Infinitesimal analysis has without doubt played a major role in the mathematical treatment of physics in the past, and will continue to do so in the future, but we must also be aware that several important aspects of the phenomenon being described, such as its geometrical and topological features, remain hidden, in using the differential formulation. This is a consequence not of performing the limit, in itself, but rather of the numerical technique used for finding the limit. In this paper, we analyze and compare the two most known techniques, the iterative technique and the application of the Cancelation Rule for limits. It is shown how the first technique, leading to the approximate solution of the algebraic formulation, preserves information on the trend of the function in the neighbourhood of the estimation point, while the second technique, leading to the exact solution of the differential formulation, does not. Under the topological point of view, this means that the algebraic formulation preserves information on the length scales associated with the solution, while the differential formulation does not. This new interpretation of the Cancelation Rule for limits is also discussed in the light of the findings of non-standard calculus, the modern application of infinitesimals, in the sense of non-standard analysis, to differential and integral calculus.
Similarities between Cell Method and Non-Standard Calculus
FERRETTI, ELENA
2014
Abstract
Infinitesimal analysis has without doubt played a major role in the mathematical treatment of physics in the past, and will continue to do so in the future, but we must also be aware that several important aspects of the phenomenon being described, such as its geometrical and topological features, remain hidden, in using the differential formulation. This is a consequence not of performing the limit, in itself, but rather of the numerical technique used for finding the limit. In this paper, we analyze and compare the two most known techniques, the iterative technique and the application of the Cancelation Rule for limits. It is shown how the first technique, leading to the approximate solution of the algebraic formulation, preserves information on the trend of the function in the neighbourhood of the estimation point, while the second technique, leading to the exact solution of the differential formulation, does not. Under the topological point of view, this means that the algebraic formulation preserves information on the length scales associated with the solution, while the differential formulation does not. This new interpretation of the Cancelation Rule for limits is also discussed in the light of the findings of non-standard calculus, the modern application of infinitesimals, in the sense of non-standard analysis, to differential and integral calculus.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
