By G. H. Hardy

ISBN-10: 0548641757

ISBN-13: 9780548641750

There should be few textbooks of arithmetic as recognized as Hardy's natural arithmetic. due to the fact that its booklet in 1908, it's been a vintage paintings to which successive generations of budding mathematicians have grew to become at first in their undergraduate classes. In its pages, Hardy combines the passion of a missionary with the rigor of a purist in his exposition of the elemental principles of the differential and essential calculus, of the homes of endless sequence and of different issues concerning the proposal of restrict.

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**Sample text**

We locally identify M 3 with an open set of Ej . Since the distribution spanned by Y\ and Y2 is integrable, there exist local coordinates u,v,w such that the integral submanifolds are obtained by putting w a constant. In particular let No be the surface obtained by putting w = 0. Denote its position vector by g{u,v) and let N(u,v) be the unit normal to this surface. Then, every point p in a neighborhood of No can be written as p(u, v,w) = g(u,v) 4- wN(u,v). So we can also consider u,v,w as local coordinates.

15) imply dB + B*821 = {l+e^\e1. 16) gives dBA*e\ + Bd(*8\) d(ac = e ^ - A 6l + (l + e^\ 1 2 l + e^]e g olA*8 Al+Bd{*B\) **? + £<*«) = = -(l+e^)d0 - ( i + 1^ ). dB1 + Bd*8l thus 2(l + e | ) ( M 1 -Bi^ +^ A « 1 = 0. 11), we obtain d*0l = O. 17), we get ! , H2 ^ -(VH, VH)/(H2 - K), we have dO1 = 0. , M is a Weingarten surface. 2 Suppose that M is a timelike Bonnet surface in L3 with H2 — K > 0 and (VH, VH) ^ -H2(H2 - K). Then M is a Weingarten surface. 1 Suppose that M is a timelike Bonnet surface in L3 with H2 — K > 0 and VH is spacelike.

17), we get ! , H2 ^ -(VH, VH)/(H2 - K), we have dO1 = 0. , M is a Weingarten surface. 2 Suppose that M is a timelike Bonnet surface in L3 with H2 — K > 0 and (VH, VH) ^ -H2(H2 - K). Then M is a Weingarten surface. 1 Suppose that M is a timelike Bonnet surface in L3 with H2 — K > 0 and VH is spacelike. Then M is a Weingarten surface. 1 Let M be a timelike Bonnet surface in L3 with H2 — K > 0 and VH is not lightlike, we already have proved that the conformal metric ^1'™' I has the Gaussian curvature —1, where / is the metric of M.

### A course of pure mathematics by G. H. Hardy

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