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Using Finite Difference Methods Instead of Standard Calculus in Teaching Physics 1. Introduction

Physics is the basis of innumerable technological applications. It has shaped the face of contemporary society and represents the paradigm of all exact sciences. For a professor, making physics more accessible to students is not only of extreme importance but also one of the most challenging and rewarding tasks. The method of teaching physics has gone remarkably unchanged for decades even when the contents of the subject are up to date. Sometimes modern teaching technologies are adopted, but most often only in laboratories, lecture delivery and presentations. In the last few decades several authors have stressed the importance of using numerical methods in introductory physics courses due to the increasing availability of personal computers and computer algebra systems. A rigorous understanding of physics presumes a rigorous understanding of standard calculus. Classical physics, however, can be reformulated using finite difference calculus instead of standard calculus. This reformulation is rigorous and in the case of classical mechanics it avoids assuming that space-time is differentiable and thus is conceptually more consistent with the intrinsic discrete nature of time and space6,7,16. On the other hand physics in its intimate nature is a discrete science and most engineering and physics problems have no analytic solutions1-7, 12-15. This raises the question of whether physics can be understood without using standard calculus. In answering this question we notice that an alternative to standard calculus is finite difference approximations, which, has been widely used to solve physics and engineering problems, especially, when we are dealing with problems that have no analytic solutions. From an educational point of view this can dramatically enlarge the base of the examples used to support courses especially in mechanics, acoustics, electrodynamics, fluid mechanics, and modern physics. This motivates our interests in the reformulation of classical mechanics, electrodynamics and acoustics by using finite difference approach instead of standard calculus.

This reformulation, based on finite differences, together with a discussion of some of the educational aspects is presented in this paper. The finite difference techniques are intuitive modeling techniques easily understood and applied by the vast majority of students. In addition to the insight that can be gained on classical mechanics, acoustics, electrodynamics as within other branches of physics there is also the possibility of studying real-world physics and engineering problems that cannot be resolved analytically10, 11, 12-14. Also this approach accords the benefit of exploring many other areas of physics that otherwise require advanced mathematical techniques and knowledge. Last but not least we have to mention that in order to take the full advantage of these potentialities the numerical methods that should be understandable to the students should be easily programmable (preferable by using computer algebra systems) and efficient so that accurate results can be obtained without excessive computational resources and time.

The paper is organized as follows. Section 1 of this paper is reserved for introduction and we will sketch the finite difference methods, section 2 and 3 are reserved for the

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Belu, R., & Belu, A. C. (2007, June), Using Finite Difference Methods Instead Of Standard Calculus In Teaching Physics Paper presented at 2007 Annual Conference & Exposition, Honolulu, Hawaii. 10.18260/1-2--1663