POINTS IN SETS → interior Å (in the set), exterior (in the complement), boundary 𝛿A (neither), closure 𝐴 (Å + δ𝐴), limit
(always surrounded), isolated (not surrounded by some 𝑟 ball).
TYPES OF SETS → open (any point has some 𝑟 ball within), bounded (contained in some 𝑟 ball), compact (closed +
bounded), convex (segment joining 2 points within), star-shaped (with point with convex prop), poligonally pathconnected (2 points connected by finite path within).
𝑓: 𝐴 ⊆ ℝ𝑁 → ℝ𝑀
VECTOR-VALUED FUNCTIONS → N=1, M=2/3 gives curve, N=2, M=3 gives surface, N=3, M=3 gives vector field.
CONTINUITY → ∀𝜖 > 0 ∃𝛿 > 0 s.t. 𝑥 ∈ 𝐴 ∩ 𝐵𝛿 (𝑥0 ) ⇒ 𝑓(𝑥) ∈ 𝐵𝜖 (𝑓(𝑥0 )), if arithm oper, composition then cont, limit
∀𝜖 > 0 ∃𝛿 > 0 s.t. 𝑥 ∈ 𝐴 ∩ 𝐵𝛿 (𝑥0 ) ∖ {𝑥0 } ⇒ 𝑓(𝑥) ∈ 𝐵𝜖 ((𝑙))
⃗⃗⃗⃗)−𝑓(𝑎
⃗⃗)
𝑓(𝑎⃗⃗+ℎ𝑒
𝑖
, gradient (∇𝑓)(𝑎⃗) ≔ (𝑓𝑥1 (𝑎⃗), 𝑓𝑥2 (𝑎⃗), … , 𝑓𝑥𝑛 (𝑎⃗)) ∈ ℝ𝑁 , directional
ℎ
⃗⃗)−𝑓(𝑎⃗⃗)
𝑓(𝑎⃗⃗+ℎ𝑣
⃗⃗) = 𝑓(𝑎⃗) + (𝑑𝑎⃗⃗ 𝑓)(ℎ) +
deriv for unit vector (𝐷𝑣⃗⃗ 𝑓)(𝑎⃗) = lim
, differentiability (all partial exist) 𝑓(𝑎⃗ + ℎ
ℎ
ℎ→0
𝛿𝑓
N, M=1 → partial deriv 𝛿𝑥 (𝑎⃗) = lim
ℎ→0
𝑖
⃗⃗|)
𝑜(|ℎ
∀𝑣⃗ ∈ ℝ𝑁 → (𝑑𝑎⃗⃗ 𝑓)(𝑣⃗) = (𝐷𝑣⃗⃗ 𝑓)(𝑎⃗) = ∇𝑓|𝑎⃗⃗ ⋅ 𝑣⃗ = |∇𝑓)(𝑎⃗)| cosθ
Tangent hyperplane equation by: (1) definition of diff for all (𝑥, 𝑦) close to given diff point (linearization of 𝑓), (2)
property that given vector is orthogonal to the level set 𝑔(𝑥, 𝑦, 𝑧) = 0 at a given point (normal vector for tangent plane).
⃗
⃗⃗
⃗
⃗⃗
⃗⃗)−(𝑑𝑎
|𝑓(𝑎⃗⃗+ℎ )−𝑓(𝑎
⃗⃗⃗ 𝑓)ℎ|
N, M → deriv linear map (𝑑𝑎⃗⃗ 𝑓⃗)𝑣⃗ = lim
= 0, Jacobian matrix (𝑑𝑎⃗⃗ 𝑓⃗)𝑣⃗ =
⃗⃗
|ℎ |
⃗⃗ →0
⃗⃗
ℎ
δ𝑓1
δ𝑥1
⋯
δ𝑓1
δ𝑥𝑛
𝑣1
⋮
⋮ |𝑥⃗→𝑎⃗⃗ [ ⋮ ]
δ𝑓𝑚
𝑣𝑛
⋯
δ𝑥𝑛 ]
⋮
δ𝑓𝑚
[ δ𝑥1
N=1, M → position of particle as a function of time, parametrized curves 𝑥⃗: 𝐼 → ℝ𝑀 : simple (one-to-one), regular (deriv
̇
𝑥⃗ (𝑡)
defined and cont, never ⃗0⃗ (unit tangent vector field 𝑇̂(𝑡) ≔ ̇ (if diff, deriv is orthogonal to it for every interior 𝑡)).
