![]() ![]() ![]() Our educational experience and the student reactions to our approach are detailed in this recent publication. Thus the students receive a significant exposure to both approaches. To respond to the recent comment, a difference between our approach and Keisler's is that we spend at least two weeks detailing the epsilon-delta approach (once the students already understand the basic concepts via their infinitesimal definitions). In fact, I did a quick straw poll in my calculus class yesterday, by presenting (A) an epsilon, delta definition and (B) an infinitesimal definition at least two-thirds of the students found definition (B) more understandable. Infinitesimals provide an alternative approach that is more accessible to the students and does not require excursions into logical complications necessitated by the epsilon, delta approach. The epsilon, delta techniques involve logical complications related to alternation of quantifiers numerous education studies suggest that they are often a formidable obstacle to learning calculus. To answer your question about the applications of infinitesimals: they are numerous (see Keisler's text) but as far as pedagogy is concerned, they are a helful alternative to the complications of the epsilon, delta techniques often used in introducing calculus concepts such as continuity. Jerome Keisler wrote a first-year-calculus textbook based to Robinson's approach.The real numbers $\mathbb$ is algebraically simplified to $2x \Delta x$ and one is puzzled by the disappearance of the infinitesimal $\Delta x$ term that produces the final answer $2x$ this is formalized mathematically in terms of the standard part function. Robinson's methods are used by only a minority of mathematicians. In the 1960s, building upon earlier work by Edwin Hewitt and Jerzy Łoś, Abraham Robinson developed rigorous mathematical explanations for Leibniz' intuitive notion of the "infinitesimal," and developed non-standard analysis based on these ideas. In that way the Leibniz notation is in harmony with dimensional analysis. In physical applications, one may for example regard f( x) as measured in meters per second, and d x in seconds, so that f( x) d x is in meters, and so is the value of its definite integral. Although the notation need not be taken literally, it is usually simpler than alternatives when the technique of separation of variables is used in the solution of differential equations. Nonetheless, Leibniz's notation is still in general use. A number of 19th century mathematicians (Cauchy, Weierstrass and others) found logically rigorous ways to treat derivatives and integrals without infinitesimals using limits as shown above. That is, mathematicians felt that the concept of infinitesimals contained logical contradictions in its development. These were quantities that are smaller than any real number, yet larger than zero. In the 19th century, mathematicians ceased to take Leibniz's notation for derivatives and integrals literally. Leibniz investigated the idea of extending the usual number system (what we call the real numbers) to a larger set containing infinitesimals. This use first appeared publicly in his paper De Geometria, published in Acta Eruditorum of June 1686, but he had been using it in private manuscripts at least since 1675. He based the character on the Latin word summa ("sum"), which he wrote ſumma with the elongated s commonly used in Germany at the time. While Newton did not have a standard notation for integration, Leibniz began using the character. The Newton-Leibniz approach to infinitesimal calculus was introduced in the 17th century. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |