By Michael Spivak

ISBN-10: 0914098810

ISBN-13: 9780914098812

Ebook by way of Michael Spivak, Spivak, Michael

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**Example text**

7). 2. The classes LI Here singularities are classified according to the rank of the first differential of the map and the ranks of its restrictions to submanifolds of singularities. x: TxMm -+ TJ(x)N n its derivative at x. X is equal to i. All the points of class :E i for f form a subset of M, called the set :E i for f and denoted by :EV). Example: For the map of Whitney pleat (Fig. 21) Fig. 21. all the critical points are of class :El, while the noncritical points are of class :Eo. Remark: In particular, the fold points and the pleat points are of the same class :E 1.

These equations are independent. Indeed consider the bordering minor, obtained by adding in the row and column which intersect at ak,l. The expansion of this minor as a Taylor series in a begins with ak,1 + O(a 2 ). Therefore the differentials of our (m - r)(n - r) minors are independent at zero. Once the differentials of the minors are independent then, by the implicit function theorem, setting the minors to zero defines a submanifold whose co dimension is equal to the number of equations, as was claimed.

For any set of natural numbers I = (i 1 , ••• , in}the set k/(f) is defined inductively as follows. Definition: Let 1;1(f) = ki' ..... ik(f) c M be a smooth manifold. Then Basic concepts 44 is the set of points where the kernel of the restriction of f to dimension i k + 1. '~/(f) has Remark: According to the definition the manifolds are embedded one inside another. Therefore the kernels of the differentials of the restrictions of f to these embedded submanifolds are also embedded one inside another.

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