Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen RevisionVorhergehende ÜberarbeitungNächste Überarbeitung | Vorhergehende Überarbeitung | ||
| spielplatz:mathematische-formeln-anleitung [2019/02/11 15:04] – [Indizes und Hochstellung] wi | spielplatz:mathematische-formeln-anleitung [2023/04/18 09:12] (aktuell) – [Mathematische Formeln in Dokuwiki] wi | ||
|---|---|---|---|
| Zeile 1: | Zeile 1: | ||
| ====== Mathematische Formeln in Dokuwiki ====== | ====== Mathematische Formeln in Dokuwiki ====== | ||
| + | < | ||
| Zusammengestellt von M. Wilfling, HTBLA Kaindorf | Zusammengestellt von M. Wilfling, HTBLA Kaindorf | ||
| - | $xxx$ | + | $ \frac{1}{x} |
| Wer möchte seine Wiki Seiten, die mathematische Formeln enthalten, professionell gestalten? | Wer möchte seine Wiki Seiten, die mathematische Formeln enthalten, professionell gestalten? | ||
| Zeile 35: | Zeile 36: | ||
| ==== Inline Formel ==== | ==== Inline Formel ==== | ||
| - | < | + | |
| Dieses ist eine wichtige Formel: $c = a + b$ | Dieses ist eine wichtige Formel: $c = a + b$ | ||
| - | < | + | |
| Für alle Zahlen $a_1, \dots, a_n$ gilt | Für alle Zahlen $a_1, \dots, a_n$ gilt | ||
| - | < | + | |
| - | und $c$ die Hypotenuse, dann gilt | + | $c=\sqrt{a^2+b^2}$ (Lehrsatz des Pythagoras). |
| - | $c=\sqrt{a^2+b^2}$ (Lehrsatz des | + | Seien $a$ und $b$ die Katheten und $c$ die Hypotenuse, dann gilt |
| - | Pythagoras).</ | + | $c=\sqrt{a^2+b^2}$ (Lehrsatz des Pythagoras). |
| - | Seien $a$ und $b$ die Katheten | + | |
| - | und $c$ die Hypotenuse, dann gilt | + | |
| - | $c=\sqrt{a^2+b^2}$ (Lehrsatz des | + | |
| - | Pythagoras). | + | |
| ==== Abgesetzte Formel ==== | ==== Abgesetzte Formel ==== | ||
| - | < | + | |
| Dieses ist eine wichtige Formel: $$y = f(x) = k~x + d$$ | Dieses ist eine wichtige Formel: $$y = f(x) = k~x + d$$ | ||
| - | < | + | |
| - | (Lehrsatz des Pythagoras).</ | + | (Lehrsatz des Pythagoras). |
| - | Seien $a$ und $b$ die Katheten und $c$ die Hypotenuse, dann gilt $$c^2=a^2+b^2$$ (Lehrsatz des Pythagoras). | + | Seien $a$ und $b$ die Katheten und $c$ die Hypotenuse, dann gilt $$c^2=a^2+b^2$$ |
| + | (Lehrsatz des Pythagoras). | ||
| - | ==== Zwischenraum, | + | ==== Zwischenraum, |
| - | $ab$, $a\,b$, $a\:b$, $a\;b$, $a\!b$ | + | $ab$, $a\,b$, $a\:b$, $a\;b$, $a\!b$, $a \quad b$, $a \qquad |
| - | $ab$, $a\,b$, $a\:b$, $a\;b$, $a\!b$ | + | $ab$, $a\,b$, $a\:b$, $a\;b$, $a\!b$, $a \quad b$, $a \qquad |
| ==== Ein nummerierter Formelblock ==== | ==== Ein nummerierter Formelblock ==== | ||
| Zeile 91: | Zeile 90: | ||
| ==== Sonderzeichen und Griechische Buchstaben ==== | ==== Sonderzeichen und Griechische Buchstaben ==== | ||
| - | < | + | |
| - | \mathrm{M}\mathrm{N}\Xi\mathrm{O}\Pi\mathrm{P}\Sigma\mathrm{T}\Phi\mathrm{X}\mathrm{Y}\Psi\Omega$</ | + | \mathrm{M}\mathrm{N}\Xi\mathrm{O}\Pi\mathrm{P}\Sigma\mathrm{T}\Phi\mathrm{X}\mathrm{Y}\Psi\Omega$ |
| - | $\mathrm{A}\mathrm{B}\Gamma\Delta\mathrm{E}\mathrm{Z}\mathrm{H}\Theta\mathrm{I}\mathrm{K}\Lambda\mathrm{M}\mathrm{N}\Xi\mathrm{O}\Pi\mathrm{P}\Sigma\mathrm{T}\Phi\mathrm{X}\mathrm{Y}\Psi\Omega$ | + | |
