Title: FS26 / Analysis II / Vorlesung
Tags: Definitions Theorems Lemmas Definition Propositions Corollary

Q: Metric Space
A: A metric space is a pair \((X,d)\), where \(X\) is a set and \(d:X\times X\to [0,\infty)\) is a function such that for all \(x,y,z\in X\) we have
- \(d(x,y)=0\) if and only if \(x=y\),
- \(d(x,y)=d(y,x)\), and
- \(d(x,z)\le d(x,y)+d(y,z)\).

Q: Triangle Inequality
A: For all \(x,y,z\in \mathbb{R}^{n}\) we have
    \[\|z-x\|\le \|z-y\|+\|y-x\|\]

Q: Cauchy-Schwarz Inequality
A: For all \(x,y\in \mathbb{R}^{n}\) we have\[x\cdot y \le \|x\|\|y\|\]

Q: Sequence
A: Given \(X\) we call a map \(x: \mathbb N\to X\) a sequence.

Q: Convergent Sequence
A: \((x_{n})_{n\ge 0}\) has limit \(x\in X\) if \(d(x_n,x)\to 0\).

Q: Subsequence
A: Given a sequence \((x_n)\) we define a subsequence as \((x_{f(k)})_{k\ge0} \) where \(f:\mathbb N\to\mathbb N\) is a strictly increasing function.

Q: Cauchy Sequence
A: A sequence is Cauchy if \(\forall \varepsilon > 0 \exists N\) such that \(d(x_n,x_m)<\varepsilon\) whenever \(n,m\ge N\).

Q: Complete Metric Space
A: Every Cauchy Sequence in \(X\) converges.

Q: Open Ball
A: Define the open ball of radius \(r>0\) centered at \(x\in X\) as
    \[B_r(x) = B(x,r) \coloneqq \{y\in X: d(x,y)<r\} .\]

Q: Interior
A: Given \(\Omega \subset X\), we define \(\mathrm{int}\Omega = \Omega^\circ = \bigcup\{U\subset \Omega \mid U\text{ is open }\}\) as interior of \(\Omega\),

Q: Closure
A: Given \(\Omega\subset X\), we define \(\overline{\Omega} = \{x\in X \mid \exists (x_n)_{n\ge 0}\subset \Omega, x_n\to x\}\) as the closure of \(\Omega\)

Q: Boundary
A: Given \(\Omega\subset X\), we define \(\partial \Omega = \overline{\Omega}\setminus \Omega^\circ\) as the boundary of \(\Omega\).

Q: Sequentially Open
A: Whenever \((x_n)\subset X\) converges to \(x\in U\), then \(x_n\in U\) eventually

Q: Sequentially Closed
A: Whenever \((x_n)\subset A\) converges to \(x\), then \(x\in A\).

Q: Epsilon-Delta Continuity
A: \(\forall x\in X, \forall\varepsilon>0, \exists \delta>0\) such that:
            \[\forall x'\in X, d_X(x,x')<\delta \implies d_Y(f(x),f(x'))<\varepsilon.\]

Q: Sequential Continuity
A: For all \((x_n)\subset X\), \[x_n\to x\implies f(x_n)\to f(x)\]

Q: Topological Continuity
A: \[\forall U\subset Y \text{ open } f^{-1}(U) \text{ is open}\]

Q: Uniform Continuity
A: \[\forall\varepsilon>0\exists \delta>0:d_X(x,x')<\delta\implies d_Y(f(x),f(x'))<\varepsilon\]

Q: Lipshitz Continuity
A: \[\exists L>0:d_Y(f(x),f(x'))\le L d_X(x,x') \forall x,x'\in X\]

Q: Banach Fixed Point Theorem
A: Let \((X,d)\) be a complete metric space and \(T:X\to X, \lambda\)-Lipschitz
    with \(\lambda\in(0,1)\). Then, \(T\) has a unique fixed point

Q: Cover
A: Given \(X\) and \(E\subset X\) we say that \(\mathcal U = \{U_i, i\in I\}\) is a cover of \(E\) if \(E\subset \bigcup \mathcal U = \bigcup_{i\in I} U_i\).

Q: Sequential Compactness
A: \[\forall (x_n)\subset K, \exists (x_{n_k}) \text{ convergent}\]

Q: Topological Compactness
A: For every open cover \(u\) of \(K\) there exists a finite subcover \(V\subset u\) of \(K\)

Q: Heine-Borel
A: \(K\subset \mathbb{R}^n\) is compact iff \(K\) is closed and bounded

Q: Disconnected Set
A: \(A\subset X\) (nonempty) is disconnected if \(\exists U,V\) open, disjoint such that \(A\subset U\cup V\) and \(A\cap U, A\cap V\) are non-empty

Q: Connected Set
A: \(A\) is connected if it is not disconnected

Q: Connected subsets of \(\mathbb R^n\) are
A: intervals

Q: Given \(f\) continuous, \(E\) connected, then \(f(E)\)
A: is connected too.

Q: Intermediate Value Theorem (Metric Space)
A: Given \(f:X\to\mathbb R\) continuous with \(f(x)=a\le b = f(y)\) then \(\exists t\in X\) such that \(f(t)=c\) for all \(c\in[a,b]\)

Q: Curve
A: \[\gamma:[0,1]\to X \text{ continuous}\]

Q: Normed Vector Space over \(\mathbb{R}\)
A: Let \(V\) be a VS over \(\mathbb{R}\). A map \(\|\cdot\|:V\to [0,\infty)\) is called a norm if \(\forall v,u\in V\) and \(\alpha\in \mathbb{R}\), the following hold:
1. Definite: \(\|v\|=0\) if and only if \(v=0\).
2. Homogeneous: \(\|\alpha v\| = |\alpha|\cdot \|v\|\)
3. Triangle inequality: \(\|v+u\|\le \|v\| + \|u\|\).

Q: Hilbert-Schmidt Norm
A: \[\|M\|_2 = \sqrt{\mathrm{tr}(M^TM)}\]

Q: Hilbert Schmidt Norm Bound
A: \[|Mx| \le \|M\|_2 \cdot |x| \,\forall x\in\mathbb{R}^n\]

Q: Inner Product Space
A: Let \(V\) be a VS over \(\mathbb{R}\). A inner product is a map \(\langle\cdot,\cdot\rangle:V\times V\to \mathbb{R}\) such that \(\forall u,v,w\in V\) and \(\alpha\in \mathbb{R}\):
1. Symmetry: \(\langle u,v\rangle = \langle v,u\rangle\).
2. Bilinearity: \(\langle \alpha u + \beta v, w\rangle = \alpha \langle u,w\rangle + \beta \langle v,w\rangle\).
3. Definite: \(\langle v,v\rangle \ge 0\) and \(\langle v,v\rangle = 0\iff v=0\).

