Analysis II EPFL

148 flashcards · Mathematics · by Athena Papkouf

scalar product
⟨x,y⟩=∑k=1nxkyk\langle\mathbf{x}, \mathbf{y}\rangle=\sum_{k=1}^{n} x_{k} y_{k}⟨x,y⟩=∑k=1nxkyk
Properties of scalar product
Positive-definiteness: ⟨x,x⟩≥0 for all x and ⟨x,x⟩=0⇔x=0\text { Positive-definiteness: }\langle\mathbf{x}, \mathbf{x}\rangle \geq 0 \text { for all } \mathbf{x} \text { and }\langle\mathbf{x}, \mathbf{x}\rangle=0 \Leftrightarrow \mathbf{x}=0 Positive-definiteness: ⟨x,x⟩≥0 for all x and ⟨x,x⟩=0⇔x=0 Symmetry: ⟨x,y⟩=⟨y,x⟩\text { Symmetry: }\langle\mathbf{x}, \mathbf{y}\rangle=\langle\mathbf{y}, \mathbf{x}\rangle Symmetry: ⟨x,y⟩=⟨y,x⟩ Linear in each argument: ⟨αx+βy,z⟩=α⟨x,z⟩+β⟨y,z⟩ for all x,y,z∈R…
dim⁡Rn\operatorname{dim} \mathbb{R}^{n}dimRn
=n<∞=n<\infty=n<∞
norm and normed space
Let EEE be a real vector space. A function ∥⋅∥:E⟶R+\|\cdot\|: E \longrightarrow \mathbb{R}_{+}∥⋅∥:E⟶R+ is called a norm on EEE if ∥⋅∥\|\cdot\|∥⋅∥ verifies the three following properties: 1. Positive-definiteness: ∥x∥≥0\|\mathbf{x}\| \geq 0∥x∥≥0 for all x∈E\mathbf{x} \in Ex∈E and ∥x∥=0⇔x=0\|\mathbf{x}\|=0 \Leftrightarrow \mathbf{x}=0∥x∥=0⇔x=0 2. Homogeneity: ∥λ⋅x∥=∣λ∣⋅∥x∥\|\lambda \cdot \mathbf{x}\|=|\lambda| \cdot\|\mathbf{x}\|∥λ⋅x∥=∣λ∣⋅∥x∥ for all λ∈R\lambda \in \mathbb{R}λ∈R and x∈E\mathbf{x…
Euclidean norm on Rn
. The function ∥⋅∥2:Rn⟶R defined by \text { . The function }\|\cdot\|_{2}: \mathbb{R}^{n} \longrightarrow \mathbb{R} \text { defined by } . The function ∥⋅∥2:Rn⟶R defined by ∥x∥2=⟨x,x⟩=(∑k=1nxk2)12\|\mathbf{x}\|_{2}=\sqrt{\langle\mathbf{x}, \mathbf{x}\rangle}=\left(\sum_{k=1}^{n} x_{k}^{2}\right)^{\frac{1}{2}}∥x∥2=⟨x,x⟩ =(∑k=1nxk2)21 is a norm in Rn
Cauchy-Schwarz inequality
⟨x,y⟩2≤⟨x,x⟩⟨y,y⟩=∣∣x∣∣2∣∣y∥2\langle\mathbf{x}, \mathbf{y}\rangle^{2} \leq\langle\mathbf{x}, \mathbf{x}\rangle\langle\mathbf{y}, \mathbf{y}\rangle=|| \mathbf{x}||^{2}|| \mathbf{y} \|^{2}⟨x,y⟩2≤⟨x,x⟩⟨y,y⟩=∣∣x∣∣2∣∣y∥2
Euclidean distance on Rn
d(x,y):=∥x−y∥d(\mathbf{x}, \mathbf{y}):=\|\mathbf{x}-\mathbf{y}\|d(x,y):=∥x−y∥
Properties of Euclidean distance
1. Positive-definiteness: 0"> d(x,y)>0d(x, y)>0d(x,y)>0 for all x,y∈M\mathbf{x,y}\in Mx,y∈M d(x,y)=0 ⟺ x=yd (\mathbf{x,y})=0 \iff \mathbf{x}=\mathbf{y}d(x,y)=0⟺x=y 2. Symmetry: d(x,y)=d(y,x)d(\mathbf{x}, \mathbf{y})=d(\mathbf{y}, \mathbf{x})d(x,y)=d(y,x) 3. Triangle inequality: d(x,y)≤d(x,z)+d(z,y)d(\mathbf{x}, \mathbf{y}) \leq d(\mathbf{x}, \mathbf{z})+d(\mathbf{z}, \mathbf{y})d(x,y)≤d(x,z)+d(z,y)
Scalar product (geometric interpretation)
⟨x,y⟩=∥x∥2∥y∥2cos⁡θ\langle\mathbf{x}, \mathbf{y}\rangle=\|\mathbf{x}\|_{2}\|\mathbf{y}\|_{2} \cos \theta⟨x,y⟩=∥x∥2∥y∥2cosθ
open ball with centre a and radius r
Let a∈Rn\mathbf{a} \in \mathbb{R}^{n}a∈Rn and 0"> r>0r>0r>0 . B(a,r)={x∈Rn:d(x,a)<r}B(\mathbf{a}, r)=\left\{\mathbf{x} \in \mathbb{R}^{n}: d(\mathbf{x}, \mathbf{a})<r\right\}B(a,r)={x∈Rn:d(x,a)<r}