| - | < | + | $ \mathrm{A}\mathrm{B}\Gamma\Delta\mathrm{E}\mathrm{Z}\mathrm{H}\Theta\mathrm{I}\mathrm{K}\Lambda \mathrm{M}\mathrm{N}\Xi\mathrm{O}\Pi\mathrm{P}\Sigma\mathrm{T}\Phi\mathrm{X}\mathrm{Y}\Psi\Omega$ |
| - | $\alpha\beta\gamma\delta\epsilon\zeta\eta\theta\iota\kappa\lambda\mu\nu\xi\mathrm{o}\pi\rho\sigma\tau\phi\chi\upsilon\psi\omega$ | + | |
| - | < | + | |
| - | $\varepsilon \quad \vartheta \quad \varpi \quad \varrho \quad \varsigma \quad \varphi$ | + | $ \alpha\beta\gamma\delta\epsilon\zeta\eta\theta\iota\kappa\lambda\mu\nu\xi\mathrm{o}\pi\rho\sigma\tau\phi\chi\upsilon\psi\omega$ |
| - | < | + | $\varepsilon \quad \vartheta \quad \varpi \quad \varrho \quad \varsigma \quad \varphi$ |
| - | $\aleph\quad\Re\quad\Im\quad\partial\quad\infty\quad\forall\quad\exists\quad\neg\quad\in\quad\heartsuit$ | + | $ \varepsilon \quad \vartheta \quad \varpi \quad \varrho \quad \varsigma \quad \varphi$ |
| - | < | + | $\aleph\quad\Re\quad\Im\quad\partial\quad\infty\quad\forall\quad\exists\quad\neg\quad\in\quad\heartsuit$ |
| - | $\forall \varepsilon> | + | $ \aleph\quad\Re\quad\Im\quad\partial\quad\infty\quad\forall\quad\exists\quad\neg\quad\in\quad\heartsuit$ |
| + | $ \forall \varepsilon> | ||
| + | $ \forall \varepsilon> | ||
| + | |||
| + | Eurozeichen mit Unicode: $ \unicode{0x20AC}$ | ||
| + | Eurozeichen mit Unicode: $ \unicode{0x20AC}$ | ||
| ==== Klammern ==== | ==== Klammern ==== | ||
| - | < | + | |
| - | $( ~ \lbrack ~ \lbrace ~ [ ~ \lfloor ~ \langle ~ \{ ~ \lceil$ | + | $ ( ~ \lbrack ~ \lbrace ~ [ ~ \lfloor ~ \langle ~ \{ ~ \lceil$ |
| - | < | + | $ ) ~ \rbrack ~ \rbrace ~ ]~ \rfloor ~ \rangle ~ \} ~ \rceil$ |
| - | $) ~ \rbrack ~ \rbrace ~ ]~ \rfloor ~ \rangle ~ \} ~ \rceil$ | + | $ ) ~ \rbrack ~ \rbrace ~ ]~ \rfloor ~ \rangle ~ \} ~ \rceil$ |
| - | < | + | $ \Bigl( (x+1) (x-1)\Bigr) ^2$ |
| - | $\Bigl( (x+1) (x-1)\Bigr) ^2$ | + | $ \Bigl( (x+1) (x-1)\Bigr) ^2$ |
| - | < | + | $ \left( (x+1) (x-1)\right) ^2$ |
| - | $\left( (x+1) (x-1)\right) ^2$ | + | $ \left( (x+1) (x-1)\right) ^2$ |
| - | < | + | $ 1 + \left(\frac{1}{1-x^2}\right)$ |
| - | $1 + \left(\frac{1}{1-x^2}\right)$ | + | $ 1 + \left(\frac{1}{1-x^2}\right)$ |
| - | < | + | $ \underbrace{\overbrace{a + b + \cdots +z}^{26} + \overbrace{A + B + \cdots+Z}^{26}}_{52}$ |
| - | $\underbrace{\overbrace{a + b + \cdots +z}^{26} + \overbrace{A + B + \cdots+Z}^{26}}_{52}$ | + | $ \underbrace{\overbrace{a + b + \cdots +z}^{26} + \overbrace{A + B + \cdots+Z}^{26}}_{52}$ |
| - | < | + | $ \overline{m+n} \qquad \underline{m+n}$ |
| - | $\overline{m+n} \qquad \underline{m+n}$ | + | $ \overline{m+n} \qquad \underline{m+n}$ |
| ==== Operatoren ==== | ==== Operatoren ==== | ||
| - | < | + | |
| $x = y > z \qquad x := y \qquad x \le y \ne z$ | $x = y > z \qquad x := y \qquad x \le y \ne z$ | ||
| - | < | + | |
| $x \sim y \simeq z \qquad x \equiv y \not\equiv z \qquad x \subset y \subseteq z$ | $x \sim y \simeq z \qquad x \equiv y \not\equiv z \qquad x \subset y \subseteq z$ | ||
| - | < | + | |