Q: Differential
A: Linear map \(L, Df_{x_0}=Df(x_0)\) such that\[\lim_{x\to 0}\frac{f(x_0+x)-f(x_0)-L(x)}{|x|}=0\]

Q: Directional Derivative
A: Let \(U\subset \mathbb{R}^{n}\) open, \(f:U\to \mathbb{R}^m\) and \(x_0\in U\). Given \(v\in\mathbb{R}^{n}\), we define the directional derivative of \(f\) at \(x_0\) along \(v\) as    \[\partial_v f(x_0) = \lim_{t \to 0} \frac{f(x_0+tv)-f(x_0)}{t},\]provided the limit exists.

Q: Partial Derivatives
A: Let \(U\subset \mathbb{R}^{n}\) open, \(f:U\to \mathbb{R}^m\) and \(x_0\in U\). Given \(i\in \{1,\dots,n\}\), we define the partial derivative of \(f\) at \(x_0\) along the \(i\)-th coordinate as\[\partial_{e_i} f(x_0) \eqqcolon \frac{\partial f}{\partial x_i}(x_0) = \partial_i f(x_0)\]

Q: Sufficient Condition for differentiability
A: Let \(U\subset \mathbb{R}^{n}\) open, \(f:U\to \mathbb{R}^m\). If \(\partial_i f(x)\) exists and is continuous for every \(i=1,\dots,n\), and every \(x\in U\), then \(f\) is differentiable at every \(x\in U\). Moreover,\[Df_{x_0}(x) = \begin{pmatrix} \partial_1 f(x_0),\dots,\partial_n f(x_0) \end{pmatrix} x.\]

Q: Jacobi Matrix
A: Given \(U\subset \mathbb{R}^{n}\) open, \(f:U\to \mathbb{R}^m\), differentiable at \(x_0\). We define the Jacobi matrix of \(f\) at \(x_0\) as  \[Df(x_0)=Jf(x_0) = (\partial_1f(x_0),\dots,\partial_n f(x_0)).\]

Q: Chain Rule
A: Given \(U\subset \mathbb{R}^{n}\) open, \(V\subset \mathbb{R}^{m}\) open and\(f:U\to V\) and \(g:V\to \mathbb{R}^k\). Assume that \(f\) is differentiable at \(x_0\in U\) and \(g\) is differentiable at \(f(x_0)\in V\). Then, \(g\circ f: U\to \mathbb{R}^k\) is differentiable at \(x_0\), and    \[D(g\circ f)_{x_0} = Dg_{f(x_0)} \circ Df_{x_0}.\]

Q: Generalized Mean Value Theorem
A: Given \(U\subset \mathbb{R}^{n}\) open, \(f\in C^1(U)\). Assume \(x_0\in U\) and \(h\in \mathbb{R}^{n}\) such that \(x_0 + th \in U\, \forall t\in [0,1]\). Then \(f(x_0+h) - f(x_0) = Df_{x_0+\theta h}(h)\) for some \(\theta\in [0,1]\).

Q: Convex Set
A: \(A\subset \mathbb{R}^{n}\) is called convex if \(\forall x,y\in A\) and \(\forall t\in [0,1]\), we have \(tx + (1-t)y \in A\).

Q: Gradient
A: Given \(U\subset \mathbb{R}^{n}\) open, \(f\in C^1(U)\). Define the gradient of \(f\) at \(x\) as the vector\[\nabla f(x) = Jf(x)^T = \begin{pmatrix} \partial_1 f(x)\\ \vdots\\ \partial_n f(x) \end{pmatrix}.\]

Q: Differentiability vs Lipschitz
A: Given \(U\subset \mathbb{R}^n\) open, \(f\in C^1(U,\mathbb{R}^m)\). Suppose, that \[\Lambda \coloneqq \sup_{x\in U} \|Jf(x)\|_{\text{2}} < \infty.\] Then, \(f\) is Lipschitz with constant \(\sqrt{m}\Lambda\).

Q: Schwarz's Theorem
A: Given \(U\subset \mathbb{R}^n\) open, \(f\in C^2(U,\mathbb{R}^m)\). Then \(\forall i,j\in \{1,\dots,n\}\), we have \(\partial_i\partial_j f = \partial_j\partial_i f\).

Q: Multi-Index
A: A multi-index is a tuple \(\alpha = (\alpha_1,\dots,\alpha_n)\subset \mathbb{N}^n\). We define \(|\alpha| = \alpha_1 + \dots + \alpha_n\) as the length of \(\alpha\). We say that \(\beta \le \alpha\) if \(\beta_i \le \alpha_i\) for all \(i=1,\dots,n\). We define the factorial \[\alpha! = \alpha_1! \cdots \alpha_n!.\]

Q: Taylor's Theorem
A: Given \(U\subset \mathbb{R}^n\) open, \(f\in C^{k+1}(U), k\ge 0\). Let \(x_0\in U\)
    and \(h\in \mathbb{R}^{n}\) such that \(x_0 + th \in U\, \forall t\in [0,1]\). Then
    \[f(x_0+h) = \sum_{|\alpha|\le k} \partial^\alpha f(x_0) \frac{h^\alpha}{\alpha!} + R_{k+1}f(x_0,h).\]
    Where the remainder term \(R_{k+1}f(x_0,h)\) is given by
    \[R_{k+1} = \int_0^1 (k+1)(1-t)^k \sum_{|\alpha|=k+1} \partial^\alpha f(x_0 + th) \frac{h^\alpha}{\alpha!} dt = O(h^{k+1}) \text{ as } h\to 0.\]

Q: Real analytic function
A: Given \(U\subset \mathbb{R}^n\) open, we say that \(f\in C^\infty(U)\) satisfies analytic estimates in \(U\) if for every \(x_0\in U\), there exists \(\rho>0\) and \(C>0\) such that \[\max_{\overline{B_\rho(x_0)}} |\partial^\alpha f| \le C|\alpha|! (n\rho)^{-|\alpha|} \text{ for all } \alpha\in \mathbb{N}^n.\]

Q: Analytic Expansion
A: Given \(U\subset \mathbb{R}^n\) open, \(f: U \to \mathbb{R}\), analytic.
Given \(x_0\in U\), let \(\rho\) and \(C\) be as in definition. Then, let \[P_{x_0, k}(h) \coloneqq \sum_{|\alpha|=1}^k \partial^\alpha f(x_0) \frac{h^\alpha}{\alpha!}.\]
    Then for all \(r\in (0,\rho)\), the power series converges absolutely, i.e. \(\forall \) \(k<l\) \[\sup_{h\in B_r(0)} \sum_{|\alpha|=k}^l \left| \partial^\alpha f(x_0) \frac{h^\alpha}{\alpha!} \right| \le \frac{C(\frac{r}{\rho})^k}{1-(\frac{r}{\rho})}.\]
    The series converges for \(h\in \overline{B_r(0)}\) and we have
    \[\sup_{h\in B_r(0)} \left| P_{x_0,k}(h)-P_{x_0}(h) \right| \le \frac{C(\frac{r}{\rho})^k}{1-(\frac{r}{\rho})}.\]
    Moreover,
    \[f(x_0+h) = P_{x_0, \infty}(h) \quad \forall h\in B_\rho(0).\]

Q: Unique Continuation Principle
A: Given \(U\subset \mathbb{R}^n\) open and connected, \(f,g\) analytic in \(U\) such that \(\exists x_0\in U\) with \[\partial^\alpha f(x_0)=\partial^\alpha g(x_0)\quad\forall\alpha\in\mathbb{N}^n.\] Then \(f=g\) on \(U\).
In particular, if \(V\subset U\) is an open subset such that \(f=g\) on \(V\), then \(f=g\) on \(U\).

