Naslov
A generalized approach to differentiability

Autori
Koceić Bilan, Nikola

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

ISBN
978-953-7155-24-7

Skup
7th Croatian Mathematical Congress

Mjesto i datum
Split, Hrvatska, 15.06.2022. - 18.06.2022

Vrsta sudjelovanja
Pozvano predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
differentiability, directional derivatives, raylike neighborhood, neighborhood ray

Sažetak
The differentiability of scalar and vector functions of multiple variables is defined only at the interior points of the domain of these functions, which results in the traditional consideration of functions only with an open domain in Rn. This significantly narrows the possibility of applying potential techniques and tools of differential calculus to a wider class of functions. Although there is a strong need for it in various problems of mathematical analysis and other mathematical branches, so far, the notion of differentiability of a function has not been considered or successfully defined at points outside the interior of domain of a function. In this talk, we will define the differentiability at all points of a domain X ⊆ Rn of a function f : X → Rm in which that notion makes sense. These are the points that admit neighborhood ray in X which is the minimum condition for the notion of linearization of a function (the essential property of differentiable functions) to make sense. In such a way, the notion of differentiability is significantly expanded, leading to a new theory of differentiable functions that offers completely unexpected phenomena and pathologies (such as the non- uniqueness of differentials, the discontinuity of differentiable functions...), but also reveals some common misconceptions. However, if one reduces this theory only to the points with particularly special properties (points that admit raylike neighborhood and a linearization space with dimensions equal to the dimension of the Euclidean space to which the domain belongs), then all properties and assertions of the extended theory remain the same. Moreover, all known theorems and techniques of the differential calculus can be successfully generalized and support the new theory, whereby the derivatives in the direction of the chosen vectors take over the role of partial derivatives. This is especially important for the functions which are differentiable at the point where there are no partial derivatives of them. If P ∈ X ⊆ Rn admits neighbourhood ray in X in the direction of some n linear independent vectors in Rn we will investigate under which conditions the existence of derivatives in the direction of those vectors at the point P implies the differentiability of a function f : X → Rm at P .

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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