| $x + y - z \qquad x * y / z \qquad x \times y \cdot z$ | $x + y - z \qquad x * y / z \qquad x \times y \cdot z$ | ||
| - | < | + | |
| $x \circ y \bullet z \qquad x \cup y \cap z \qquad x \sqcup y \sqcap z$ | $x \circ y \bullet z \qquad x \cup y \cap z \qquad x \sqcup y \sqcap z$ | ||
| - | < | + | |
| $x \vee y \wedge z\qquad x \pm y \mp z$ | $x \vee y \wedge z\qquad x \pm y \mp z$ | ||
| ==== Akzente ==== | ==== Akzente ==== | ||
| - | < | + | |
| - | $\hat a\qquad\check b\qquad\tilde c \qquad \acute d \qquad \grave$ | + | $ \hat a \qquad \check b \qquad \tilde c \qquad \acute d \qquad \grave |
| - | < | + | $ \dot f\qquad\ddot g\qquad\breve h \qquad \bar k \qquad \vec l$ |
| - | $\dot f\qquad\ddot g\qquad\breve h \qquad \bar k \qquad \vec l$ | + | $ \dot f\qquad\ddot g\qquad\breve h \qquad \bar k \qquad \vec l$ |
| - | < | + | $ \hat\imath \qquad \check\jmath$ |
| - | $\hat\imath \qquad \check\jmath$ | + | $ \hat\imath \qquad \check\jmath$ |
| - | < | + | $ \widehat x \qquad \widehat{xy} \qquad \widehat{xyz}$ |
| - | $\widehat x \qquad \widehat{xy} \qquad \widehat{xyz}$ | + | $ \widehat x \qquad \widehat{xy} \qquad \widehat{xyz}$ |
| - | < | + | $ \widetilde x \qquad \widetilde{xy} \qquad \widetilde{xyz}$ |
| - | $\widetilde x \qquad \widetilde{xy} \qquad \widetilde{xyz}$ | + | $ \widetilde x \qquad \widetilde{xy} \qquad \widetilde{xyz}$ |
| ==== Vektoren ==== | ==== Vektoren ==== | ||
| - | < | + | |
| - | $\alpha \cdot(\vec x + \vec y) = \alpha \cdot \vec x + \alpha \cdot \vec y$ | + | $ \alpha \cdot(\vec x + \vec y) = \alpha \cdot \vec x + \alpha \cdot \vec y$ |
| - | < | + | $ \vec x \cdot (\vec y \cdot \vec z) \not=(\vec x\cdot\vec y)\cdot \vec z$ |
| - | $\vec x \cdot (\vec y \cdot \vec z) \not=(\vec x\cdot\vec y)\cdot \vec z$ | + | $ \vec x \cdot (\vec y \cdot \vec z) \not=(\vec x\cdot\vec y)\cdot \vec z$ |
| - | < | + | $ \vec x\times (\vec y\times \vec z) \not= (\vec x \times \vec y) \times \vec z$ |
| - | $\vec x\times (\vec y\times \vec z) \not= (\vec x \times \vec y) \times \vec z$ | + | $ \vec x\times (\vec y\times \vec z) \not= (\vec x \times \vec y) \times \vec z$ |
| ==== Pfeile ==== | ==== Pfeile ==== | ||
| - | < | + | |
| - | \Leftrightarrow \qquad \uparrow\qquad\downarrow\qquad\nearrow$</ | + | \Leftrightarrow \qquad \uparrow\qquad\downarrow\qquad\nearrow$ |
| - | $\leftarrow \qquad \Leftarrow \qquad \leftrightarrow \qquad \Leftrightarrow \qquad \uparrow\qquad\downarrow\qquad\nearrow$ | + | $ \leftarrow \qquad \Leftarrow \qquad \leftrightarrow \qquad \Leftrightarrow \qquad \uparrow\qquad\downarrow\qquad\nearrow$ |
| - | < | + | $\longleftarrow\qquad\leftharpoonup \qquad \mapsto \qquad \leadsto$ |
| - | $\longleftarrow\qquad\leftharpoonup \qquad \mapsto \qquad \leadsto$ | + | $ \longleftarrow\qquad\leftharpoonup \qquad \mapsto \qquad \leadsto$ |
| - | < | + | $ (\mathcal{A} \Rightarrow \mathcal{B}) \Longleftrightarrow |
| - | (\lnot \mathcal{B} \Rightarrow \lnot \mathcal{A})$</ | + | (\lnot \mathcal{B} \Rightarrow \lnot \mathcal{A})$ |
