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

Revision as of 01:10, 15 February 2021

Revision as of 01:11, 15 February 2021

Line 33:

Throughout, let <math>\Omega</math> be an open subset of <math>\R^n,</math> where <math>n \geq 1</math> is an integer.

Suppose <math>t = \left( t_1, \ldots, t_n \right) \in \Omega</math> and <math>f : \operatorname{Dom___domain} f \to Y</math> is a function such that <math>t \in \operatorname{Dom___domain} f</math> with <math>t</math> an accumulation point of <math>\operatorname{Dom___domain} f.</math> Then <math>f</math> is '''{{em|differentiable at <math>t</math>}}'''{{sfn|Trèves|2006|pp=412–419}} if there exist <math>n</math> vectors <math>e_1, \ldots, e_n</math> in <math>Y,</math> called the '''{{em|partial derivatives of <math>f</math> at <math>t</math>}}''', such that

::<math>\lim_{\stackrel{p \to t}{t \neq p \in \operatorname{Dom___domain} f}} \frac{f(p) - f(t) - \sum_{i=1}^n \left(p_i - t_i \right) e_i}{\|p - t\|_2} = 0</math> in <math>Y</math>

where <math>p = \left(p_1, \ldots, p_n\right).</math>

Line 43:

For <math>k \in \N,</math> having defined what it means for a function <math>f</math> to be <math>C^k</math> (or <math>k</math> times continuously differentiable), say that <math>f</math> is '''{{em|<math>k + 1</math> times continuously differentiable}}''' or that '''{{em|<math>f</math> is <math>C^{k+1}</math>}}''' if <math>f</math> is continuously differentiable and each of its partial derivatives is <math>C^k.</math>

Say that <math>f</math> is <math>C^{\infty},</math> '''{{em|smooth}}''', <math>C^\infty,</math> or '''{{em|infinitely differentiable}}''' if <math>f</math> is <math>C^k</math> for all <math>k = 0, 1, \ldots.</math>

The '''{{em|[[Support (mathematics)|support]]}}''' of a function <math>f</math> is the [[Closure (topology)|closure]] (taken in its ___domain <math>\operatorname{Dom___domain} f</math>) of the set <math>\{ x \in \operatorname{Dom___domain} f : f(x) \neq 0 \}.</math>

== Spaces of ''C''^''k'' vector-valued functions ==