ODEs together PDEs and Vector Fields in the SoftAge

At this moment when we can employ software, mathematics education has to be reviewed. In this article, we point out that ODEs should be taught from a geometric and qualitative point of view together with an introduction to PDEs and vector fields. This would increase the skills of the future mathematics user, not only to obtain explicit solutions from a straight command like DSolve, but also in the situations where this command does not help. The geometric interpretation and the concept of direction fields with images generated by software will give us a good understanding of possible system evolutions.

PROBLEMS OF DIFFERENTIAL AND TOPOLOGICAL DIAGNOSTICS. PART 5. THE CASE OF TRAJECTORIAL MEASUREMENTS WITH ERROR

Proposed work is the fifth work of the cycle on differential and topological diagnostics. The article gives an estimate of the errors method of direction fields in the case of not accurate trajectory measurements, but trajectory measurements with an error are limited by the modulus of a given a smooth function of time, and in case this error is a random variable distributed according to the normal law with fixed parameters. We show that in these more complex cases, you can specify the best number of required trajectory measurements, inwhich the proposed algorithms of diagnostics will work constructively, and the malfunction will be determined unambiguously.

Frame and direction mappings for surfaces in ℝ3

We study frames in ℝ3 and mapping from a surface M in ℝ3 to the space of frames. We consider in detail mapping frames determined by a unit tangent principal or asymptotic direction field U and the normal field N. We obtain their generic local singularities as well as the generic singularities of the direction field itself. We show, for instance, that the cross-cap singularities of the principal frame map occur precisely at the intersection points of the parabolic and subparabilic curves of different colours. We study the images of the asymptotic and principal foliations on the unit sphere by their associated unit direction fields. We show that these curves are solutions of certain first order differential equations and point out a duality in the unit sphere between some of their configurations.