| - | $(\mathcal{A} \Rightarrow \mathcal{B}) \Longleftrightarrow (\lnot \mathcal{B} \Rightarrow \lnot \mathcal{A})$ | + | $ (\mathcal{A} \Rightarrow \mathcal{B}) \Longleftrightarrow (\lnot \mathcal{B} \Rightarrow \lnot \mathcal{A})$ |
| - | < | + | $(\mathcal{A} \Longleftrightarrow \mathcal{B}) \Longleftrightarrow |
| - | (\mathcal{A}\Rightarrow \mathcal{B}) \wedge (\mathcal{B}\Rightarrow\mathcal{A})$ | + | (\mathcal{A}\Rightarrow \mathcal{B}) \wedge (\mathcal{B}\Rightarrow\mathcal{A})$ |
| - | </ | + | |
| - | $(\mathcal{A} \Longleftrightarrow \mathcal{B}) \Longleftrightarrow | + | $ (\mathcal{A} \Longleftrightarrow \mathcal{B}) \Longleftrightarrow |
| (\mathcal{A}\Rightarrow \mathcal{B}) \wedge (\mathcal{B}\Rightarrow\mathcal{A})$ | (\mathcal{A}\Rightarrow \mathcal{B}) \wedge (\mathcal{B}\Rightarrow\mathcal{A})$ | ||
| ==== Schriftartwechsel ==== | ==== Schriftartwechsel ==== | ||
| - | < | + | |
| - | $\forall x\in\mathbf{R}: | + | $ \forall x\in\mathbf{R}: |
| < | < | ||
| \mathbf{A} \cdot \mathbf{x} & = & | \mathbf{A} \cdot \mathbf{x} & = & | ||
| Zeile 184: | Zeile 188: | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| ==== Indizes und Hochstellung ==== | ==== Indizes und Hochstellung ==== | ||
| - | < | + | |
| - | $\displaystyle a_i \qquad a_{\displaystyle i_j} $ | + | $ \displaystyle a_i \qquad a_{\displaystyle i_j} $ |
| - | < | + | $ \displaystyle a^{\displaystyle i} \qquad a^{\displaystyle i^j} $ |
| - | $\displaystyle a^{\displaystyle i} \qquad a^{\displaystyle i^j} $ | + | $ \displaystyle a^{\displaystyle i} \qquad a^{\displaystyle i^j} $ |
| - | < | + | $ _1x_2 \qquad x^3_4 \qquad a^{b^\alpha_\beta}_{c^\gamma_\delta} \qquad F^1_2 \qquad F{}^1_2$ |
| - | $$x^{y^z} \qquad x^{{\displaystyle y}^{\displaystyle z}}}$$ | + | $ _1x_2 \qquad x^3_4 \qquad a^{b^\alpha_\beta}_{c^\gamma_\delta} \qquad F^1_2 \qquad F{}^1_2$ |
| - | < | + | |
| - | $_1x_2 \qquad x^3_4 \qquad a^{b^\alpha_\beta}_{c^\gamma_\delta} \qquad F^1_2 \qquad F{}^1_2$ | + | |
| - | < | + | Vergleiche |
| - | $\displaystyle _1x_2 = -\frac{p}{2}\pm \sqrt{\left(\frac{p}{2}\right)^2-q}$ | + | mit $ _1x_2 = -\frac{p}{2}\pm \sqrt{\left(\frac{p}{2}\right)^2-q}$ |
| ==== Brüche ==== | ==== Brüche ==== | ||
| - | < | + | |
| - | $\displaystyle \frac{1}{2} \qquad \frac{n+1}{3}$ | + | $ \displaystyle \frac{1}{2} \qquad \frac{n+1}{3}$ |
| - | < | + | $ \displaystyle |
| - | $\displaystyle \frac{x+y^2}{m+1} \qquad \frac{x+y^2}{m} + 1 \qquad x + \frac{y^2}{m}+1$ | + | $ \displaystyle \frac{x+y^2}{m+1} \qquad \frac{x+y^2}{m} + 1 \qquad x + \frac{y^2}{m}+1$ |
| - | < | + | $x + \frac{\displaystyle |
| $x + \frac{\displaystyle y^2}{\displaystyle m+1} \qquad x + y^\frac{\displaystyle 2}{\displaystyle m+1}$ | $x + \frac{\displaystyle y^2}{\displaystyle m+1} \qquad x + y^\frac{\displaystyle 2}{\displaystyle m+1}$ | ||
| - | < | + | |
| - | $\displaystyle x + \frac{y^2}{m+1} \qquad x + y^\frac{2}{m+1}$ | + | $ \displaystyle x + \frac{y^2}{m+1} \qquad x + y^\frac{2}{m+1}$ |
| - | < | + | $ \displaystyle \frac{\frac{x}{y}}{2} \qquad \frac{x}{\frac{y}{2}}$ |
| - | $\displaystyle \frac{\frac{x}{y}}{2} \qquad \frac{x}{\frac{y}{2}}$ | + | $ \displaystyle \frac{\frac{x}{y}}{2} \qquad \frac{x}{\frac{y}{2}}$ |