Q: Spectral Theorem
A: If \(A\) is \(n\times n\) symmetric \((A^T = A)\), with coefficients in \(\mathbb{R}\), then \(A\) diagonalizes in some orthonormal basis and has real eigenvalues.
Also, for \(O = (v_1|\dots|v_n)\), where \(v_1,\dots,v_n\) is an orthonormal basis of eigenvectors of \(A\), we have \[O^TO = I_n, O^{-1} = O^T, O^{-1}AO = \Lambda = \begin{pmatrix}\lambda_1 & \dots \\\vdots & \ddots \\\end{pmatrix}.\]

Q: Local Minimum / Maximum
A: Given \(U\subset \mathbb{R}^n\) open, \(f\in C^1(U)\). We say that \(x_0\) is a
local minimum of \(f\) if \(\exists r>0\) such that \[f(x_0)\le f(x) \text{ for all } x\in B_r(x_0).\]
local maximum of \(f\) if \(\exists r>0\) such that \[f(x_0)\ge f(x) \text{ for all } x\in B_r(x_0).\]
strict local minimum of \(f\) if \(\exists r>0\) such that \[f(x_0) < f(x) \text{ for all } x\in B_r(x_0)\setminus \{x_0\}.\]

Q: Critical Point
A: Given \(U\subset \mathbb{R}^n\) open, \(f\in C^1(U)\). Then \(x_0\in U\) is a critical point of \(f\) if \(\nabla f(x_0) = 0\).

Q: Lagrange Multipliers
A: Given \(U\subset \mathbb{R}^n\) open, \(f,g_j \in C^1(U)\) for \(j=1,\dots,m\). Let \(M=\{x\in U \mid g_1(x)=\dots=g_m(x)=0\}\) be the constraint set. If \(x_0\) is a local minimum of \(f|_M\), then \(\exists \lambda_*, \lambda_1,\dots,\lambda_m\in \mathbb{R}\), with \(\lambda_*^2 + \lambda_1^2 + \dots + \lambda_m^2 = 1\), such that \[\lambda_* \nabla f(x_0) + \sum_{j=1}^m \lambda_j \nabla g_j(x_0) = 0.\]\(\lambda_*, \lambda_1,\dots,\lambda_m\) are called the Lagrange multipliers.

Q: Hessian
A: Given \(U\subset \mathbb{R}^n\) open, \(f: U \to \mathbb{R} \in C^2\). The Hessian Matrix of \(f\) at \(x\in U\) is the \(n\times n\) matrix defined by \[H_{ij}f(x) = \partial_i\partial_j f(x).\] By Schwarz's theorem, the Hessian is a symmetric matrix. It is also denoted \(D^2 f(x)\). The Laplacian of \(f\) is the trace of the Hessian:    \[\Delta f(x) = \mathrm{tr}(Hf(x)) = \sum_{i=1}^n \partial_{ii} f(x).\]

Q: non-negative definite
A: \(A\in M_{n\times n}(\mathbb{R})\) is non-negative definite if \(\forall x\in \mathbb{R}^n, x\cdot Ax \ge 0\).

Q: Local Minimum and Hessian
A: Given \(U\subset \mathbb{R}^n\) open, \(f\in C^3(U)\). If \(f\) has a local minimum at \(x_0\in U\), then \(Hf(x_0)\) is non-negative definite.

Q: Lipschitz Perturbation
A: Let \(U\subset \mathbb{R}^{n}\) be open, \(F:U\to \mathbb{R}^{n}\) a function of the form \[F(x) = x +\phi(x),\] where \(\phi\) is a \(\lambda\)-Lipschitz function for \(\lambda\in (0,1)\). Then
1) \(\forall x_0\in U\), and \(r>0\) such that \(B_r(x_0)\subset U\), \[B_{(1-\lambda)r}(F(x_0)) \subset F(B_r(x_0)) \subset F(U).\] In particular, \(F(U)\) is open.
2) \(F:U\to F(U)\) is bijective and \(F^{-1}\) is \(\frac{1}{1-\lambda}\)-Lipschitz.

Q: Automatic differentiability of the inverse
A: Let \(U,V\subset \mathbb{R}^{n}\) be open, \(f:U\to V, g:V\to U\) be bijective, such that \(f(g(y)) = y \,\forall y\in V\). Assume, that \(f\) is differentiable at \(x_0\in U\) and \(Df_{x_0}:\mathbb{R}^{n}\to \mathbb{R}^{n}\) is invertible and \(g\) is Lipschitz.
Then, \(g\) is differentiable at \(y_0 = f(x_0)\) and \(Dg_{y_0} = (Df_{x_0})^{-1}\).

Q: Inverse Function Theorem
A: Let \(U\subset \mathbb{R}^{n}\) be open, \(f:U\to \mathbb{R}^{n}\in C^1(U)\). If \(Df_{x_0}\) is invertible for some \(x_0\in U\), then there exists \(U_0\subset U\) open, such that
1. \(f|_{U_0}\) is injective,
2. \(f(U_0)\coloneqq V\) is open,
3. \(g=f|_{U_0}^{-1}:V\to U_0\) is \(C^1\) and \(Dg_{f(x)} = (Df_{x})^{-1}\) for all \(x\in U_0\).
Moreover, if \(f\in C^k(U)\) for some \(k\ge 1\), then \(g\in C^k(V)\).

Q: Diffeomorphism
A: Let \(U,V\subset \mathbb{R}^{n}\) be open. A map \(f:U\to V\in C^{-1}\) is called a diffeomorphism if \(f\) is bijective and \(f^{-1}:V\to U\) is \(C^1\).
If \(f\) and \(f^{-1}\) are \(C^k\), then \(f\) is called a \(C^k\)-diffeomorphism.

