Analysis MOC

Differentiability

A map 𝑓 :𝐷 ⊆ℝ →𝑅 ⊆ℝ is 𝑛-differentiable at a point 𝑥0 ∈𝑈 iff it has an 𝑛-th derivative at that point, and thus all derivatives up to 𝑛. anal Moreover 𝑓 is called 𝑛-differentiable if it is differentiable at every 𝑥0 ∈𝑈. 𝐶𝑛 is the set of all 𝑛-differentiable functions with a continuous 𝑛th derivative, and is called a differentiability class, and 𝑛 ≤𝑚 ⟹ 𝐶𝑛 ⊆𝐶𝑚. In particular,

• 𝐶0 is the class of all continuous functions;

• 𝐶𝜔 of analytic functions; and

• 𝐶∞ of infinitely differentiable functions¹.

Generalizations

Complex functions

In complex analysis all differentiable functions are analytic and infinitely differentiable. Such a function is called holomorphic.

Open subsets of real coördinate space

Differentiability generalizes naturally to higher dimensional Real coördinate space (and open subsets thereof). A function 𝑓 :ℝ𝑁 →ℝ𝑀 is 𝐶𝑛 iff it has all 𝑛-th order partial derivatives.

Arbitrary subsets of real coördinate space

Let 𝑋 ⊆ℝ𝑁 be inhabited. A function 𝑓 :𝑋 →ℝ𝑀 is 𝐶𝑘 iff every 𝑥 ∈𝑋 has an open neighbourhood 𝑈 ⊆ℝ𝑁 with a 𝐶𝑘 extension 𝐹 :𝑈 →ℝ𝑀 such that 𝐹(𝑦) =𝑓(𝑦) for all 𝑦 ∈𝑋 ∩𝑈. diff

By considering real submanifolds, this yields the notion of differentiability for maps between such manifolds.

Map between manifolds

Let 𝑓 :𝑀 →𝑁 be a map between manifolds of dimension 𝑚 and 𝑛 respectively. The 𝐶𝑘 class is only well defined if 𝑀 and 𝑁 are 𝐶𝑘 differentiable manifolds. Let 𝑝 ∈𝑀. 𝑓 is called 𝑘-differentiable at 𝑝 ∈𝑀 iff there exists a chart (𝑈,𝜑) containing 𝑝 and (𝑉,𝜓) containing 𝑓(𝑝) such that 𝜓𝑓𝜑−1 is 𝑘-differentiable at 𝑝. diff 𝑓 is called 𝐶𝑘 iff it is 𝑘-differentiable everywhere.

---

tidy | en | SemBr

Footnotes

1. Often called smooth, however the exact meaning of this term varies between authors, e.g. sometimes 𝐶1 functions are referred to as smooth. ↩