Open subset
A subset S⊂RnS \subset \R^nS⊂Rn is open if ∀x∈S\forall x \in S∀x∈S there exists a neigh- borhood B(x,ϵ)B(\mathbf{x}, \epsilon)B(x,ϵ) with 0"> ϵ>0\epsilon > 0ϵ>0 such that B(x,ϵ)B(\mathbf{x}, \epsilon)B(x,ϵ) ⊂S\subset S⊂S . The empty set ∅\emptyset∅ and the space Rn\R^nRn are open. Any open ball B B(a,r)B(\mathbf{a}, r)B(a,r) is an open space. Any union of open sets is open. Any finite intersection of open sets is open. SSS is open  ⟺ \iff⟺ S=S∘S=\stackrel{\circ}{S}S=…
Closed subset
A subset S⊂RnS \subset \R^nS⊂Rn is closed if Rn\R^nRn \ SSS is open. The empty set ∅\emptyset∅ and the space Rn\R^nRn are closed (and open). SSS is closed  ⟺ S=Sˉ\iff S=\bar{S}⟺S=Sˉ
Interior of a set
Let S⊂Rn S \subset \R^n S⊂Rn and a∈Sa \in Sa∈S . We say aaa is in the interior of SSS if there exists a neighborhood B(a,ϵ)B(\mathbf{a}, \epsilon)B(a,ϵ) with 0"> ϵ>0\epsilon > 0ϵ>0 such that B(a,ϵ) B(\mathbf{a}, \epsilon) B(a,ϵ) ⊂S\subset S⊂S . The set of all interior points of SSS is called the interior of SSS and written S˚\mathring{S}S˚ .
boundary of set
A point a∈Rna \in \R^na∈Rn is called a boundary point of SSS if any neighborhood B(a,ϵ) B(\mathbf{a}, \epsilon) B(a,ϵ) contains points of SSS and points of RnR^nRn \ SSS . The set of all boundary points of SSS is called the boundary of SSS and written ∂S\partial S∂S .
closure of set
Let S⊂RnS \subset \R^nS⊂Rn and a∈Rna \in \R^na∈Rn . We say aaa is a point of closure of SSS if for any neighbourhood B(a,ϵ)B(\mathbf{a}, \epsilon)B(a,ϵ) : B(a,ϵ)∩S≠0B(\mathbf{a}, \epsilon) \cap S \neq 0B(a,ϵ)∩S̸=0 The set of all points of closure of S is called the closure of S and written S̄ Sˉ=S∘∪∂S\bar{S}=\stackrel{\circ}{S} \cup \partial SSˉ=S∘∪∂S S⊂S⊂SˉS \subset S \subset \bar{S}S⊂S⊂Sˉ
Sequence
A sequence of elements of Rn\R^nRn is a function f:N→Rnf : \N \to \R^nf:N→Rn , which associates an element xk=f(k)∈Rn\mathbf{x}_k = f(k) \in \R^nxk=f(k)∈Rn for every k∈Nk \in \Nk∈N . Sequences are noted (xk)k∈N (\mathbf{x_k})_{k\in\N} (xk)k∈N
Convergent sequence
A sequence (xk)k∈N (\mathbf{x_k})_{k\in\N} (xk)k∈N converges toward x∈Rnx \in \R^nx∈Rn , if for every 0"> ϵ>0\epsilon > 0ϵ>0 , we can associate an integer Nϵ\N_{\epsilon}Nϵ such that for every k≥Nϵk\geq \N_{\epsilon}k≥Nϵ on a d2(xk,x)<ϵd_2(\mathbf{x_k, x}) < \epsilond2(xk,x)<ϵ We then write lim⁡k→+∞xk=x\lim _{k \rightarrow+\infty} x_{k}=xlimk→+∞xk=x  ⟺ lim⁡k→+∞d2(xk,x)=0\iff \lim _{k \rightarrow+\infty} d_{2}\left(x_{k}, x\right)=0⟺limk→+∞d2(xk,x)=0
Cauchy sequences
A sequence (xk)k∈N (\mathbf{x_k})_{k\in\N} (xk)k∈N is a Cauchy sequence if for every 0"> ϵ>0 \epsilon > 0 ϵ>0 , we can associate an N= Nϵ∈NN = \N_{\epsilon} \in \NN= Nϵ∈N such that k,l≥Nk, l \geq Nk,l≥N implies d2(xk,xl)<ϵd_2(\mathbf{x_k, x_l}) < \epsilond2(xk,xl)<ϵ . Any convergent sequence is a Cauchy sequence.