Q: Implicit Function Theorem
A: Let \(0<d<n, k\ge 1 \in \mathbb{N}\) and let \(U\subset \mathbb{R}^{n}\) be open, \(f\in C^1(U,\mathbb{R}^{n-d})\). We write a point in \(\mathbb{R}^{n}\times \mathbb{R}^{n-d}\) as \((x,y)\), where \(x\in \mathbb{R}^{d}, y\in \mathbb{R}^{n-d}\). If there exists \((x_0,y_0)\in U\) such that \(f(x_0,y_0) = 0\) and \(J_y f_{(x_0,y_0)}\) is invertible, then for sufficiently small \(r,s>0\) there exists \(g: B_r(x_0)\subset \mathbb{R}^{d} \to B_s(y_0)\subset \mathbb{R}^{n-d}\) such that for all \((x,y)\in B_r(x_0)\times B_s(y_0)\), \(f(x,y) = 0\) iff \(y = g(x)\).
Moreover, \(\forall x\in B_r(x_0)\), \[Jg(x) = -((J_y f)(x, g(x)))^{-1} (J_x f)(x, g(x)).\] Furthermore, if \(f\in C^k(U, \mathbb{R}^{n-d})\) for some \(k\ge 1\), then \(g\in C^k(B_r(x_0), \mathbb{R}^{n-d})\).

Q: Submanifold
A: Given \(0<d<n\) and \(k\ge 1 \in \mathbb{N}\). We say that \(M\subset \mathbb{R}^{n}\) nonempty is a d-dimensional submanifold of \(\mathbb{R}^{n}\) of class \(C^k\) if, for every point \(p_0\in M, \exists U\subset \mathbb{R}^{n}\) open containing \(p_0\), \(V\subset \mathbb{R}^{n}\) open containing 0, and a \(C^k\)-diffeomorphism \(\Psi:U\to V\), such that: \[\Psi(M\cap U) = \{y\in V | y_{d+1} = \dots + y_n = 0\}.\]The map \(\Psi\) is called a submanifold chart.

Q: Implicit Definition of Submanifold
A: \(\forall p\in M\), \(\exists U\subset \mathbb{R}^{n}\) open, such that \(p\in U\) and \(\exists f:U\to \mathbb{R}^{n-d}\in C^k(U)\) such that \(Df_p\) has rank \(n-d\) and \(M\cap U = \{x\in U | f(x) = 0\}\).

Q: Parametrized Definition of Submanifold
A: \(\forall p\in M\), \(\exists V\subset \mathbb{R}^d\) open, \(q\in V\), \(G:V\to \mathbb{R}^{n}\in C^k(V)\) such that \(DG_q\) has rank \(d\), \(G(q) = p\) and \(\forall V'\subset V\) open \(\exists U' \subset \mathbb{R}^{n}\) open with \(M\cap U' = \{G(y) \mid y\in V\}\).

Q: Graphical Definition of Submanifold
A: \(\forall p\in M\), \(\exists U\subset \mathbb{R}^{n}\) open with \(p\in U\), \(V\subset \mathbb{R}^d\) open, g: \(V\to \mathbb{R}^{n-d}\in C^k(V)\) with \[M\cap U = \{x\in \mathbb{R}^{n} \mid (x_{d+1},\dots,x_n) = g(x_1,\dots,x_d)\}.\]

Q: Permutation of Coordinates
A: \(P:\mathbb{R}^{n}\to \mathbb{R}^{n}\) is a permutation of coordinates if \[P(x_1,\dots,x_n) = (x_{\sigma(1)},\dots,x_{\sigma(n)}) \text{ for some } \sigma\in S_n.\]

Q: Tangent Vectors
A: Given \(M\subset \mathbb{R}^{n}\), a \(d\)-dimensional submanifold of \(\mathbb{R}^{n}\), then \(\tau\in \mathbb{R}^{n}\) is tangent to \(M\) at \(p\in M\) if \(\exists p_k\in M, r_k>0\) such that \(\frac{p_k-p}{r_k}\to \tau\), where \(p_k\to p\) and \(r_k\to 0\) as \(k\to \infty\).

Q: Normal Vectors
A: \(\nu\in \mathbb{R}^{n}\) is normal to \(M\) at \(p\in M\) if \(\nu\cdot \tau = 0\) for all \(\tau\in T_pM\), the set of tangent vectors to \(M\) at \(p\).

Q: Connection between Differential and Tangent Vectors
A: \(\tau\in T_pM\) iff \(\tau \in (D\psi)^{-1}_{\psi(p)}(E)\), where \(E = \{y\in \mathbb{R}^{n} \mid y_{d+1} = \dots = y_n = 0\}\).
In particular, \(T_pM \subset \mathbb{R}^{n}\) is a \(d\)-dimensional vector subspace.

Q: Parametrized Surface
A: Let \(V\subset \mathbb{R}^{2}\) be open, \(\phi: V\to \mathbb{R}^{3}\) a \(C^k\) map with \(k\ge 1\) and \(\mathrm{rank} D\phi_y = 2\) for all \(y\in V\) and every \(V'\subset V\) is open, then \(\exists U'\subset \mathbb{R}^{3}\) open such that \[\phi(V) \cap U' = \phi(V').\]

Q: Dyadic Cube
A: Given \(p\in \mathbb{N}\) we say that \(Q\subset \mathbb{R}^{n}\) is a dyadic cube of length \(2^{-p}\) if \(\exists \alpha \in \mathbb{Z}^n\) such that \(Q = 2^{-p} (\alpha + [0,1)^n)\). \(2^{-p}\) is called the pixel size.

Q: Dyadic Subset
A: \(F\subset \mathbb{R}^{n}\) is a dyadic subset if it is a finite union of disjoint dyadic cubes, i.e. \(\exists p\in \mathbb{N}\), \(N\in \mathbb{N}\), and a map \(\alpha:\{1,\dots,N\}\to \mathbb{Z}^n\) injective, such that
    \[F = \bigcup_{i=1}^N 2^{-p}(\alpha(i) + [0,1)^n).\]

Q: Volume of a dyadic subset
A: Let \(F\subset \mathbb{R}^{n}\) be a dyadic set with \(F = \bigcup_{i=1}^N 2^{-p}(\alpha(i) + [0,1)^n)\). Then we define the volume of \(F\) as\[\mu(F) = N\cdot 2^{-pn}.\]

Q: Properties of \(\mu\)
A: \(\mu\) defined on dyadic set satisfies:
1) additive: \(F_1, F_2\) disjoint: \(\mu(F_1\cup F_2) = \mu(F_1) + \mu(F_2)\).
2) normalisation: \(\mu([0,1)^n) = 1\).
3) translation invariance: \(\mu(F+\alpha) = \mu(F)\) for all \(\alpha\in \mathbb{Z}^{n}\).

Q: Outer / Inner Volume
A: Let \(E\subset \mathbb{R}^{n}\). We define the outer volume of \(E\) as\[\mu_\text{out}(E) = \inf \{\mu(f) \mid E\subset G, G \text{ diadic }\}.\] We define the inner volume of \(E\) as\[\mu_\text{in}(E) = \sup \{\mu(F) \mid F\subset E, F \text{ diadic }\}.\]

Q: Jordan Measurable (and the volume)
A: \(E\subset \mathbb{R}^{n}\) bounded is Jordan measurable if we have \(\mu_\text{out}(E) = \mu_\text{in}(E)\).
This number is denoted \(\mu(E) = \mathrm{vol}_n(E)\coloneqq \mu_\text{in}(E)\)and is called the volume of \(E\).

