T_{p}(M) = {v : G(p) → R | v is linear and satisfies the Leibniz rule } G(p) = {[g] | g^{smooth} : U → R, for some U^{open} such that p ∈ U ⊂ M} v ∈ T_{p}(M) is called a derivation Thm: Let M be an m-manifold, then T_{p}(M) is an mdimensional real vector space. Take a chart (U, ϕ) at p where ϕ(p) = 0, The standard basis for T_{p}(M) w.r.t.(U, ϕ) = {v_{1}, ..., v_{m}}, where v_{i} = D_{α_{i}}, α_{i} : (−ϵ, ϵ) → M, α_{i}(t) = ϕ^{−1}(0, ..., t, ..., 0) for some ϵ > 0, and the directional derivative of [g] in direction [α_{i}] =

D_{α_{i}}g = \frac{d(g ∘ α_{i})}{dt}|_{t=0} ∈ R

Suppose f^{smooth} : M → N, f(p) = q. The tangent (or differential) map, df_{p} : T_{p}M → T_{q}N, (df_{p}(v)) = the derivation which takes [g] ∈ G(q) to the real number v([g ∘ f]) I.e., df_{p} takes the derivation (v : G(p) → R) ∈ T_{p}M to the derivation in T_{q}N which takes the germ [g] ∈ G(q) to the real number v([g ∘ f]). Proposition: With respect to this choice of basis, the matrix of d_{p}f: T_{p}M → T_{q}N is (\frac{∂F_{i}}{∂x_{j}})_{φ_{(p)}}, where x_{j} are coordinates in R^{m} and F_{i} = (F_{1}, ⋯, F_{n}) in coordinates for R^{n}.

: Properties: D_{α} is linear and satisfies the Liebniz rule. i.e, 1.) D_{α}(g + h) = D_{α}g + D_{α}h 2.) D_{α}(cg) = cD_{α}g 3.) D_{α}(g · h) = D_{α}g · h(p) + g(p) · D_{α}h Here are some more properties of d_{p}f and T_{p}M: Let f: M → N be smooth. 1.) If f: M → N is a diffeomorphism, the d_{p}f is an isomorphism, for all p ∈ M. 2a.) If d_{p}f = 0 for all p ∈ M iff f is a constant map. 2b.) If v is a derivation, and f is constant, then v(f) = 0 3.) If N ≅ M/G, where G is a discrete Lie group acting properly discontinuously on M, and f: M → M/G is the orbit map, the d_{p}f is an isomorphism for all p. 4.) T_{(p,q)}(M × N) ≅ T_{p}(M) × T_{q}(N).

Rank Theorem: Suppose A_{0} ⊂ R^{n}, B_{0} ⊂ R^{m}, F : A_{0} → B_{0} ∈ C^{1} a ∈ A_{0}, b ∈ B_{0}. Suppose rank F = k.

Then there exists A^{open} ⊂ A_{0} such that a ∈ A and B^{open} ⊂ B_{0} such that b ∈ B and G, H, C^{r} diffeomorphisms

such that G : A → U^{open} ⊂ R^{n}, H : B → V^{open} ⊂ R^{m} and

H ∘ F ∘ G^{−1}(x_{1}, ..., x_{n}) = (x_{1}, ..., x_{k}, 0, ..., 0)

Proposition: rank DF (a) = k implies there exists V open such that a ∈ V and DF (x) ≥ k for all x ∈ V

Proof:

Rank of A = dim(span{r_{1}, ..., r_{m}}) = dim(span{c_{1}, ..., c_{m}}) = maximum order of any nonvanishing minor determinant.

Use determinant is a continuous function.

Thm 6.4 (Inverse Function Theorem): Suppose F : W^{open} ⊂ R^{n} → R^{n} ∈ C^{r}. Suppose for a ∈ W, det(DF_{a}) ≠ 0. Then there exists U such that a ∈ U^{open}, V = F (U) is open, and F : U → V is a C^{r}-diffeomorphism. Moreover, for x ∈ U and y = F (x), DF_{y}^{−1} = (DF_{x})^{−1}