| - | < | + | |
| - | \frac{1}{x_2 + | + | |
| - | \frac{1}{x_3 + | + | \frac{1}{x_2 + |
| - | \frac{1}{x_4}}}}$</ | + | \frac{1}{x_3 + |
| + | \frac{1}{x_4}}}}$ | ||
| $x_0 + \frac{1}{x_1 + | $x_0 + \frac{1}{x_1 + | ||
| \frac{1}{x_2 + | \frac{1}{x_2 + | ||
| \frac{1}{x_3 + | \frac{1}{x_3 + | ||
| \frac{1}{x_4}}}}$ | \frac{1}{x_4}}}}$ | ||
| - | < | + | |
| - | \frac{\strut 1}{\displaystyle x_2 + | + | |
| - | \frac{\strut 1}{\displaystyle x_3 + | + | \frac{\strut 1}{\displaystyle x_2 + |
| - | \frac{\strut 1}{\displaystyle x_4}}}}$</ | + | \frac{\strut 1}{\displaystyle x_3 + |
| - | $\displaystyle x_0+\frac{1}{\displaystyle x_1 + | + | \frac{\strut 1}{\displaystyle x_4}}}}$ |
| + | |||
| + | $ \displaystyle x_0+\frac{1}{\displaystyle x_1 + | ||
| \frac{\strut 1}{\displaystyle x_2 + | \frac{\strut 1}{\displaystyle x_2 + | ||
| \frac{\strut 1}{\displaystyle x_3 + | \frac{\strut 1}{\displaystyle x_3 + | ||
| \frac{\strut 1}{\displaystyle x_4}}}}$ | \frac{\strut 1}{\displaystyle x_4}}}}$ | ||
| ==== Wurzeln ==== | ==== Wurzeln ==== | ||
| - | < | + | |
| - | $\sqrt 2 \qquad \sqrt{\displaystyle x^2-y^2} \qquad \sqrt{\alpha^2 + \beta^2 - \gamma^2}$ | + | $ \sqrt 2 \qquad \sqrt{\displaystyle x^2-y^2} \qquad \sqrt{\alpha^2 + \beta^2 - \gamma^2}$ |
| - | < | + | $ \sqrt[3]{2} \qquad \sqrt[n]{\sqrt\alpha + \sqrt\beta} \qquad \sqrt[n+1]{a^n + b^n}$ |
| - | $\sqrt[3]{2} \qquad \sqrt[n]{\sqrt\alpha + \sqrt\beta} \qquad \sqrt[n+1]{a^n + b^n}$ | + | $ \sqrt[3]{2} \qquad \sqrt[n]{\sqrt\alpha + \sqrt\beta} \qquad \sqrt[n+1]{a^n + b^n}$ |
| - | < | + | $ \sqrt a +\sqrt{b^2} +\sqrt c \qquad \sqrt{\mathstrut a} + \sqrt{\mathstrut b^2} + \sqrt{\mathstrut c}$ |
| - | $\sqrt a +\sqrt{b^2} +\sqrt c \qquad \sqrt{\mathstrut a} + \sqrt{\mathstrut b^2} + \sqrt{\mathstrut c}$ | + | $ \sqrt a +\sqrt{b^2} +\sqrt c \qquad \sqrt{\mathstrut a} + \sqrt{\mathstrut b^2} + \sqrt{\mathstrut c}$ |
| - | < | + | $ \sqrt{1+ \sqrt{1+ \sqrt{1+ \sqrt{1+ \sqrt{1 + \sqrt{1+x}}}}}}$ |
| - | $\sqrt{1+ \sqrt{1+ \sqrt{1+ \sqrt{1+ \sqrt{1 + \sqrt{1+x}}}}}}$ | + | $ \sqrt{1+ \sqrt{1+ \sqrt{1+ \sqrt{1+ \sqrt{1 + \sqrt{1+x}}}}}}$ |
| ==== Binominalkoeffizienten ==== | ==== Binominalkoeffizienten ==== | ||
| - | < | + | |
| ${n \choose 2} \qquad {n+1 \choose k} \qquad \frac{\displaystyle{n \choose k}}{\displaystyle 2}$ | ${n \choose 2} \qquad {n+1 \choose k} \qquad \frac{\displaystyle{n \choose k}}{\displaystyle 2}$ | ||
| - | < | + | |
| ${x \atop a+b} \qquad {n \atop k+1}$ | ${x \atop a+b} \qquad {n \atop k+1}$ | ||
| - | < | + | |
| - | $\displaystyle \sum_{{\scriptstyle 1 \le i \le p \atop \scriptstyle 1 \le j \le q}} a_{ij} b_{ji}$ | + | $ \displaystyle \sum_{{\scriptstyle 1 \le i \le p \atop \scriptstyle 1 \le j \le q}} a_{ij} b_{ji}$ |
| ==== Limes, Ableitungen ==== | ==== Limes, Ableitungen ==== | ||
| - | < | + | |
| - | $\displaystyle f\prime(x) = \lim_{\Delta x \to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$ | + | $ \displaystyle f\prime(x) = \lim_{\Delta x \to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$ |
