Published in Scientific Papers. Series E. Land Reclamation, Earth Observation & Surveying, Environmental Engineering, Vol. XII
Written by Cosmin-Constantin NIŢU
In mathematics, the term "differential" refers to several related notions derived from the early days of mathematical analysis, which became rigorous later, such as infinitesimal differences and the derivatives of functions. The notion is used in various areas of mathematics such as algebraic geometry, algebraic topology, calculus, differential geometry etc. The term differential is used non rigorously in calculus referring to an infinitely small ("infinitesimal") variation change in a quantity. For example, if one considers x as a variable, then a "bigger" change in the value of x is often denoted by Δx. The differential dx is an infinitesimal change of the variable x. The concept of an infinitely small or infinitely slow change is very useful, and there are several of mathematical tools to make it precise. Using calculus, it is possible to relate the infinitesimal changes of several variables to each other using function derivatives. In this article we present the notions of Gateaux and Fréchet differentials of a multivariable function with their properties, geometric interpretation and applications to function approximations.
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