By William M. Boothby
Nice introductory differential geometry textual content! I used this e-book to assist me move my qualifying examination. Yay Boothby!
Read or Download An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised, Volume 120, Second Edition (Pure and Applied Mathematics) PDF
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Extra info for An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised, Volume 120, Second Edition (Pure and Applied Mathematics)
I, Chap. 7. - ~ T f(Mtp )dt T 0 tends (as T- oo),to a certain constant; at any rate for all initial configurations P not lying in a set e in the phase surface whose measure /-tee) is zero; /-t here is the (natural) Lebesgue measure in the phase surface. Thus, the familiar argument continues, we should not expect to observe a configuration in which the long-time average of such a function f is not elose to its equilibrium value. Here I may conveniently use a bit of mathematical prestidigitation of the very sort to which I object, thus paradoxically making an argument serve the purpose of its own denunciation.
Unfortunately however, an absurdity in uniform is far more persuasive than an absurdity unelad. The very fact that a theory appears in mathematical form, that, for instance, a theory has provided the occasion for the application of a fixed-point theorem, or of a result about difference equations, somehow makes us more ready to take it seriously. And the mathematical-intellectual effort of applying the theorem fixes in us the particular point of view of the theory with which we deal, making us blind to whatever appears neither as adependent nor as an independent parameter in its mathematical formulation.
Mathematics adjusts only with reluctance to the extern al , and vitally necessary, approximating of the scientists, and shudders each time a batch of small terms is cavalierly erased. Only with difficulty does it find its way to the scientist's ready grasp of the relative importance ofmany factors. Quite typically, science leaps ahead and mathematics plods behind. Related to this deficiency of mathematics, and perhaps more productive of rueful consequence, is the simple-mindedness of mathematics - its willingness, like that of a computing machine, to elaborate upon any idea, however absurd; to dress scientific brilliancies and scientific absurdities alike in the impressive uniform offormulae and theorems.
An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised, Volume 120, Second Edition (Pure and Applied Mathematics) by William M. Boothby