Q: Properties of inner and outer volume
A: 1) \(\mu_\text{out}\) is subadditive, i.e. for \(E\subset \bigcup_{i=1}^N E_i\) we have
\[\mu_\text{out}(E) \le \sum_{i=1}^N \mu_\text{out}(E_i) .\]
2) \(\mu_\text{in}\) is superadditive, i.e. for \(E\supset \bigcup_{i=1}^N E_i\) with \(E_i\) disjoint, we have
\[\mu_\text{in}(E) \ge \sum_{i=1}^N \mu_\text{in}(E_i) .\]

Q: Volume of a box
A: \(E = (a_1,b_1) \times \dots \times (a_n,b_n) \subset \mathbb{R}^{n}\) with \(a_i<b_i\) and \(\overline{E}\) are
Jordan measurable and
\[\mathrm{vol}_n(E) = \mathrm{vol}_n(\overline{E}) = \prod_{i=1}^n (b_i - a_i) .\]

Q: Sandwich Lemma (Jordan Measure)
A: Given \(E\subset \mathbb{R}^{n}\). If \(\forall \varepsilon>0\) there exists\(F,G\) Jordan measurable such that \(F\subset E \subset G\) and\(\mu(G) - \mu(F) < \varepsilon\), then \(E\) is Jordan measurable.

Q: Jordan null set
A: \(E\subset \mathbb{R}^{n}\) is called null if \(\mu_\text{out}(E) = 0\).

Q: Jordan measurability Criterion
A: \(E\subset \mathbb{R}^{n}\) is Jordan measurable iff \(E\) is bounded and \(\partial E\) is null.

Q: Lipschitz map applied to a null set
A: Given \(E\subset \mathbb{R}^{n}\) null and \(f:E\to \mathbb{R}^{n}\) Lipschitz, then \(f(E)\) is null as well.

Q: Volume of a uniformly continuous graph
A: Given \(E\subset \mathbb{R}^{n-1}\) bounded, \(f:E\to \mathbb{R}\) uniformly continuous, then the graph of \(f\) is null.
The graph of \(f\) is defined as \(\Gamma_f = \{(x,f(x)) \mid x\in E\}\).

Q: Translation and Dilatation of Jordan Measurable Sets
A: Given \(E\subset \mathbb{R}^{n}\) Jordan measurable, then \(\rho E + a\) is Jordan measurable and \(\mu(\rho E + a) = \rho^n \mu(E)\).

Q: Volume change of linear maps
A: Given \(L:\mathbb{R}^{n}\to \mathbb{R}^{n}\) linear. If \(E\) is Jordan measurable, then \(L(E)\) is Jordan measurable and\[\mu(L(E)) = |\det(L)| \cdot \mu(E).\]

Q: Indicator Function
A: Let \(A\subset \mathbb{R}^{n}\) be a set. The \vocab{indicator function}
    or \vocab{characteristic function} of \(A\) is defined as
    \[1_A(x) \coloneqq \begin{cases} 1 & x\in A\\ 0 & x\notin A \end{cases}.\]

Q: Dyadic Step Function
A: A function \(g:\mathbb{R}^{n}\to \mathbb{R}\) is a dyadic step function if\[g(x) = \sum_{l=1}^{N} g_l 1_{Q_p(a_l)}(x).\] where \(g_l\in \mathbb{R}\), \(a_l\in 2^{-p}\mathbb{Z}^n\) and \(p\in \mathbb{N}\).

Q: Integral of a step function
A: Let \(g:\mathbb{R}^{n}\to \mathbb{R}\) be a dyadic step function. We define the integral of \(g\) as\[\int g(x) = \sum_{l=1}^N g_l \cdot \mu_n(Q_p(a_l)) = 2^{-pn} \sum_{l=1}^N g_l.\]

Q: Riemann Integrability
A: We say that \(f:A\subset \mathbb{R}^{n}\to \mathbb{R}\) is Riemann integrable over \(A\) if the upper and lower sums defined below coincide.\[\mathcal L_A = \sup \left\{\int g\mid g\le f\cdot 1_A, g \text{ step function}\right\} \]\[\mathcal U_A = \inf \left\{\int g \mid g\ge f\cdot 1_A, g \text{ step function}\right\}.\]

Q: Integral of a constant
A: Given \(A\subset \mathbb{R}^{n}\) Jordan measurable and \(c:A\to \mathbb{R}\) constant, then \(1_A\) is Riemann integrable and \[ \int_A c = c\int_A 1_A = c\mu_n(A).\]

Q: Riemann Integrability is equivalent to:
A: \(f:A\subset \mathbb{R}^{n}\to [0,\infty)\) is Riemann integrable, iff\[\Gamma_f = \{(x, x_{n+1})|x\in A, 0\le x_{n+1}\le f(x)\} \subset \mathbb{R}^{n+1},\] is Jordan measurable. In this case,\[\mu_{n+1}(\Gamma_f) = \int_A f.\]

Q: Which Continuity Type is Riemann Integrable?
A: Given \(A\subset \mathbb{R}^{n}\) Jordan measurable and \(f:A\to [0,\infty)\) uniformly continuous. Then, \(f\) is Riemann integrable over \(A\).

Q: Change of Variables
A: Let \(\Phi:U\subset \mathbb{R}^{n} \to V\subset \mathbb{R}^{n}\) be a \(C^1\)-diffeomorphism. Let \(A\subset U\) be Jordan measurable, such that \(\overline{A}\subset U\). Given \(f:\overline{A}\to [0,\infty)\) continuous, then \(f\) is Riemann integrable over \(A\) and \[\int_A f(x)\mathrm{d} x = \int_{\Phi(A)} \frac{f(\Phi^{-1}(y))}{|\det J\Phi(\Phi^{-1}(y))|} \mathrm{d} y.\] Letting \(\Psi = \Phi^{-1}\) we write this as
\[\int_{A} f(x) \mathrm d x = \int_{\Psi^{-1}(A)} f(\Psi(y)) |\det J\Psi(y)| \mathrm d y.\]

Q: Cavalieri's Principle
A: For \(\mathbb{R}^{n}=\mathbb{R}^k \times \mathbb{R}^{n-k}\), let\((x,y)\in\mathbb{R}^{n}\). Given \(A\subset \mathbb{R}^{n}\) Jordan measurable, \(A\subset (-C,C)^n\) for some \(C>0\), define the slices for \(x\in (-C,C)^k\) as
\[A_x = \{y\in \mathbb{R}^{n-k} \mid (x,y)\in A\}.\]
Let also \(\overline\varphi(x) = \mu_{n-k, \text{out}} (A_x)\),\(\underline{\varphi}(x) = \mu_{n-k, \text{in}}(A_x)\).
Then, \(\overline\varphi, \underline{\varphi}:[-C,C]^k \to \mathbb{R}\) are Riemann integrable and\[\int_{(-C,C)^k} \underline{\varphi}(x) \mathrm d x = \mu_n(A) = \int_{(-C,C)^k} \overline\varphi(x) \mathrm d x.\]

