mirror of
https://github.com/janishutz/eth-summaries.git
synced 2026-01-12 20:28:31 +00:00
122 lines
4.7 KiB
TeX
122 lines
4.7 KiB
TeX
\subtext{Treating functions $f: X \subset \R^n \to \R / \C / \R^m,\quad m,n \geq 1$}
|
|
|
|
\notation $f(x)$ for $f: I \subset \R^n \to \R^m$ means:\\
|
|
$x = (x_1, \ldots, x_n),\quad f(x) = f\bigl( f_1(x), \ldots, f_m(x) \bigr)$
|
|
|
|
\subsection{Multivariate functions}
|
|
|
|
\definition \textbf{Linear map} $f: \R^n \to \R^m$\\
|
|
\subtext{In other words: $f(x) = \textbf{A}x,\quad \textbf{A} \in \C^{m \times n}$}
|
|
|
|
Linear Maps are continuous
|
|
|
|
\definition \textbf{Affine Linear map} $f(x) \mapsto \textbf{A}x + c$
|
|
|
|
\definition \textbf{Quadratic form} $Q: \R^n \to \R$\\
|
|
\subtext{In other words: $Q(x) = \sum_{i=0}^{n}\sum_{j=0}^{m}\left( a_{i,j}x_i x_j \right)$}
|
|
|
|
\definition \textbf{Monomials} $M(x): \R^n \to \R \mapsto \alpha x_1^{d_1}\cdots x_n^{d_n}$\\
|
|
\subtext{For example: $f(x, y, z) = 16x^2yz^5$}
|
|
|
|
\definition $\deg(M) := e = \sum_{i=1}^{n} d_i$\\
|
|
\subtext{For example: $\deg(16x^2yz^5) = 8$}
|
|
|
|
\definition \textbf{Polynomials} $P(x) := \sum_{i=0}^{n} M_i(x)$\\
|
|
\subtext{For example: $P(x,y,z) = x^3 + 25x^2y^6z + xy$}
|
|
|
|
Polynomials are continuous.
|
|
|
|
\definition $\deg(P) := d \geq \max \{ \deg(M_i) \sep M_i \text{ in } P \}$\\
|
|
\subtext{For example: $\deg(x^3 + 25x^2y^6z + xy) = 9$}
|
|
|
|
Visualisations for some function types:
|
|
|
|
\definition \textbf{Graph} $G_f := \{(x,y,z) \in \R^3 \sep z = f(x,y) \}$\\
|
|
\subtext{Only for $f: \R^2 \to \R$. Visually, this is a surface in $\R^3$}
|
|
|
|
\definition \textbf{Vector Plots} for $f: \R^2 \to \R^2$\\
|
|
\subtext{Points in $(x,y) \in \R^2$ are displayed as vectors $f(x,y)$}
|
|
|
|
\newpage
|
|
\subsection{Sequences in $\R^n$}
|
|
|
|
\definition \textbf{Sequences in $\R^n$}\\
|
|
$(x_k)_{k \geq 1}$ s.t. $x_k \in \R^n$ where $x_k = \bigl( x_{k,1},\ldots x_{k,n} \bigr)$
|
|
|
|
\definition \textbf{Convergence in $\R^n$}\\
|
|
$$
|
|
\lim_{k \to \infty} \Bigl( x_k \Bigr) = y \iff \forall \epsilon > 0, \exists N \geq 1: \forall k \geq N:\quad || x_k - y || < \epsilon
|
|
$$
|
|
|
|
Using this definition preserves many familiar results:
|
|
|
|
\lemma \textbf{Equivalent conditions to Convergence}\\
|
|
$
|
|
\begin{array}{ll}
|
|
(i) & \forall i \text{ s.t. } 1 \leq i \leq n:\quad \underset{k \to \infty}{\lim} \Bigl(x_{k,i}\Bigr) = y_i \\
|
|
(ii) & \underset{k \to \infty}{\lim} \Big\| x_k - y \Big\| = 0
|
|
\end{array}
|
|
$
|
|
|
|
\definition \textbf{Continuity in $\R^n$}\\
|
|
$f \text{ continuous at } x_0 \in X \iffdef \forall \epsilon > 0, \exists \delta > 0:\\$
