The rôle of categorical structures in infinitesimal calculus

• Autorzy: Steingartner W. , Galinec D.
• Języki publikacji: EN

Abstrakty

EN

The development of mathematics stands as one of the most important achievements of humanity, and the development of the calculus, differential calculus and integral calculus is one of the most important achievements in mathematics. Differential calculus is about finding the slope of a tangent to the graph of a function, or equivalently, differential calculus is about finding the rate of change of one quantity with respect to another quantity. On the other hand, integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus. Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. The category theory is a mathematical approach to the study of algebraic structure that has become an important tool in theoretical computing science, particularly for semantics-based research. The notion of a limit in category theory generalizes various types of universal constructions that occur in diverse areas of mathematics. In our paper we illustrate how to represent some parts of infinitesimal calculus in categorical structures.

DE

Die Theorie von Kategorien ist der Bereich von Mathematik und sie dient vor allem fur das Studium der algebraischen Strukturen. Sie wird aber sehr oft auch in Informatik geltend gemacht. Manche bedeutende mathematische Bereiche kann man mithilfe der Kategorien darstellen und das ermöglicht mit den mathematischen Strukturen viel einfacher zu arbeiten als ohne Anwendung der Kategorien. Der Grund der Infinitesimalrechnung bilden zwei duale Bereiche - Differential- und Integralrechnung. In unserem Beitrag orientieren wir uns auf die Konstruktion des Diagramms von Stammfunktionen zur Funktion Kosinus. Von diesen Funktionen konstruieren wir den kommutativen Kegel und wir finden seinen Grenzwert. In dem zweiten Teil des Artikels zeigen wir den Ausdruck der Derivationen von Funktionen in der Kommakategorie und wir konstruieren den kodomänen Funktor zwischen der Kategorie der Derivationen und der Kategorie der Mengen für differenzierbare Funktionen.

Słowa kluczowe

EN

infinitesimal calculus; integral calculus; differential calculus

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2013
• Tom: Vol. 12, nr 1
• Strony: 107--119
• Opis fizyczny: Bibliogr. 13 poz., rys., tab.

Twórcy

autor

Steingartner W.

• Department of Computers and Informatics, Technical University of Kosice, Kosice, Slovakia

autor

Galinec D.

• Department of Informatics and Computing, Zagreb Polytechnic for Technical Sciences Zagreb, Croatia

Bibliografia

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