342-0 - Introduction to Differential Geometry List of Formulas PDF

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342-0 - Introduction to Differential Geometry Northwestern University List of formulas which you will have to know for the final It is up to you to reconstruct the background of each formula, which is given without definitions, etc. This list is not complete, in the sense that it will not include some basics (like the equation of a circle, line, etc.) which you will also need to know. Also, these are just formulas, you will also need to know definitions of key concepts and problem-solving procedures. In other words, this list is just supposed to help you remember these formulas, it is not an exhaustive list of what you need to know for the final. • L(γ) = • s(t) =

∫

∫

b

∥γ ′ (t)∥dt

a

t

∥γ ′ (τ )∥dτ,

h = s−1 ,

γ˜ = γ ◦ h

a

• If ∥γ ′ (t)∥ = 1 then κ(t) = ∥γ ′′(t)∥ • T = • If ∥γ ′ (t)∥ = 1 then T = γ ′, • In R3 , T ′ = κN,

N ′ = −κT − τ B,

• In R2 , T ′ = κ ˜ N,

N ′ = −˜ κT,

κ ˜=

N =

γ ′′ , ∥γ ′′∥

B =T ×N

B′ = τ N 1 ∥γ ′ ∥3

det(γ ′ , γ ′′)

• A6 •

γ′ ∥γ ′ ∥

ℓ2 4π

Tp S = Im(dφq : R2 → R3 )

• Tp (f −1 (a)) = ∇f(p)⊥ , and N = • ∂1 =

∂φ , ∂u

∂2 =

∇f ∥∇f ∥

∂φ ∂v

• Ip (v) = ⟨v, v⟩p ,

E = ∥∂1 ∥2 , F = ⟨∂1 , ∂2 ⟩, G = ∥∂2 ∥2

• v = v1 ∂1 + v2 ∂2 ,

I(v) = Ev12 + 2F v1 v2 + Gv22

• IIp (v) = −⟨dNp (v), v⟩p , e = −⟨∂1 , dN (∂1 )⟩, f = −⟨∂1 , dN (∂2 )⟩ = −⟨∂2 , dN (∂1 )⟩, g = −⟨∂2 , dN (∂2 )⟩

1

• e=

⟨

N,

∂2φ ⟩ ∂u2

,f =

• v = v1 ∂1 + v2 ∂2 ,

⟨

∂2φ N, ∂u∂v

⟩ ∂2φ N, 2 ∂v

II(v) = k1 cos2 θ + k2 sin2 θ

v = cos θe1 + sin θe2 ,

H=

,g =

⟨

II(v) = ev 12 + 2f v1 v2 + gv22

• •

⟩

k1 + k2 , 2

eg − f 2 EG − F 2

K = k1 k2 =

• k12 = H ±

√

H2 − K

• ∥v∥ = 1, • •

DV (t) = Πγ(t) (V ′ (t)) = V ′ (t) − ⟨V ′ (t), N (t)⟩N (t) d ⟨V (t), W (t)⟩ = ⟨DV (t), W (t)⟩ + ⟨V (t), DW (t)⟩ dt

• • •

κn (v) = II(v )

Dγ ′ = γ ′′ − ⟨γ ′′, N ⟩N ∥γ ′ ∥ = 1, ∫

Dγ ′ (t) = κg (t)(N (t) × γ ′ (t)),

f dA =

R

∫ ∫

(f ◦ φ−1 )

∫

∑∫

ϕ−1 (R)

• R simple region,

KdA +

R

√

κ2 = κ2g + κn2

EG − F 2 dudv

ti+1

κg (s)ds +

ti

i

∑

αi = 2π

i

• χ = #F − #E + #V • R regular region,

∫

KdA +

R

∑∫ i

• S closed and bounded,

κg (s)ds +

Ci

∫

∑

αi = 2πχ(R)

i

KdA = 2πχ(S)

S

• χ(Σg ) = 2 − 2g • S1 #S2 , connected sum, χ(S1 #S2 ) = χ(S1 ) + χ(S2 ) − 2 • S minimal if H = 0

2...