By V.I. Arnold, Alexander Varchenko, S.M. Gusein-Zade
... there's not anything so enchanting, so grandiose, not anything that stuns or captivates the human soul really lots as a primary path in a technology. After the 1st 5 - 6 lectures one already holds the brightest hopes, already sees oneself as a seeker after fact. I too have wholeheartedly pursued technology passionately, as one might a loved girl. i used to be a slave, and sought no different sunlight in my existence. Day and evening I stuffed myself, bending my again, ruining myself over my books; I wept whilst I beheld others exploiting technology fot own achieve. yet i used to be no longer lengthy enthralled. in fact each technology has a starting, yet by no means an finish - they cross on for ever like periodic fractions. Zoology, for instance, has came across thirty-five thousand kinds of existence ... A. P. Chekhov. "On the line" during this booklet a commence is made to the "zoology" of the singularities of differentiable maps. This conception is a tender department of study which at present occupies a crucial position in arithmetic; it's the crossroads of paths major from very summary corners of arithmetic (such as algebraic and differential geometry and topology, Lie teams and algebras, advanced manifolds, commutative algebra and so on) to the main utilized parts (such as differential equations and dynamical platforms, optimum regulate, the idea of bifurcations and catastrophes, short-wave and saddle-point asymptotics and geometrical and wave optics).
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Additional resources for Singularities of Differentiable Maps: Volume I: The Classification of Critical Points Caustics and Wave Fronts
Consider the critical points at which one of the second derivatives, say o2f/ox 2, is different from zero. The equation of/ox = 0 determines a smooth curve (by the implicit function theorem). The critical points Basic concepts 34 of interest to us lie on this curve and are critical for the restriction off to this curve. By Proposition 1 the measure of the set of values offat these critical points is equal to zero. For the other second derivatives the argument is the same. Proposition 4 is thus proved.
The classes I/ 35 Proof of the weak transversality theorem: We consider the special case where C is a linear subspace in B. We embed the given map f:A --. e(x) = f(x) - S (SE B). Let p:B --. e is transversal to C. This proves the assertion concerning everywhere denseness for the transversality theorem in the special case considered. The general case may be reduced to the special case, something that we do not stop to do here; the openness of the set of maps of a closed manifold, transversal to a closed submanifold, is obvious.
The map f is said to be transversal to C at the point a of A if either f(a) does not belong to C or (Fig. a is transversal to the tangent space to C: The classes 1/ 31 f A a ---0-- Fig. 23. The map f is said to be transversal to C if it is transversal to C at every point of A. Proposition: Iff: A ~ B is transversal to C then f - l( C) is a smooth submanifold in A, having the same codimension in A as C has in B. Example: Let C be a curve in a three-dimensional space B and let A be onedimensional.