Normed complete space
A normed space (E, || · ||) is complete if any Cauchy sequence converges relatively to this metric. A normed complete space is called a Banach space. Any Cauchy sequence in Rn\R^nRn converges. Therefore the normed space ( Rn\R^nRn , ∣∣⋅∣∣2||\cdot ||_2∣∣⋅∣∣2 ) is complete.
Bounded sequence
A sequence (xk)k∈N (\mathbf{x_k})_{k\in\N} (xk)k∈N is bounded in ( Rn\R^nRn , ∣∣⋅∣∣2||\cdot ||_2∣∣⋅∣∣2 ) if there exists a constant 0"> C>0C > 0C>0 such that ∣∣xk∣∣2≤C ∀k∈N||x_k ||_2 \leq C ~\forall k\in\N∣∣xk∣∣2≤C ∀k∈N
Bolzano-Weierstrass theorem in Rn\R^nRn
Each bounded sequence (xk)k∈N (\mathbf{x_k})_{k\in\N} (xk)k∈N in Rn\R^nRn has a convergent subsequence (xkj)j∈N (\mathbf{x_{k_j}})_{j\in\N} (xkj)j∈N
boundary definition from QCM1.1
Convergence of sequences in Rn
A sequence (xk)k∈N(x_k)_{k\in N}(xk)k∈N converges in the normed space (Rn,∣∣⋅∣∣2)(R^n , || · ||_2 )(Rn,∣∣⋅∣∣2) if and only if the n numerical sequences (x1,k)k,...,(xn,k)k(x_{1,k})_k , . . . , (x_{n,k})_k(x1,k)k,...,(xn,k)k converge. The following theorem follows: Any Cauchy sequence in R n converges. Therefore the normed space is complete.
Sets and sequences in Rn
Continuous functions
uniformly continuous functions
If for each ϵ\epsilonϵ , we can choose a δ that works for each x ∈ A, then the function f is uniformly continuous. If a function f is continuous on A, a closed bounded set, then automatically f is uniformly continuous. If f is continuous but A is not closed or not bounded then maybe f is uniformly continuous, maybe not. Trick: If f is uniformly continuous on B and A ⊂ B, then f is uniformly continuous on A.
Max and min of continuous functions
Banach fixed point theorem
Equivalent norms
Theorem about equivalent norms
Space of continuous functions
Continuity of multiplication and division of functions
continuity of linear functions
A bound on linear functions
curve definition
differentiability of curves
curve derivative
smoothness of curves
Intersection of smooth curves
segment of curve
Length of a curve segment
Rectifiable curve
arc-length of curve
The length of the arc does not depend on the parametrization that has been chosen.
inequality for arc length of curve
curvature of curve r(t)
graph of curve
derivative of graph of curve
length of graph of curve
graph of a function
level set c of f
limit of a function
Continuous function
Given a continuous function f defined on a domain D a function g is a continuous extension of f if
Parametrized version of functions
Partial derivative
gradient of f
partial derivatives and continuity
differentiability of functions
sufficient and necessary conditions for function differentiability
Directional derivative
first order approximation of a function
Functions of class C1
Mean value theorem
The continuity of the partial derivatives at a point a implies the continuity of the function at a. It follows that any function of class C1 (U) is differentiable (and therefore continuous) at each a ∈ U
Equation of a hyperplane
Tangent hyperplane
equation of tangent hyperplane
derivative of composition of functions with curves
derivative of composition of function with curve where
derivative of composition of functions
function of class Cm
partial derivatives of higher order
Hessian matrix
number of partial derivatives of order m
Laplacian of f
second order approximation
Jacobian matrix
Jacobian iff
partial derivatives of a vector field and Jacobian matrix
first order approximation of vector field
composition rule for vector fields
divergence of the field v
Invertibility of vector fields
Inversibility - necessary condition
Inverse function theorem
Linear functions invertibility
necessary and sufficient condition so that the matrix A is invertible
detA≠0det A \neq 0detA=0
invertibility of 2x 2 matrix
Implicit function theorem
Generalization of the implicit function theorem
Derivatives of higher order - implicit function theorem