Q: Fubini's Theorem
A: Let \(A\subset \mathbb{R}^{n}\) be bounded. Write \(\mathbb{R}^{n} = \mathbb{R}^k \times \mathbb{R}^{n-k}\) and let \((x,y)\in \mathbb{R}^{n}\) be the coordinates. Given \(f:A\to \mathbb{R}\) Riemann integrable over \(A\), then let \[\tilde f = f\cdot 1_A = \begin{cases} f(x,y) & (x,y)\in A\\ 0 & x\notin A \end{cases}.\]
For \(x\in \mathbb{R}^k\), define the marginals\[\underline \varphi(x) = \underline\int \tilde f_x(y) \mathrm d y \quad \text{and} \quad \overline\varphi(x) = \overline\int \tilde f_x(y) \mathrm d y.\]
Where \(\tilde f_x(y) = \tilde f(x,y)\). Then,\[\int_A f = \int_{\mathbb{R}^k} \underline\varphi(x) \mathrm d x = \int_{\mathbb{R}^k} \overline\varphi(x) \mathrm d x.\]

Q: Integral over a box
A: Given \(K=[a_1,b_1]\times \dots \times [a_n,b_n]\subset \mathbb{R}^{n}\) and\(f:K\to \mathbb{R}\) continuous. Then, \(f\) is Riemann integrable over \(K\) and\[\int_K f(x) \mathrm d x = \int_{a_1}^{b_1}\dots \int_{a_n}^{b_n} f(x_1,\dots,x_n) \mathrm d x_n \dots \mathrm d x_1.\]

Q: Differentiation under the Integral
A: Given \(K\subset \mathbb{R}^{n}\) compact and \(f:K\times [a,b]\to \mathbb{R}\) continuous, \(f=f(x,t)\), such that \(\partial_t f\) exists and is continuous. Then, \(\forall t_0\in (a,b)\),\[\left.\frac{\mathrm d}{\mathrm d t}\right\rvert_{t_0} \int_K f(x,t) \mathrm d x = \int_K \partial_t f(x,t_0) \mathrm d x.\]

Q: Improper Integral
A: Given \(U\subset \mathbb{R}^{n}\) open and \(f:U\to \mathbb{R}\) continuous, nonnegative. We define\[\int_U f(x) \mathrm d x = \sup \left\{\int_K \mid K\subset U, K \text{compact, Jordan Measurable}\right\}.\]

Q: Gaussian Integral
A: \[\int_{-\infty}^{\infty} e^{-x^2} \mathrm dx = \sqrt{\pi}\]

Q: Length
A: Given \([a,b]\subset V\), then we define the length of \(\phi([a,b])\) as\[L(\phi([a,b])) = \int_a^b |\phi'(t)| \mathrm d t.\]

Q: d-Volume on parametrized submanifolds
A: Given \(V\subset \mathbb{R}^{d}\) open and \(\phi: V\to \mathbb{R}^{n}\) a parametrized \(d\)-dimensional submanifold, and \(E\subset V\) Jordan measurable with \(\overline{E}\subset V\), we define the \vocab{\(d\)-volume} as \[\mathrm{vol}_d(\phi(E)) = \int_E \sqrt{\det((D\phi(x))^T D\phi(x))} \mathrm d x.\]
Here, \(\det((D\phi(x))^T D\phi(x))\) is called the \vocab{Gram determinant}.

Q: Properties of the d-volume
A: The d-volume is
1. Independent of the chose parametrization
2. Invariant under euclidean isometries
3. Additive
4. Normalized

Q: Support of a function
A: Given \(V\subset \mathbb{R}^m\) open, let \(f:V\to \mathbb{R}^{n}\) be a continuous function. We define the \vocab{support} of \(f\) as \[\mathrm{spt}(f) = \overline{\{x\in V|f(x)\neq 0\}} \subset \overline{V} \subset \mathbb{R}^m.\] We say that the \(f\) has compact support if \(\mathrm{spt}(f)\) is compact (i.e. bounded)

Q: Integral over parametrized submanifolds
A: Given \(\phi:V\to \mathbb{R}^{n}\) a parametrized submanifold and \(f:\phi(V)\to \mathbb{R}\) is continuous such that \(\mathrm{spt}(f\circ\phi)\) is compact, and contained in \(V\). Let \(M=\phi(V)\), then we define the integral of \(f\) over \(\phi(V)\) as\[\int_M f(p) \mathrm d \mathrm{vol}_M(p) = \int_V (f\circ \phi) \sqrt{\det((D\phi)^T D\phi)} \mathrm d x.\]

Q: \(C^k\) function on a general set
A: Given \(A\subset \mathbb{R}^{n}\), we say that \(f:A\to \mathbb{R}^m\) is \(C^k\) if there exists \(U\supset A\) open and \(\tilde f:U\to \mathbb{R}^m\) \(C^k\) such that \(\tilde f|_A = f\).

Q: Divergence
A: Given \(F=(F_1,\dots,F_n):\overline{\Omega}\to \mathbb{R}^{n}\), we define the divergence of \(F\) as\[\mathrm{div} F = \sum_{i=1}^n \partial_i F_i.\]

Q: Partition of Unity
A: Given \(K\subset \mathbb{R}^{n}\) compact and \(B_{p_l}(r_l)\) for \(l=1,\dots,N\) open balls, such that \(K\subset \bigcup_{l=1}^N B_{p_l}(r_l)\), then \(\exists U\) open such that \(K\subset U\), \(N\) functions \( \eta_1,\dots, \eta_N: \mathbb{R}^{n}\to [0,\infty) \) and one background function \(\tilde\eta:\mathbb{R}^{n}\to [0,\infty)\) all \(C^\infty\) such that\[1=\sum_{l=1}^{N} \eta_l + \tilde \eta.\]And\[\mathrm{spt}(\eta_l) \subset B_{p_l}(r_l), \forall l=1,\dots,N, \mathrm{spt}(\tilde \eta) \subset \mathbb{R}^{n}\setminus U.\]

Q: \(C^k\) domain
A: Bounded open set \(\Omega \subset \mathbb{R}^n\) such that \(M\coloneqq \partial \Omega\) is a \((n-1)\)-dimensional submanifold of class \(C^k\)

Q: Divergence Theorem
A: Let \(\Omega \subset \mathbb{R}^{n}\) be a bounded \(C^1\) domain and \(G\in C^1(\overline{\Omega},\mathbb{R}^{n})\). Then, \[\int_\Omega \mathrm{div} G \mathrm dx = \int_{\partial \Omega} G\cdot \nu \mathrm d \mathrm{vol}_{\partial \Omega}.\]

Q: Greens Theorem
A: In \(\mathbb{R}^{2}\), given \(F\in C^1(\overline{\Omega},\mathbb{R}^{2})\), \(\partial \Omega\) is a curve. Then,\[\int_{\partial \Omega} \vec{F}\cdot \tau \mathrm d L = \int_\Omega \mathrm{rot}(\vec{F}) \mathrm d x,\]where \(\mathrm{rot}(\vec{F}) = \partial_2 F_1 - \partial_1 F_2\) is the rotation of \(F\).

