Aspects of Integrability of Differential Systems and Fields (1511.01788v2)

Abstract: This monograph, written for educational purposes, serves as an introduction to the concept of integrability as it applies to systems of differential equations (both ordinary and partial) as well as to vector-valued fields. The general cases of integrability examined are: path-independence of line integrals of vector fields on the plane and in space; integration of systems of ordinary differential equations by using first integrals; and integrable systems of partial differential equations. Special topics include the integration of analytic functions on the complex plane and some elements from the geometric theory of differential systems. Certain more advanced subjects, such as Lax pairs and B\"acklund transformations, are also discussed.