| < | < | ||
| f(x) & = & \cos x \\ | f(x) & = & \cos x \\ | ||
| Zeile 251: | Zeile 259: | ||
| f’’(x) & = & -\cos x \\ | f’’(x) & = & -\cos x \\ | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| + | |||
| < | < | ||
| h(x) & = & f(x) \cdot g(x)\\ | h(x) & = & f(x) \cdot g(x)\\ | ||
| Zeile 259: | Zeile 268: | ||
| \frac{h(x)}{\mathrm{d}x} & = & f(x)\cdot\frac{g(x)}{\mathrm{d}x}+\frac{f(x)}{\mathrm{d}x} \cdot g(x) | \frac{h(x)}{\mathrm{d}x} & = & f(x)\cdot\frac{g(x)}{\mathrm{d}x}+\frac{f(x)}{\mathrm{d}x} \cdot g(x) | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| + | |||
| < | < | ||
| \mathbf{x} & = & \frac{1}{2} \mathbf{k} \cdot t^2 + \mathbf{v_0} \cdot t + \mathbf{x_0}\\ | \mathbf{x} & = & \frac{1}{2} \mathbf{k} \cdot t^2 + \mathbf{v_0} \cdot t + \mathbf{x_0}\\ | ||
| Zeile 264: | Zeile 274: | ||
| \ddot \mathbf{x} & = & \mathbf{k} | \ddot \mathbf{x} & = & \mathbf{k} | ||
| \end{eqnarray*}</ | \end{eqnarray*}</ | ||
| + | |||
| < | < | ||
| \mathbf{x} & = & \frac{1}{2} \mathbf{k} \cdot t^2 + \mathbf{v_0} \cdot t + x_0\\ | \mathbf{x} & = & \frac{1}{2} \mathbf{k} \cdot t^2 + \mathbf{v_0} \cdot t + x_0\\ | ||
| Zeile 274: | Zeile 285: | ||
| \ddot{\mathbf{x}} & = & \mathbf{k} | \ddot{\mathbf{x}} & = & \mathbf{k} | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| + | |||
| < | < | ||
| z (x, y) & = & xy\\ | z (x, y) & = & xy\\ | ||
| Zeile 284: | Zeile 296: | ||
| \frac{\partial z}{\partial y} & = & x | \frac{\partial z}{\partial y} & = & x | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| + | |||
| < | < | ||
| z (x, y) & = & \frac{xy}{x^2+y^2} \quad(\forall x, | z (x, y) & = & \frac{xy}{x^2+y^2} \quad(\forall x, | ||
| Zeile 295: | Zeile 308: | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| ==== Summen ==== | ==== Summen ==== | ||
| - | < | + | |
| - | \displaystyle{\frac{1}{4}-\frac{1}{9}+\frac{1}{16}-\frac{1}{25}+}~\cdots$</ | + | |
| - | $\sum\limits_{i=2}^{\infty} \frac{\displaystyle(-1)^i}{\displaystyle i^2}=\displaystyle{\frac{1}{4}-\frac{1}{9}+\frac{1}{16}-\frac{1}{25}+}~\cdots$ | + | \displaystyle{\frac{1}{4}-\frac{1}{9}+\frac{1}{16}-\frac{1}{25}+}~\cdots$ |
| - | < | + | |
| - | $\sum\limits_{i=1}^p \sum\limits_{j=1}^q\sum\limits_{k=1}^r a_{ij}b_{jk}c_{ki}$ | + | $ \sum\limits_{i=2}^{\infty} \frac{\displaystyle(-1)^i}{\displaystyle i^2}=\displaystyle{\frac{1}{4}-\frac{1}{9}+\frac{1}{16}-\frac{1}{25}+}~\cdots$ |
| - | < | + | |
| - | \sum\limits_{{\scriptstyle 1 \le i \le p \atop \scriptstyle 1 \le j \le q} \atop \scriptstyle 1 \le k \le r} | + | < |
| - | a_{ij} b_{jk} c_{ki}$</ | + | $ \sum\limits_{i=1}^p \sum\limits_{j=1}^q\sum\limits_{k=1}^r a_{ij}b_{jk}c_{ki}$ |
| - | $\sum\limits_{i=1}^p \sum\limits_{j=1}^q\sum\limits_{k=1}^r a_{ij}b_{jk}c_{ki} \qquad | + | < |
| \sum\limits_{{\scriptstyle 1 \le i \le p \atop \scriptstyle 1 \le j \le q} \atop \scriptstyle 1 \le k \le r} | \sum\limits_{{\scriptstyle 1 \le i \le p \atop \scriptstyle 1 \le j \le q} \atop \scriptstyle 1 \le k \le r} | ||
| a_{ij} b_{jk} c_{ki}$ | a_{ij} b_{jk} c_{ki}$ | ||
| + | </ | ||