Q: 1-Form
A: Given \(U\subset \mathbb{R}^{n}\) open, a map \(\alpha\) assigning to each point \(x\in U\) a covector\[x\mapsto (\alpha_1(x),\dots,\alpha_n(x)) = \alpha_x.\]is called a 1-form. (\(C^k\) 1-form if \(\alpha_i\in C^k(U)\) \(\forall i\))

Q: Differential 1-Form
A: Given \(f\in C^\infty(U)\), we define \[df: U \to \mathbb{R}^{n*}, \quad df_x = Df_x.\]

Q: Line Integral
A: Given \(\alpha:U\to \mathbb{R}^{n*}\), \(C^1\) is a 1-form, then we define the integral of \(\alpha\) over a curve \(\gamma:[a,b]\to U\) as\[\int_{\gamma} \alpha \coloneqq \int_a^b \alpha_{\gamma(t)}(\gamma'(t)) \mathrm d t = \int_a^b \alpha(\gamma(t))\cdot \gamma'(t) \mathrm d t.\]

Q: Conservative Vector Field / Exact Differential Form
A: Given \(U\subset \mathbb{R}^{n}\) connected, open, we say that \(F: U\to \mathbb{R}^{n}, C^1\) is conservative if \(\exists f:U\to \mathbb{R}\), \(C^2\) such that \(\mathrm{grad} f = F\). \(f\) is called the potential of \(F\).
Equivalently, \(\alpha:U\to \mathbb{R}^{n*}\), \(C^1\) is exact if \(\exists f:U\to \mathbb{R}\), \(C^2\) such that \(df = \alpha\).

Q: Path Integral of an exact 1-form
A: If \(\alpha\) is an exact 1-form, then \(\int_\gamma \alpha\) only depends on the endpoints of \(\gamma\). In particular, if \(\alpha = \mathrm df\),\[\int_\gamma \alpha = f(\gamma(b)) - f(\gamma(a)).\]

Q: Integrability Conditions
A: Given \(\alpha:U\to \mathbb{R}^{n*}\) exact, then setting \(\alpha_x =(\alpha_1(x),\dots,\alpha_n(x))\), we have\[\partial_j \alpha_i = \partial_i \alpha_j, \forall i,j=1,\dots,n.\]

Q: Integrability Condition =>
A: Assume that \(U\subset \mathbb{R}^{n}\) is convex and let \(\alpha:U\to \mathbb{R}^{n*}\) satisfying integrability conditions. Then, \(\exists f:C^2(U)\) such that \(\mathrm d f = \alpha\).

Q: k-Form in \(\mathbb{R}^n\)
A: A map \(p\in U \subset \mathbb{R}^{n}\) assigns \(\alpha_p = \alpha_p(T)\),\[\alpha_p: \mathbb{R}^{n\times k} \to \mathbb{R}.\] is called a \(k\)-form if \(\alpha_p\) satisfies the following properties:
1. \(\alpha_p(TS) = \alpha_p(T)\cdot \det S\)
2. \(\alpha_p\) is multilinear (linear in each column of \(T\))

Q: Explicit Form of a k-Form
A: \[\alpha(T) = \sum_{1<i_1<\dots<i_k<n} c{i_1,\dots,i_k}(p) \cdot \det(E_{i_1,\dots,i_k}^T T)
.\]

Q: Wedge Product
A: Let \(\alpha\) be a \(k\)-form and \(\beta\) be a \(m\)-form. If \(k+m \le n\), and\[\alpha(T)=\det(AT) \quad \beta(S) = \det(BS),\]then we define for \(X\in \mathbb{R}^{n\times (k+m)}\),\[\alpha\wedge \beta(X) = \det\left(\begin{pmatrix}A\\B\end{pmatrix}\cdot X\right).\]

Q: Exterior Derivative
A: Let \(0\le k\le n\) and \(\alpha\in \Omega^k(U)\) be a \(k\)-form such that \[\alpha_x(T) = \sum_{I\in \mathcal I_k} c_I(x) \underbrace{\det(e_I^T T)}_{\coloneqq w_I}.\]We define the \vocab{differential} of \(\alpha\) as the \((k+1)\)-form \[\beta \coloneqq \mathrm d\alpha, \quad \beta_x(S) = \sum_{I} \mathrm d c_I(x) \wedge w_I(S).\]

Q: Double Exterior Derivative
A: \(\forall \alpha\in \Omega^k(U)\), \(\mathrm d(\mathrm d \alpha) = 0 \in \Omega^{k+2}(U)\). \[\mathrm d^2 = 0.\]

Q: Leibniz Rule
A: Given \(\alpha\in \Omega^k(U)\) and \(\beta\in \Omega^l(U)\), then\[\mathrm d(\alpha\wedge \beta) = \mathrm d\alpha \wedge \beta + (-1)^k \alpha \wedge \mathrm d\beta.\]

Q: Pull-back
A: Given \(\alpha\in \Omega^k(U)\), \(U\subset \mathbb{R}^{n}\), \(V\subset \mathbb{R}^m\) and a map \(F: V\to U, C^\infty\); we define the pull-back of \(\alpha\) by \(F\), denoted as \(F^*(\alpha)\in \Omega^k(V)\) as follows: \[(F^*\alpha)_x(A) = \alpha_{F(x)}(DF_x A).\]

Q: Pullback, Wedge and Exterior Derivative
A: Given \(U\subset \mathbb{R}^{n}\), \(V\subset \mathbb{R}^m\) open and \(F: V\to U, C^\infty\), then:
1. \(\forall \alpha\in \Omega^k(U)\), \(\beta\in \Omega^l(U)\), \[F^*(\alpha\wedge \beta) = F^*(\alpha)\wedge F^*(\beta).\]
2. \(\forall \alpha\in \Omega^k(U)\), \[F^*(\mathrm d \alpha) = \mathrm d F^*(\alpha).\]

Q: Integral of a k-form over a submanifold
A: Let \(\Phi: V\to U\) be a \(k\)-dimensional parametrized submanifold with \(M=\Phi(V)\). Let \(\alpha\) be a \(k\)-form on \(U\). We define the integral of \(\alpha\) over \(M\) as\[\int_M \alpha \coloneqq \int_V \alpha_{\Phi(x)}(D\Phi_x) \mathrm d x.\]

Q: Orientability
A: A \(k\)-dimensional submanifold \(M\subset \mathbb{R}^{n}\) is orientable ifi it is covered by local parametrizations with transition maps that have positive determinant.

