Algebraic geometry 01 Algebraic curves, algebraic manifolds by I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov

By I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov

"... To sum up, this e-book is helping to profit algebraic geometry very quickly, its concrete sort is pleasing for college students and divulges the great thing about mathematics." --Acta Scientiarum Mathematicarum

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Example text

N are not considered functions of anything else, the idea of dl is not meant to apply to them, and they are more often regarded as infinitesimal numbers of some kind. In calculus on manifolds both 1 and ¢l, ... ,¢n are interpreted as species of the same kind, namely functions on a given set, and in particular the definition of the differential of a function applies equally well to the co-ordinate functions. One should always translate the letter d in this context as 'the rate of change of', rather than as 'an infinitesimal increment of' so that dl translates as 'the rate of change of f', and dl( v) as 'the rate of change of 1 along (the tangent vector) v'.

3 The chain rule, velocities and tangent vectors The derivative of a function under a variation involves only onedimensional calculus and a fortiori the same is true of partial derivatives. The first genuinely multi-dimensional result is the chain rule. If, is any path through a point p in a set P, and ¢I, ... , ¢n is a co-ordinate system, then we can formulate the chain rule as (J 0 ,)'(0) of (p)(¢l 0,)'(0) + ... 8. It is more familiar in an abbreviated form in which, is suppressed. For example, if P is a plane, say with a polar co-ordinate system (r, 0), then a path is usually called a curve in parametric form and described as (r(t), O(t)), rather than (r(-y(t)),O('Y(t))) whereupon the chain rule is expressed 0' f ' -- of, or r + of 00 The two versions of the chain rule therefore correspond by replacing f, r, and 0 in the expressions f', r' and 0' with f 0 " r 0 , and 00,.

The effort of calculation is replaced by the effort of learning some geometric concepts. However the dividend is that we can then hope to extract statistical significance from geometric significance. For instance, without geometry as a guide it would be hard to imagine why one would propose the vanishing of 'Y as a criterion for the exponentiality of a family. Along the way we will also discover the geometric significance of quantities such as the score and the Fisher information which are not invariant but nevertheless seem to transform in an orderly fashion when we change co-ordinates.

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