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} is an algebra over R. v ∈ T_{p}(M) is called a derivation The directional derivative of [g] in direction [α] =

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

Properties: D_{α} is linear and satisfies the Leibniz rule. i.e, 1a.) D_{α}(g + h) = D_{α}g + D_{α}h 1b.) D_{α}(cg) = cD_{α}g 2.) D_{α}(g · h) = D_{α}g · h(p) + g(p) · D_{α}h Thus D_{α} ∈ T_{p}(M) Thm: Let M be an m-manifold, then T_{p}(M) is an m-dimensional real vector space. [(c_{1}v + c_{2}w)(f) = c_{1}v(f) + c_{2}w(f)]. 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}} and α_{i} : (−ϵ, ϵ) → M, α_{i}(t) = ϕ^{−1}(0, ..., t, ..., 0) for some ϵ > 0.

Prop: {v_{1}, ..., v_{m}} are linearly independent. Proof: Evaluate v_{i} at a “projection map”. Thm: {v_{1}, ..., v_{m}} span T_{p}(M). If v ∈ T_{p}(M), then v = Σ_{i=1}^{m}a_{i}v_{i} where a_{i} = v([π_{i} ∘ ϕ]) Prop: If v is a derivation, and f is constant, then v(f) = 0.

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]).

Suppose f : M → N is smooth. Then f^{∗} : G(q) → G(p) is a homomorphism. Let V be a real vector space. Then the dual space, V^{∗} = {f : V → R | f is linear } 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}.

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. 2.) If d_{p}f = 0 for all p ∈ M iff f is a constant map. 3.) If id : M → M, id(x) = x, then d_{p}(id) = I_{m} 4.) d_{r}(f ∘ h) = d_{p}(f) ∘ d_{r}(h) where p = h(r). 5.) 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. 6.) 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}