Differentiable vector-valued functions from Euclidean space: Difference between revisions

Revision as of 01:36, 15 February 2021

Revision as of 19:22, 8 March 2021

Line 1:

In the mathematical discipline of [[functional analysis]], it is possible to generalize the notion of [[derivativeDerivative (mathematics)|derivative]] to arbitrary (i.e. [[Dimension (vector space)|infinite dimensional]]) [[topological vector space]]s (TVSs) in multiple ways.

But when the ___domain of a TVS-value functionsfunction is a subset of finite-dimensional [[Euclidean space]] then the number of generalizations of the derivative is much more limited and derivatives are more well behaved.

This article presents the theory of <math>k</math>-times continuously differentiable functions on an open subset <math>\Omega</math> of Euclidean space <math>\R^n</math> (<math>1 \leq n < \infty</math>), which is an important special case of [[Differentiation (mathematics)|differentiation]] between arbitrary TVSs.

All vector spaces will be assumed to be over the field <math>\mathbb{F},</math> where <math>\mathbb{F}</math> is either the [[real numbers]] <math>\R</math> or the [[complex numbers]] <math>\C.</math>