|
|
$$
|
|
\big\| x - x_0 \big\| < \delta \implies \big\| f(x) - f(x_0) \big\| < \epsilon\\
|
|
$$
|
|
$f$ continuous $\iffdef \forall x \in X: f$ continuous at $x$\\
|
|
\subtext{$X \subset \R^n,\quad f:X \to \R^m$}
|
|
|
|
\lemma \textbf{Continuitiy using Sequences}\\
|
|
$f$ continuous at $x_0$ if and only if:
|
|
$$
|
|
\forall (x_k)_{k \geq 1}:\quad \underset{k \to \infty}{\lim} \Bigl( x_k \Bigr) = x_0 \implies \underset{k \to \infty}{\lim}\Bigl(f(x_k)\Bigr) = f(x_0)
|
|
$$
|
|
\subtext{$X \subset \R^n,\quad f:X \to \R^m$}
|
|
|
|
\definition \textbf{Limits at points}
|
|
\begin{align*}
|
|
& \underset{x \neq x_0 \to x_0}{\lim} \Bigl( f(x) \Bigr) = y \iffdef \forall \epsilon > 0, \exists \delta > 0: \\
|
|
& \forall x \neq x_0 \in X: \big\| x - x_0 \big\| < \delta \implies \big\| f(x) - y \big\| < \epsilon
|
|
\end{align*}
|
|
\subtext{$X \subset \R^n,\quad f:X \to \R^m,\quad x_0 \in X,\quad y \in \R^m$}
|
|
|
|
The sequence test for Continuity works for point-limits too.
|
|
|
|
\lemma \textbf{Continuity of Compositions}\\
|
|
$f: X \to Y,\ g: Y \to \R^p \text{ continuous } \implies g \circ f \text{ continuous}$\\
|
|
\subtext{$X \subset \R^n,\quad Y \subset \R^m,\quad p \geq 1$}
|
|
|
|
\lemma \textbf{Continuity using Coordinate Functions}\\
|
|
$f: \R^n \to \R^m$ continuous $\iff \forall i \leq m: f_i$ continuous
|
|
|
|
\subsection{Subsets of $\R^n$}
|
|
|
|
\definition \textbf{Bounded}\\
|
|
$X \subset \R^n$ bounded $\iffdef \Bigl\{ \big\| x \big\| \sep x \in X \Bigr\} \subset \R$ bounded.\\
|
|
\subtext{Example: The open disc $D = \{ x \in \R^n \sep \big\| x - x_0 \big\| < r \}$ is bounded.}
|
|
|
|
\definition \textbf{Closed}\\
|
|
$X \subset \R^n$ closed $\iffdef \forall (x_k)_{k\geq 1} \in X:\quad \underset{x \to \infty}{\lim}\Bigl( x_k \Bigr) \in X$\\
|
|
\subtext{Example: $\emptyset$, $\R^n$ are closed.}
|
|
|
|
\definition \textbf{Compact} if closed and bounded.\\
|
|
\subtext{Example: The closed Disc $\Lambda = \{ x \in \R^n \sep \big\| x - x_0 \big\| \leq r \}$ is compact.}
|
|
|
|
\lemma The Cartesian Product preserves these properties.
|
|
|
|
\lemma \textbf{Continous functions preserve closedness}
|
|
$$
|
|
\forall \text{ closed } Y:\quad f^{-1}(Y) = \bigl\{ x \in \R^n \sep f(x) \in Y \bigr\} \text{ is closed.}
|
|
$$
|
|
\subtext{$f: \R^n \to \R^m$ is continuous,$\quad Y \subset \R^m$}
|
|
|
|
\begin{subbox}{Min-Max Theorem}
|
|
\smalltext{For compact, non-empty $X \subset \R^n$, continuous $f: X \to \R$:}
|
|
$$
|
|
\exists x_1,x_2 \in X :\quad f(x_1) = \underset{x \in X}{\sup} f(x),\quad f(x_2) = \underset{x \in X}{\inf} f(x)
|
|
$$
|
|
\end{subbox}
|
|
|
|
\subsection{Partial Derivatives}
|
|
|