By Luther Pfahler Eisenhart

Created particularly for graduate scholars by way of a number one author on arithmetic, this advent to the geometry of curves and surfaces concentrates on difficulties that scholars will locate such a lot worthwhile.

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I'n are defined as follows: follows : each ¢i CPi 0O,j is constant and equal to ¢i(a) cpi(a) for i i'l'j -:f:. j, and ¢j CPj 0lj(t) = ¢j(a) cpj(a) + t. These arcs ares are transformed by the diffeomorphism 0 'l'j(t) = 28 1 Local Inversion f :; U --+ f(U) c Rn that is associated to the curvilinear coordinate system into the n straight lines parallel to the coordinate eoordinate axes and passing through f (a). In particular, we have (i'j (0), drPi (a)) == Oi,j (1'j(O), d4Ji(a)) di,j ; thus the 1'j(O) form a basis for E and the drPi(a) d4Ji(a) are the dual basis of E*.

Ym) Ym) 1--7 1-+ Yi -

16 , who This question has played an important historical role. In 1874, Cantor 16 the year before had proved the impossibility of constructing a bijection between N and R, posed the problem of dimension. After having tried unsuccessfully for three years to prove the nonexistence of a bijection between Rand R Rnn for n > 1, he succeeded, to his own amazement, to establish such a bijection : he wrote to 17 "I see it but I don't believe it". In reply, Dedekind suggested that it Dedekind 17 ought to be feasible to prove the impossibility of a bicontinuous bijection between m Rn =I- m.