| + | $ \sum\limits_{{\scriptstyle 1 \le i \le p \atop \scriptstyle 1 \le j \le q} \atop \scriptstyle 1 \le k \le r} a_{ij} b_{jk} c_{ki} $ | ||
| + | |||
| + | |||
| ==== Reihen ==== | ==== Reihen ==== | ||
| - | < | + | < |
| + | \begin{eqnarray*} | ||
| \sum_{i=0}^{\infty}(-1)^i \frac{1}{2i+1} & = & 1 - \frac{1}{3}+\frac{1}{5}-\cdots\\ | \sum_{i=0}^{\infty}(-1)^i \frac{1}{2i+1} & = & 1 - \frac{1}{3}+\frac{1}{5}-\cdots\\ | ||
| & = & \frac{\pi}{4} | & = & \frac{\pi}{4} | ||
| \end{eqnarray*}</ | \end{eqnarray*}</ | ||
| + | |||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| \sum_{i=0}^{\infty}(-1)^i \frac{1}{2i+1} & = & 1 - \frac{1}{3}+\frac{1}{5}-\cdots\\ | \sum_{i=0}^{\infty}(-1)^i \frac{1}{2i+1} & = & 1 - \frac{1}{3}+\frac{1}{5}-\cdots\\ | ||
| & = & \frac{\pi}{4} | & = & \frac{\pi}{4} | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| - | < | + | |
| + | < | ||
| + | \begin{eqnarray*} | ||
| \sum_{i=1}^{\infty}(-1)^{i+1} \frac{1}{i^2} & = & 1-\frac{1}{2^2} + \frac{1}{3^2} - \cdots\\ | \sum_{i=1}^{\infty}(-1)^{i+1} \frac{1}{i^2} & = & 1-\frac{1}{2^2} + \frac{1}{3^2} - \cdots\\ | ||
| & = & \frac{\pi^2}{12} | & = & \frac{\pi^2}{12} | ||
| - | \end{eqnarray*}</ | + | \end{eqnarray*} |
| + | </ | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| \sum_{i=1}^{\infty}(-1)^{i+1} \frac{1}{i^2} & = & 1-\frac{1}{2^2} + \frac{1}{3^2} - \cdots\\ | \sum_{i=1}^{\infty}(-1)^{i+1} \frac{1}{i^2} & = & 1-\frac{1}{2^2} + \frac{1}{3^2} - \cdots\\ | ||
| & = & \frac{\pi^2}{12} | & = & \frac{\pi^2}{12} | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| - | < | + | |
| + | < | ||
| + | \begin{eqnarray*} | ||
| \forall x \in \mathbf{R}: | \forall x \in \mathbf{R}: | ||
| 1 - x + \frac{x^2}{2!} - | 1 - x + \frac{x^2}{2!} - | ||
| Zeile 330: | Zeile 354: | ||
| (-1)^i\frac{x^i}{i!} | (-1)^i\frac{x^i}{i!} | ||
| \end{eqnarray*}</ | \end{eqnarray*}</ | ||
| + | |||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| \forall x \in \mathbf{R}: | \forall x \in \mathbf{R}: | ||
| Zeile 337: | Zeile 362: | ||
| (-1)^i\frac{x^i}{i!} | (-1)^i\frac{x^i}{i!} | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| - | < | + | |
| + | < | ||
| + | \begin{eqnarray*} | ||
| \forall x \in \mathbf{R}: | \forall x \in \mathbf{R}: | ||
| 1 + x + \frac{x^2}{2!} + | 1 + x + \frac{x^2}{2!} + | ||
| Zeile 349: | Zeile 376: | ||
| & | & | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| - | < | + | |
| - | $\displaystyle \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k^2} = \frac{\pi^2}{6}$ | + | < |
| + | </ | ||
| + | $ \displaystyle \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k^2} = \frac{\pi^2}{6}$ | ||
| ==== Integrale ==== | ==== Integrale ==== | ||
| - | < | + | < |
| - | $\int\limits_0^{\frac{\pi}{3}}\cos x~\mathrm{d}x \qquad$ | + | $ \int\limits_0^{\frac{\pi}{3}}\cos x~\mathrm{d}x \qquad$ |
| - | $\int\int_D\limits f(x, y) \mathrm{d}x \mathrm{d}y \qquad | + | $ \int\int_D\limits f(x, y) \mathrm{d}x \mathrm{d}y \qquad |
| \int\!\!\!\int_D\limits f(x, y)\, | \int\!\!\!\int_D\limits f(x, y)\, | ||
| </ | </ | ||
