The Geometry of Vector Fields and Two Dimensional Heat Equation

Year 2023, Volume: 16 Issue: 1, 73 - 80, 30.04.2023

Abdigappar Narmanov Eldor Rajabov

https://doi.org/10.36890/iejg.1230873

Abstract

The geometry of orbits of families of smooth vector fields was studied by many mathematicians due
to its importance in applications in the theory of control systems, in dynamic systems, in geometry
and in the theory of foliations.
In this paper it is studied geometry of orbits of vector fields in four dimensional Euclidean space. It is shown that orbits generate
singular foliation every regular leaf of which is a surface of negative Gauss curvature and zero normal torsion.

In addition, the invariant functions of the considered vector fields are used to find solutions of the two-dimensional heat equation that are invariant under the groups of transformations generated by these vector fields.

Keywords

Vector field, orbit of vector fields, foliation, infinitesimal generator, heat equation, symmetry group

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