2.5 Let U ⊂ R^{n} Defn: A vector field is a function, V : U → ∪_{a∈U}T_{a}(R^{n}), such that V : (a) ∈ T_{a}(R^{n}) Defn: A vector field is smooth if its components relative to the canonical basis {E_{ia} | i = 1, ..., n} are smooth. Ex: V : R^{2} → ∪_{a∈U}T_{a}(R^{2}), V(x, y) = (2x, −y) = 2xE_{1a} − yE_{2a}. Defn: A field of frames is a set of vector fields {V_{1}, ..., V_{2}} such that {V_{1}(a), ..., V_{2}(a)} forms a basis for T_{a}(R^{n}) for all a. Ex: {E_{1a}, ..., E_{na}} on R^{n}. Ex: {xE_{1a} + yE_{2a}, yE_{1a} − xE_{2a}} on R^{n} − {0}.

Let V = Σ_{i=1}^{n}α_{i}(a)E_{ia} be a smooth vector field on U. V : C^{∞} → C^{∞} V(f) = Σ_{i=1}^{n}α_{i}(a)\frac{∂f}{∂x_{i}} (a) is a derivation.

Thm 5.1: Let F^{closed} ⊂ R^{n}, K^{compact} ⊂ R^{n}, F ∩ K = ∅. There there is a C^{∞} function σ : R^{n} → [0, 1] such that σ(K) = {1}, σ(F) = {0}

Show that h(t) = \{ \begin{matrix}0 & t ≤ 0 \\ e^{\frac{−1}{t}} & t > 0 \end{matrix} is C^{∞}, (but not C^{ω}).

Let \overline{g}(x) = \frac{h(ϵ^{2}−||x||^{2})}{h(ϵ^{2}−||x||^{2})+h(||x||^{2}− \frac{ϵ^{2}}{4} )}

\overline{g}(x) = \{ \begin{matrix}1 & 0 ≤ ||x|| ≤ \frac{ϵ}{2} \\ positive & \frac{ϵ}{2} ≤ ||x|| < ϵ \\ 0 & ||x|| ≥ ϵ \end{matrix}

Let g(x) = \overline{g}(x − a)

σ(x) = 1 − Π_{i=1}^{k}(1 − g_{i})

where K ⊂ ∪_{i=1}^{k}B\frac{ϵ}{2} (a_{i}) and B_{ϵ}(a_{i}) ⊂ R^{n} − F.