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题名/责任者:

The mathematical mechanic : using physical reasoning to solve problems / Mark Levi.

出版发行项:

Princeton : Princeton University Press, 2009.

出版发行项:

Princeton : Princeton University Press, 2009.

出版发行项:

Princeton : Princeton University Press, 2009.

ISBN:

9780691154565:

ISBN:

0691154562

载体形态项:

viii, 186 pages : illustrations ; 25 cm

个人责任者:

Levi, Mark, 1951- author.

论题主题:

Problem solving.

论题主题:

Mathematical physics.

中图法分类号:

O411

一般附注:

1st paperback printing, 2012.

书目附注:

Includes bibliographical references (pages 183-184) and index.

内容附注:

1. Math versus physics -- 2. The Pythagorean theorem -- The "fish tank" proof of the Pythagorean theorem -- Converting a physical argument into a rigorous proof -- Fundamental theorem of calculus -- The determinant by sweeping -- The Pythagorean theorem by rotation -- Still water runs deep -- A three-dimensional Pythagorean theorem -- A surprising equilibrium -- Pythagorean theorem by springs -- More geometry with springs -- A kinetic energy proof : Pythagoras on ice -- Pythagoras and Einstein? -- 3. Minima and maxima -- The optical property of ellipses -- Linear regression via springs -- The polygon of least area -- The pyramid of least volume -- A theorem on centroids -- A isoperimetric problem -- The inscribed angle -- Fermat's Principle and Snell's Law -- The least sum of squares to a point -- Why does a triangle balance on the point of intersection of the medians? -- The least sum of distances to four points in space -- Shortest distance to the sides of an angle -- Minimal-perimeter triangles -- An ellipse in the corner -- 4. Inequalities by electric shorting -- The arithmetic mean is greater than the geometric mean by throwing a switch -- 5. Center of mass : proofs and solutions -- Center of mass of a semicircle by conservation of energy -- Center of mass of a half-disk -- Center of mass of a hanging chain -- Pappus's centroid theorems -- Ceva's theorem -- 6. Geometry and motion -- An equal-volumes theorem -- 7. Computing integrals using mechanics -- 8. The Euler-Lagrange equation via stretched springs -- Energy conservation by sliding a spring -- 9. Lenses, telescopes, and Hamiltonian mechanics -- Area-preserving mappings of the plane -- Mechanics and maps -- A hand-waving "proof" of area preservation -- The generating function -- A table of analogies between mechanics and analysis -- "The uncertainty principle" -- Area preservation in optics -- Telescopes and are preservation -- 10. A bicycle wheel and the Gauss-Bonnet theorem -- The dual-cones theorem -- A bicycle wheel and the dual cones -- The area of a country -- 11. Complex variables made simple(r) -- How a complex number could have been invented -- Functions as ideal fluid flows -- A physical meaning of the complex integral -- The Cauchy integral formula via fluid flow -- Heat flow and analytic functions -- Riemann mapping by heat flow -- Euler's sum via fluid flow -- Appendix. Physical background -- Springs -- Soap films -- Compressed gas -- Vacuum -- Torque -- The equilibrium of a rigid body -- Angular momentum -- The center of mass -- The moment of inertia -- Current -- Voltage -- Kirchhoff's Laws -- Resistance and Ohm's Law -- Resistors in parallel -- Resistors in series -- Power dissipated in a resistor -- Capacitors and capacitance -- The inductance : inertia of the current -- An electrical-plumbing analogy.

摘要附注:

In this delightful book, 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.

豆瓣简介：