Using MAPLE to Model Engineering Systems in Cylindrical and Spherical Coordinates

摘要

Students have difficulty with visualizing and solving problems in three dimensions, especially when the appropriate coordinate system is not rectangular. Not only do they have a hard time working in these coordinate systems, but the solutions, such a Bessel functions and Legendre polynomials, are alien to them upon first exposure. In order to assist the students in working in cylindrical and spherical coordinates, MAPLE is used to solve problems and plot the solutions in these coordinate systems. An introduction to these, and other, orthogonal functions is first presented. Once the students understand how various functions can be represented by series of orthogonal functions, other than a Fourier series of trigonometric functions, solutions to partial differential equations can be found. MAPLE is extremely useful here in that it can solve these equations, with help from the user, and the "strange" functions that result are handled and plotted quite nicely by MAPLE. Thus, an understanding of engineering systems characterized in these coordinates is made easier with the use of MAPLE. This paper begins with a description of orthogonal functions and generalized Fourier series of these functions to fit known functions. Specifically, Bessel functions and Legendre polynomials are illustrated. This is followed by the solution of partial differential equations in cylindrical and spherical coordinates. Finally, these solutions are plotted in order to get a feel for the results. In all cases, MAPLE is used to the extent possible. Because MAPLE does not have a boundary value problem solver, MAPLE needs assistance from the user. How this is best accomplished is illustrated. The students were very receptive to this approach and much of this paper is an illustration of their homework solutions or excerpts from their project reports.