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

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The definition given above for curves are now extended from functions valued on subsets of <math>\R</math> to functions valued on open subsets of finite-dimensional [[Euclidean space]]s.

Throughout, let <math>k \inOmega</math> \{be 0,an 1,open \ldots,subset \inftyof <math>\}R^n,</math> and letwhere <math>n \Omegageq 1</math> beis an integer. either:

Suppose <math>t = \left( t_1, \ldots, t_n \right) \in \Omega</math> and <math>f : \operatorname{Dom} f \to Y</math> is a function such that <math>t \in \operatorname{Dom} f</math> with <math>t</math> aan limitaccumulation point of <math>\operatorname{Dom} 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

# an open subset of <math>\R^n,</math> where <math>n \geq 1</math> is an integer, or else

# a [[locally compact]] topological space, in which case <math>k</math> can only be <math>0,</math> and let <math>Y</math> be a [[topological vector space]] (TVS).

::<math>\lim_{\stackrel{p \to t}{t \neq p \in \operatorname{Dom} 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>

Suppose <math>t = \left( t_1, \ldots, t_n \right) \in \Omega</math> and <math>f : \operatorname{Dom} f \to Y</math> is a function such that <math>t \in \operatorname{Dom} f</math> with <math>t</math> a limit point of <math>\operatorname{Dom} 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>}}''', such that

::<math>\lim_{\stackrel{p \to t}{p \in \operatorname{Dom} 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>

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

If <math>f</math> is differentiable at a point then it is continuous at that point.{{sfn|Trèves|2006|pp=412–419}}

If <math>f</math> is differentiable at every point in some subset <math>S</math> of its ___domain then <math>f</math> is said to be ('''{{em|once}}''' or '''{{em|<math>1</math>-time}}''') '''{{em|differentiable in <math>S</math>}}''', where if the subset <math>S</math> is not mentioned then it is this means that it is differentiable at every point in its ___domain.

Say that <math>f</math> is <math>C^0</math> if it is continuous.

If <math>f</math> is differentiable atand everyif pointeach inof someits setpartial derivatives is a continuous function then <math>S \subseteq \Omegaf</math> thenis wesaid sayto thatbe ('''{{em|once}}''' or '''{{em|<math>f1</math> is-time}}''') '''{{em|continuously differentiable}}''' inor '''{{em|<math>SC^1.</math>}}'''.{{sfn|Trèves|2006|pp=412–419}}

IfFor <math>fk \in \N,</math> ishaving differentiabledefined atwhat everyit pointmeans offor itsa ___domainfunction and<math>f</math> ifto eachbe of<math>C^k</math> its(or partial<math>k</math> derivativestimes iscontinuously a continuous function then wedifferentiable), 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>}}'''{{sfn|Trèves|2006|pp=412–419}} if <math>f</math> is continuously differentiable and each of its partial derivatives is <math>C^k.</math>

Having defined what it means for a function <math>f</math> to be <math>C^k</math> (or {{mvar|k}} 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}}''' or '''{{em|infinitely differentiable}}''' if <math>f</math> is <math>C^k</math> for all <math>k = 0, 1, \ldots.</math>

If <math>f : \Omega \to Y</math> is any function then itsThe '''{{em|[[Support (mathematics)|support]]}}''' of a function <math>f</math> is the [[Closure (topology)|closure]] (taken in its ___domain <math>\Omegaoperatorname{Dom} f</math>) of the set <math>\{ x \in \operatorname{Dom} f : f(x) \neq 0 \}.</math>

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