|𝑥⃗ (𝑡)|
N, M → chain rule functions as multiplication of matrices 𝑑𝑎⃗⃗ (𝑓 ∘ 𝑔) = (𝑑𝑔⃗⃗(𝑎⃗⃗) 𝑓⃗)(𝑑𝑎⃗⃗ 𝑔⃗), parametrized curve for 𝑓 and
𝑔: ℝ → ℝ𝑁 diff at 0 with derivative β̇(0) = (𝑑𝑎 𝑓)(𝑔̇ (0)) that takes velocity vec of a curve through a point to the
velocity vec of the image curve through the image point.
𝑖 𝑡ℎ column of the Jacobian matrix representing 𝑑𝑎 𝑓 can be interpreted as the velocity at 𝑓(𝑎)of the image curve under f
of the 𝑖 𝑡ℎ “coordinate line” through 𝑎 (parameterized at constant unit speed in the positive 𝑒𝑖 direction).
PARAMETRIZED SURFACE → map ϕ: 𝐴 ⊆ ℝ2 → ℝ𝑛 (picking 𝐴 and deforming it): regular if each component function is
diff with cont partial deriv at every point (columns of Jacobian matrix independent).
δ𝑥 δ𝑦 δ𝑧
𝛿𝑥 𝛿𝑦 𝛿𝑧
Freezing components and writing ϕ𝑢 = (δ𝑢 , δ𝑢 , δ𝑢) = (𝑑ϕ)𝑒1 , 𝜙𝑣 = (𝛿𝑣 , 𝛿𝑣 , 𝛿𝑣) = (𝑑𝜙)𝑒2 for 𝑎 = (𝑎1 , 𝑎2 ), vectors
ϕ𝑢×ϕ
span tangent plane for ϕ. Unit normal 𝑁(𝑢, 𝑣) = |ϕ𝑢×ϕ𝑣 |.
𝑣
PARAMETRIZATION OF GRAPHS → ϕ: (𝑢, 𝑣) ↦ (𝑢, 𝑣, 𝑓(𝑢, 𝑣)): regular, cross-product ϕ𝑢 × ϕ𝑣 = (−𝑓𝑢 , −𝑓𝑣 , 1), unit
normal 𝑁(𝑢, 𝑣) =
(−𝑓𝑢 ,−𝑓𝑣 ,1)
√1+𝑓𝑢2 +𝑓𝑣2
=
(−𝑓𝑢 ,−𝑓𝑣 ,1)
√1+|∇𝑓|2
.
HIGHER DERIVATIVES → Parametric curve α regular if 𝑑α nonzero and α ∈ 𝐶 1 (𝐼), parametric surface σ regular if 𝑑σ has
δ𝑓
δ𝑓
rank 2 and σ ∈ 𝐶 1 (𝐴). Clairaut’s/Schwarz’s 𝑓: 𝐴 ⊆ ℝ𝑛 → ℝ partial deriv continuous: δ𝑥 δ𝑥 = δ𝑥 δ𝑥 (order of deriv not
𝑖
important).
𝑗
𝑗
𝑖
Second deriv given by Hessian: (𝐷 2 𝑓)𝑏 = 𝑑𝑏 ∇𝑓 = [
𝑓𝑥1 𝑥1 (𝑏) ⋯ 𝑓𝑥1 𝑥𝑛 (𝑏)
⋮
⋮
⋮
]
(𝑏)
(𝑏)
𝑓𝑥𝑛 𝑥1
⋯ 𝑓𝑥𝑛 𝑥𝑛
TAYLOR EXPANSION FOR 𝑪𝟐
1
𝑓(𝑎 + ℎ) = 𝑓(𝑎) + (𝑑𝑎 𝑓)ℎ + ℎ𝑇 (𝐷 2 𝑓)𝑎 ℎ + 𝑜(|ℎ|2 )
2
ℎ1
1
= 𝑓(𝑎) + [𝑓𝑥1 (𝑎) ⋯ 𝑓𝑥𝑛 (𝑎)] [ ⋮ ] + [ℎ1
2
ℎ𝑛
⋯
𝑓𝑥1 𝑥1 (𝑎) ⋯ 𝑓𝑥1 𝑥𝑛 (𝑎) ℎ1
ℎ𝑛 ] [
⋮
⋮
⋮
] [ ⋮ ] + 𝑜(|ℎ|2 )
𝑓𝑥𝑛 𝑥1 (𝑎) ⋯ 𝑓𝑥𝑛 𝑥𝑛 (𝑎) ℎ𝑛
GLOBAL EXTREMIZERS (N, M=1) → interior by Fermat’s: 𝑓 not differentiable at 𝑎 or ∇𝑓|𝑎 = (0, … ,0), exterior by
Lagrange: 𝑔(𝑥1 , … , 𝑥𝑛 ) = 𝐶 → 𝑓 and 𝑔 diff at 𝑎, (∇𝑔|𝑎 ≠ 0, … ,0), 𝑎 is extremizer of 𝑓 with constraint 𝑔 then there
exists ∇𝑓|𝑎 = λ∇𝑔|𝑎 . Finding min and max: (1) interior (2) exterior by Lagrange or parametrizing the boundary (3)
compare.
