Charts for S^{2}:

Stereographic projection:

U_{N} = S^{n} − {N = (1, 0, ..., 0)}. ϕ_{N} : U_{N} → R^{n}, ϕ_{N}(x) = \frac{x}{(1−x_{0})}

U_{S} = S^{n} − {S = (−1, 0, ..., 0)}. ϕ_{S} : U_{S} → R^{n}, ϕ_{S}(x) = \frac{x}{(1+x_{0})}

Pre-atlas for S^{n}: A_{1} = {(U_{N}, ϕ_{N}), (U_{S}, ϕ_{S})},

Projection:

U_{zp} = {(x, y, z) ∈ S^{2} | z > 0}, φ_{zp} : U_{zp} → {(x, y) | x^{2} + y^{2} < 1}, φ_{zp}(x, y, z) = (x, y)

U_{yp} = {(x, y, z) ∈ S^{2} | y > 0}, φ_{yp} : U_{yp} → {(x, z) | x^{2} + z^{2} < 1}, φ_{yp}(x, y, z) = (x, z)

U_{xp} = {(x, y, z) ∈ S^{2} | x > 0}, φ_{xp} : U_{xp} → {(y, z) | y^{2} + z^{2} < 1}, φ_{xp}(x, y, z) = (y, z)

U_{zn} = {(x, y, z) ∈ S^{2} | z < 0}, φ_{zn} : U_{zn} → {(x, y) | x^{2} + y^{2} < 1}, φ_{zn}(x, y, z) = (x, y)

U_{yn} = {(x, y, z) ∈ S^{2} | y < 0}, φ_{yn} : U_{yn} → {(x, z) | x^{2} + z^{2} < 1}, φ_{yn}(x, y, z) = (x, z)

U_{xn} = {(x, y, z) ∈ S^{2} | x < 0}, φ_{xn} : U_{xn} → {(y, z) | y^{2} + z^{2} < 1}, φ_{xn}(x, y, z) = (y, z)

Compatibility: φ_{zp} ∘ φ_{yn}^{−1} : φ_{yn}(U_{zp} ∩ U_{yn}) → φ_{zp}U_{zp} ∩ U_{yn}). U_{zp} ∩ U_{yn} = {(x, y, z) ∈ S^{2} | z > 0 and y < 0}. φ_{yn}(U_{zp} ∩ U_{yn}) = {(x, z) | z > 0 and x^{2} + z^{2} < 1} φ_{zp} ∘ φ_{yn}^{−1}(x, z) = φ_{zp}(x, −(1 − x^{2} − z^{2})^{1/2}, z) = (x, −(1 − x^{2} − z^{2})^{1/2}). Pre-atlas for S^{2}: A_{2} = {(U_{zp}, φ_{zp}), ..., (U_{xn}, φ_{xn})}, Polar coordinates (x, y) = (rcosθ, rsinθ), r ≥ 0, θ ∈ [0, 2π), (r, θ) = (\sqrt{x^{2} + y^{2}}, tan^{−1}(y/x) U_{γ} = S^{1} − {(1, γ)}, ϕ_{(1,γ)} : U_{p} → (0, 2π) ϕ_{(1,γ)}(r, θ) = θ − γ mod 2π Pre-atlas for S^{1}: {(U_{γ}, ϕ_{(1,γ)} | γ ∈ S^{1}} Spherical coordinates (x, y, z) = (rsinϕcosθ, rsinϕsinθ, rcosϕ), r ≥ 0, θ ∈ [0, 2π), ϕ ∈ [0, π] (r, θ, ϕ) = (\sqrt{x^{2} + y^{2} + z^{2}}, tan^{−1}(y/x), tan^{−1}(\sqrt{x^{2} + y^{2}}/z)). U_{p} = S^{2} − {p}, ϕ_{p} : U_{p} → ϕ_{p}(U) ϕ_{p}(r, θ, ϕ) = (θ, ϕ) Cylindrical: (x, y, z) = (rcosθ, rsinθ, z), r ≥ 0, θ ∈ [0, 2π), z ∈ R (r, theta, z) = (\sqrt{x^{2} + y^{2}}, tan^{−1}(y/x), z).

: G(p, N) = {[g] | g^{smooth} : U → N, for some U^{open} such that p ∈ U ⊂ M} G(p) = G(p, R) Let α : I → M where I = an interval ⊂ R, α(0) = p. Note [α] ∈ G[0, M] Directional derivative of [g] in direction [α] =

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

T_{p}(M) = {v : G(p) → R | v is linear and satisfies the Leibniz rule } v ∈ T_{p}(M) is called a derivation Given a chart (U, ϕ) at p where ϕ(p) = 0, the standard basis for T_{p}(M) = {v_{1}, ..., v_{m}}, where v_{i} = D_{α_{i}} and for some ϵ > 0, α_{i} : (−ϵ, ϵ) → M, α_{i}(t) = ϕ^{−1}(0, ..., t, ..., 0) If v ∈ T_{p}(M), then v = Σ_{i=1}^{m}a_{i}v_{i} where a_{i} = v([π_{i} ∘ ϕ]) 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]).