Metric Spaces

Question 1

Define differentiable in a metric space

Answer 1

We say that a function f: Ω ⊆ Rn → Rm is differentiable at a ∈ Ω if there exists a linear map L: Rn → Rm such that lim(h→0) [f(a + h) − f(a)]/|h| = L.

Question 2

Define a directional derivative

Answer 2

We say that a function f : Ω ⊆ Rn → Rm is differentiable at a ∈ Ω in direction n if lim(λ→0)[f(a+λn)−f(a)]/λ exists.

Question 3

Define a path

Answer 3

Let U ⊆ R2 and a0, a1 ∈ U. A path in U between a0 and a1 is a continuous map γ : [0, 1] → U with γ(0) = a0 and γ(1) = a1.

Question 4

Define path connected

Answer 4

U is path connected if for every a0, a1 ∈ U there exists a path between a0 and a1.

Question 5

Define homotopy

Answer 5

Let U ⊆ R2 and a0, a1 ∈ U. Let γ0 and γ1 be paths in U connecting a0 and a1. We say that the paths γ0 and γ1 are homotopic if there exists a continuous map Γ : [0, 1] × [0, 1] → U with Γ(s, 0) = a0 and Γ(s, 1) = a1 for all s ∈ [0, 1], and Γ(0, t) = γ0(t) and Γ(1, t) = γ1(t) for all t ∈ [0, 1].

Question 6

Define simply connected

Answer 6

U ⊆ R2 is simply connected if it is path-connected and given any points a0, a1 ∈ U then any two paths in U between a0 and a1 are homotopic.