Exact Differential Equations Studied in the Context of Vector Fields

Charles Wakefield
University of Texas of the Permian Basin

In this presentation I will use Mathematica to illustrate the concepts of path dependency and path independency as studied in a vector field. For the path-independent example I will solve the differential equation for the potential function and then show the graphics, which illustrate that the work required is indeed path independent. I will calculate the work for several different paths, showing that the results are all equal. Then in one graphic I will illustrate the initial and final positions of the particle along with all the paths considered superimposed on the vector field. This presentation is a part of one module of a larger project; about 15 modules are planned, funded by a grant from the Telecommunications and Information Technology Division of the University of Texas System. I will illustrate in similar fashion the path dependency for a second example, showing this time that the work of moving the particle is different for the different paths. This presentation is primarily to illustrate how Mathematica can be effectively used to teach these concepts more clearly to engineering physics students, Calculus III students, and finally differential equations students.