Q: Orientability of \((n-1)\) dimenisonal submanifolds
A: A \((n-1)\) dimensional submanifold \(M\subset \mathbb{R}^{n}\) is orientable iff \(\exists \nu: M\to \mathbb{S}^{n-1}\) such that \(\nu(p)\perp T_p M\) and \(\nu\) is continuous.

Q: General Integration of k-Forms over Submanifolds
A: Let \(M\) be compact, \(m\)-dim, and oriented and let \(\Phi_i:V_i \to M\cap U_i\) be a system of parametrizations where \(U_i\) are Balles. Take \(\eta_i\) be the partition of unity such that \(\mathrm{spt} \eta_i \subset U_i\). Then, we can define for \(\alpha\in \Omega^m(U)\),\[\int_M \alpha \coloneqq \sum_{i} \int_{V_i} \alpha(D\Phi_i)\eta_i(\Phi(y)) \mathrm d y.\] Using pull-back, we can write \[\int_M \alpha = \sum_i \int_{V_i} \Phi_i^*(\alpha \eta_i)(e_1, \dots, e_m) \mathrm d y.\]

Q: Divergence Theorem for k-forms
A: \(\forall \alpha\in \Omega^{n-1}(\mathbb{R}^{n})\), we have \[\int_U \mathrm d \alpha = \int_{\partial U} \alpha,\] where \(\partial U\) is oriented according to the exterior unit normal.

Q: Parametrized bounded submanifold with boundary
A: Given \(U\subset V\subset \mathbb{R}^m\), and \(\Phi:V\to \mathbb{R}^{n}\) a parametrized submanifold with \(\Phi(V) = M\). We call \(\Phi(U)\) a parametrized bounded submanifold with boundary.
The boundary \(\partial M \coloneqq \Phi(\partial U)\) is a parametrized submanifold ofdimension \(m-1\).

Q: Generalized Stokes Theorem
A: Given a parametrized bounded submanifold with boundary \(M\), and \(\alpha\in \Omega^{m-1}(\mathbb{R}^{n})\), we have\[\int_{\partial M} \alpha = \int_M \mathrm d \alpha,\] where the orientation of \(\partial M\) is induced by the exterior unit normal to \(M\).

Q: Stokes Theorem in 2D
A: Given \(\Sigma\subset \mathbb{R}^{3}\) a smoothly bounded 2 dimensional surface, and \(F\in C^\infty(\mathbb{R}^{3}, \mathbb{R}^{3})\) a vector field, then \[\int_\Sigma \mathrm{rot} F \cdot \nu \mathrm d s = \int_{\partial \Sigma} F\cdot \mathrm d L,\]where \(\mathrm{rot} F = (\partial x, \partial y, \partial z) \wedge F\) is the curl of \(F\).

Q: Characteristic Polynomial
A: Given an ODE, we define the characteristic polynomial as
    \[p(x) = \sum_{k=0}^{m} a_k x^k.\]

Q: Homogenous ODE
A: The equation \(Lz = 0\), where \(z:I\to \mathbb{C}\) is called the homogeneous ODE associated to \(L\).

Q: Basis for \(Lz=0\)
A: If \(\alpha\) is a root of \(p\) with multiplicity \(m\), then the functions \[p(x) = \prod_{j=1}^k (x-\alpha_j)^{m_j}.\]where \(\alpha_1,\dots,\alpha_k\) are the distinct roots of \(p\), and \(m_j\) is the multiplicity of \(\alpha_j\), then the functions\[p(\frac{\mathrm d}{\mathrm d t}) (t^k \exp(\alpha_j t)) = 0, \forall k=0,\dots,m_j-1.\]

Q: Pulse at 0
A: \[\begin{cases}Lz = 0\\z^{(k)}(0) = 0 \forall k=0,\dots,m-2\\z^{(m-1)}(0) = 1\end{cases}.\]We denote the unique solution \(z(t)\) to this IVP \(P(t)\) and call it pulse at 0 for \(L\).

Q: Duhamels Principle
A: Given the ODE\[\begin{cases}Ly(t) = f(t)\\y(t_0) = y'(t_0) = \dots = y^{(m-1)}(t_0) = 0\end{cases}.\] Then, the solution is given by\[y(t) = \int_{t_0}^t f(s) P(t-s) \mathrm d s.\]Where $P$ is a pulse at 0.

Q: Matrix Exponential
A: Given \(A\in M_{n\times n}(\mathbb{R})\), we define\[\exp(A) = \sum_{k=0}^{\infty} \frac{A^k}{k!}.\]

Q: Homogenous System of ODEs
A: The unique solution of\[\begin{cases}u'(t) = Au(t)\\u(t_0) = x_0\end{cases},\] is given by\[u(t) = \exp(A(t-t_0))x_0.\]

Q: Continuous Bounded Functions Space
A: If \((Y,d)\) is a metric space, we define\[C_b(Y,\mathbb{R}^{n}) \coloneqq \{f:Y\to \mathbb{R}^{n}: f \text{ cont. and bdd.}\}.\]This space is a normed vector space with the norm\[\|f\|_\infty = \sup_{y\in Y} |f(y)|.\]

Q: Uniform Convergence
A: Given \(f, f_k: Y\to \mathbb{R}^{n}\) continuous we say that \[f_k \xrightarrow{unif.} f,\] if \(\|f_k - f\|_\infty \to 0\) as \(k\to \infty\).

Q: Cauchy-Lipschitz-Piccard-Lindelöff
A: Given \(U\subset \mathbb{R}\times \mathbb{R}^{n}\) open, \(F:U\to \mathbb{R}^{n}\) \(C^0\) in \(t\) and \(C^1\) in \(x\) (\(F=F(t,x)\)). Let \((t_0,x_0)\in U\). Then, there exists a maximal interval \(I=(a,b)\ni t_0\) and a unique \(u:I\to \mathbb{R}^{n}\) such that \((t,u(t))\subset U \forall t\in I\) and \(u\in C^1\) satisfies\[\begin{cases}u'(t) = F(t,u(t)) & \forall t\in I\\u(t_0) = x_0\end{cases}.\]

Q: Ascoli-Arzelà
A: Let \(K\subset \mathbb{R}^{n}\) be compact and \(f_k: K\to \mathbb{R}^{m}\) be a sequence of functions in \(C_b(K,\mathbb{R}^{m})\), that is:
bounded: \(\sup_K \|f_k\|_\infty < \infty\).
equicontinuous: \(\forall \varepsilon>0 \exists \delta>0\) such that \(\forall k, \forall x\in K, f(B_\delta(x)\cap K) \subset B_\varepsilon(f(x))\),
then, it has a convergent subsequence in \(C_b(K,\mathbb{R}^{m})\).