
In the last chapter, aimed at more
advanced readers, the Hamiltonian and the momentum are compared to
forces in a certain static problem. This gives a palpable physical
meaning to some seemingly abstract concepts and theorems.
minimal prerequisites consisting of basic calculus and basic
undergraduate physics, this book is suitable for courses from an
undergraduate to a beginning graduate level, and for a mixed audience
of mathematics, physics and engineering students. Much of the enjoyment
of the subject lies in solving almost 200 problems in this book.