By Stein M.R., Dennis R.K. (eds.)

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**Example text**

5 we give higher-dimensional analogues of Rolle's theorem. An inte- = ak = 0 gral manifold I" of an ordered system of Pfaff equations a, = on a manifold M is called a separating solution of this system if there exists a chain of submanifolds M D r-1 D . D r'k such that the manifold r, is a separating solution of the Pfaff equation ai = 0 on the manifold M, the manifold r" is a separating solution of the equation aZ = 0 on the manifold r, , etc. The sequence of forms fi, = a, , fi, = a, A a l , ...

On the other hand, on each oval it has a maximum and a minimum. There are examples of algebraic curves of degree n + 1 that have n(n - 1)/2 compact components. COROLLARY 5. A P-curve of degree n has at most (3n - 1) (4n - 1) inflexion points (there may be straight lines among the components of the P-curve). Indeed, at the inflexion points of the trajectory of the vector field F the vectors x = F(x) and 9 = (OF(x)/dx)F(x) are collinear. The determinant of the matrix with the vectors F and (8F/8x)F as rows is equal to zero in these points.

Depending on whether the induced orientation under the inclusion of the coorientation of the origin in Mk coincides or not with the original orientation of Mk). III. SEPARATING SOLUTIONS OF PFAFF EQUATIONS 34 A coorientation of a submanifold 14 of codimension k in a manifold M is a coorientation of the tangent space of 1"k at each point in the tangent space of M at the same point that depends continuously on the point. A coorientation may be given by fixing a differential k-form on M at the points of the submanifold that is locally equal to the exterior product of independent I-forms that vanish when restricted to the submanifold.