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7d-8&U6b2q4>xxX_n`oHy;Zy`OS+Lg;ePfD zASv7n_fZ{cPYJ1_BT2O-mJxl1^0XYPgq=HQsBk<5J1nw{Ocv@b?Z5>+9=n z$0(BHoueLSpTYrF`=NGqkt4Dw;sMa_LQ&cVn3??F{bOR-4a68C=q%Hn+( diff --git a/Band4/Band4.md b/Band4/Band4.md index 079272c..8b7d7a1 100644 --- a/Band4/Band4.md +++ b/Band4/Band4.md @@ -35,5 +35,25 @@ $$ Für eine Ausdehnung der Theorie auf Funktionen von *mehreren* unabhängigen Veränderlichen ist es erforderlich, neue Grundmengen heranzuziehen. Wir betrachten zunächst Paare von reellen Zahlen und legen für je zwei reelle Zahlen $a, b$ eine Reihenfolge fest. Soll $a$ die erste und $b$ die zweite Zahl sein, so schreiben wir $x_1=a$ und $x_2=b$ und fassen beide Zahlen durch Klammern in der Weise zu einem Paar zusammen, daß wir innerhalb der Klammern $x_1$ an die erste und $x_2$ an die zweite Stelle setzen. Wir schreiben also ( $x_1, x_2$ ) und bezeichnen ( $x_1, x_2$ ) als **geordnetes Zahlenpaar**. Die Zahlen 3 und -1 können also zu dem Paar $(3,-1)$ oder zu dem Paar $(-1,3)$ zusammengefaßt werden. Zwei geordnete Zahlenpaare ( $x_1, x_2$ ) und ( $y_1, y_2$ ) nennen wir gleich, wenn innerhalb der Klammern an der jeweils entsprechenden Stelle die gleiche Zahl steht. Wir setzen also $\left(x_1, x_2\right)=\left(y_1, y_2\right)$ genau dann, wenn $x_1=y_1$ und $x_2=y_2$ gilt. Somit ist $(3,-1) \neq(-1,3)$ und auch $(3,-1) \neq(3,0)$ wegen $-1 \neq 0$. -Eine geometrische Veranschaulichung von geordneten Zahlenpaaren ist in einer mit einem kartesischen Koordinatensystem versehenen Ebene möglich. Man erkennt an Bild 1.1, daß man das Zahlenpaar ( $x_1, x_2$ ) durch den Punkt $X$ mit den Koordinaten $x_1$ und $x_2$ oder durch den Vektor $\mathbf{x}$ mit den Koordinaten $x_1$ und $x_2$ veranschaulichen +Eine geometrische Veranschaulichung von geordneten Zahlenpaaren ist in einer mit einem kartesischen Koordinatensystem versehenen Ebene möglich. Man erkennt an Bild 1.1, daß man das Zahlenpaar ( $x_1, x_2$ ) durch den Punkt $X$ mit den Koordinaten $x_1$ und $x_2$ oder durch den Vektor $\mathbf{x}$ mit den Koordinaten $x_1$ und $x_2$ veranschaulichen kann. Gleichbedeutend +Band4_1.1sprechen wir in diesem Zusammenhang also von dem **geordneten Zahlenpaar** ( $x_1, x_2$ ), dem **Punkt** $X\left(x_1, x_2\right)$ oder dem **Vektor** $\mathbf{x}=x_1 \mathbf{e}_1+x_2 \mathbf{e}_2$. Eine Menge von geordneten Zahlenpaaren nennen wir demzufolge auch eine Punktmenge. Unter dem $R^2$ verstehen wir die Menge aller geordneten Zahlenpaare ( $x_1, x_2$ ). +Sind $X\left(x_1, x_2\right)$ und $Y\left(y_1, y_2\right)$ zwei beliebige Punkte des $R^2$, so bezeichnet man unter Beachtung des Satzes von Pythagoras die Zahl +$$ +d(X, Y)=\sqrt{\left(x_1-y_1\right)^2+\left(x_2-y_2\right)^2} +$$ +![Band4_1.2](/Volumes/Daten02/maing01_06/Band4/media/Band4_1.2.png) +als Abstand der Punkte $X$ und $Y$ (vgl. Bild 1.2). Bei Benutzung der Vektorschreibweise $\mathbf{x}=x_1 \mathbf{e}_1+x_2 \mathbf{e}_2$ und $\mathbf{y}=y_1 \mathbf{e}_1+y_2 \mathbf{e}_2$ bzw. der Koordinatenschreibweise hätte man zu formulieren: + +​ $d(\mathbf{x}, \mathbf{y})=\sqrt{\left(x_1-y_1\right)^2+\left(x_2-y_2\right)^2}$ + +bzw. + +​ $d\left(\left(x_1, x_2\right),\left(y_1, y_2\right)\right)=\sqrt{\left(x_1-y_1\right)^2+\left(x_2-y_2\right)^2} .$ + +Speziell für $X(-1,3)$ und $Y(5,-4)$ erhält man + +​ $d(X, Y)=\sqrt{(-1-5)^2+(3+4)^2}=\sqrt{36+49}=\sqrt{85}.$ + + +Je zwei Punkten $X$ und $Y$ ist also eine nichtnegative reelle Zahl $d(X, Y)$ als Abstand zwischen $X$ und $Y$ zugeordnet. Man sagt für diesen Sachverhalt auch, daß im $R^2$ eine Abstandsfunktion oder Metrik erklärt ist. 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