DIAGONALIZATION → any 𝑛 × 𝑛 matrix can be diagonalized such that 𝐵𝑣𝑖 = λ𝑖 𝑣𝑖 , giving 𝑃 ≔ [𝑣1
λ1 0 0
𝑃−1 𝐵𝑃 = 𝑃𝑇 𝐵𝑃 = [ 0 λ2 0 ].
0 0 λ𝑛
⋯ 𝑣𝑛 ] and
SIGN OF SYMMETRIC MATRIX → positive semidefinite ℎ𝑇 𝐵ℎ ≥ 0 positive definite ℎ𝑇 𝐵ℎ > 0, negative semidefinite
ℎ𝑇 𝐵ℎ ≤ 0, negative definite ℎ𝑇 𝐵ℎ < 0, indefinite (1 strictly positive and 1 strictly negative eigenvalue), degenerate (0
an eigenvalue).
LOCAL EXTREMIZER (N, M=1) → necessary: if 𝑎 is a local minimizer of 𝑢, then (𝐷^2𝑢)|_𝑎 is positive semidefinite, If 𝑎 is a
local maximizer of 𝑢, then (𝐷^2𝑢)|_𝑎 is negative semidefinite; sufficient:
𝑎
Saddle point
Local minimum
Local maximum
(𝐷 2 𝑢)|𝑎
Indefinite
Positive definite
Negative definite
det(𝐷 2 𝑢)|𝑎
+
+
𝑢𝑥𝑥 (𝑎)
+
-
IMPLICIT FUNCTION THEOREM → 𝐹(𝑥1 , … , 𝑥𝑛 , 𝑦) ∈ 𝐶 1 : ℝ𝑛+1 → ℝ, 𝑎 = (𝑎1 , … , 𝑎𝑛 ) ∈ ℝ𝑛 , 𝑦0 ∈ ℝ s.t 𝐹(𝑎, 𝑦0 ) =
0, 𝐹𝑦 (𝑎, 𝑦0 ) ≠ 0 ⇒ connected open set 𝑈 ⊆ ℝ𝑛 with 𝑎 ∈ 𝑈, open interval 𝐼 ⊆ ℝ with 𝑦0 ∈ 𝐼, unique 𝐶 1 function 𝑔: 𝑈 →
ℝ such that 𝐹(𝑥, 𝑔(𝑥)) = 0 ∀𝑥 ∈ 𝑈;
Every (𝑥, 𝑦) ∈ ℝ𝑛+1 satisfying 𝑥 ∈ 𝑈, 𝑦 ∈ 𝐼 and 𝑓(𝑥, 𝑦) = 0 also satisfies 𝑦 = 𝑓(𝑥).
δ𝐹
δ𝑔
δ𝑥𝑖
| =−
|
δ𝐹 (𝑥,𝑔(𝑥))
δ𝑥𝑖 𝑥
δ𝑦
∀𝑥 ∈ 𝑈, 𝑖 ∈ {1, … , 𝑛}
𝐹(𝑥, 𝑔(𝑥)) = 0
𝛿
𝛿
(𝐹(𝑥, 𝑔(𝑥)) =
(0)
𝛿𝑥𝑖
𝛿𝑥𝑖
δ𝐹 δ𝐹 δ𝑔
+
=0
δ𝑥𝑖 δ𝑦 δ𝑥𝑖