| - | $\int\limits_{-\infty}^{\infty}\displaystyle{\frac{1}{1 + x^2}} \mathrm{d}x \qquad$ | + | $ \int\limits_{-\infty}^{\infty}\displaystyle{\frac{1}{1 + x^2}} \mathrm{d}x \qquad$ |
| - | $\int\limits_0^{\frac{\pi}{3}}\cos x~\mathrm{d}x \qquad$ | + | $ \int\limits_0^{\frac{\pi}{3}}\cos x~\mathrm{d}x \qquad$ |
| - | $\int\int_D\limits f(x, y) \mathrm{d}x \mathrm{d}y \qquad | + | $ \int\int_D\limits f(x, y) \mathrm{d}x \mathrm{d}y \qquad |
| \int\!\!\!\int_D\limits f(x, y)\, | \int\!\!\!\int_D\limits f(x, y)\, | ||
| - | < | + | |
| - | $\prod_{i=1}^n i = n! \qquad \prod\limits_{i=1}^n i = n! \qquad \prod\nolimits_{i=1}^n i = n!$ | + | |
| - | < | + | $ \prod_{i=1}^n i = n! \qquad \prod\limits_{i=1}^n i = n! \qquad \prod\nolimits_{i=1}^n i = n!$ |
| - | $\displaystyle{{n \choose k}} = \frac{\displaystyle\prod_{i=1}^n i} {\displaystyle\prod_{i=1}^k i\cdot \prod_{i=1}^{n-k} i}$ | + | |
| + | | ||
| + | $ \displaystyle{{n \choose k}} = \frac{\displaystyle\prod_{i=1}^n i} {\displaystyle\prod_{i=1}^k i\cdot \prod_{i=1}^{n-k} i}$ | ||
| ==== Funktionen ==== | ==== Funktionen ==== | ||
| - | < | + | |
| - | $\displaystyle \lim_{x \to 0} \frac{\sin x}{x}=1$ | + | $ \displaystyle \lim_{x \to 0} \frac{\sin x}{x}=1$ |
| - | $\displaystyle \int \frac{\mathrm{d}x} {\sin a x \cos a x} = \frac{1}{a} \ln \tan a x$ | + | $ \displaystyle \int \frac{\mathrm{d}x} {\sin a x \cos a x} = \frac{1}{a} \ln \tan a x$ |
| - | < | + | $ \displaystyle \int \frac{\mathrm{d}x} {\sin a x \cos a x} = \frac{1}{a} \ln \tan a x$ |
| - | $\arcsin x = \left[ \arccos \sqrt{1 - x^2}\right]$ | + | |
| + | $ \arcsin x = \left[ \arccos \sqrt{1 - x^2}\right]$ | ||
| ==== Matrizen ==== | ==== Matrizen ==== | ||
| - | < | + | < |
| - | a_{11} & a_{12} & \cdots & a_{1n} \\ | + | a_{11} & a_{12} & \cdots & a_{1n} |
| - | a_{21} & a_{22} & \cdots & a_{21} \\ | + | a_{21} & a_{22} & \cdots & a_{21} |
| - | \vdots & \vdots & \ddots & \vdots \\ | + | \vdots & \vdots & \ddots & \vdots |
| a_{m1} & a_{m2} & \cdots & a_{mn} | a_{m1} & a_{m2} & \cdots & a_{mn} | ||
| \end{array}$</ | \end{array}$</ | ||
| - | $\begin{array}{|cccc|} | + | $ \begin{array}{|cccc|} |
| - | a_{11} & a_{12} & \cdots & a_{1n} \\ | + | a_{11} & a_{12} & \cdots & a_{1n} |
| - | a_{21} & a_{22} & \cdots & a_{21} \\ | + | a_{21} & a_{22} & \cdots & a_{21} |
| - | \vdots & \vdots & \ddots & \vdots \\ | + | \vdots & \vdots & \ddots & \vdots |
| a_{m1} & a_{m2} & \cdots & a_{mn} | a_{m1} & a_{m2} & \cdots & a_{mn} | ||
| \end{array}$ | \end{array}$ | ||
| - | < | + | |
| - | \left\{\begin{array}{cccc} | + | < |
| - | \Gamma_{11} & \Gamma_{12} & \cdots & | + | \Gamma_{11} & \Gamma_{12} & \cdots & \Gamma_{1n}\\\\ |
| - | \Gamma_{1n}\\ | + | \Gamma_{21} & \Gamma_{22} & \cdots & \Gamma_{2n}\\\\ |
| - | \Gamma_{21} & \Gamma_{22} & \cdots & | + | \vdots & \vdots & \ddots & \vdots\\\\ |
| - | \Gamma_{2n}\\ | + | \Gamma_{m1} & \Gamma_{m2} & \cdots & \Gamma_{mn} |
| - | \vdots & \vdots & \ddots & | + | \end{array}\right\}$ |
| - | \vdots\\ | + | </ |
| - | \Gamma_{m1} & \Gamma_{m2} & \cdots & | + | |
| - | \Gamma_{mn} | + | $ \left\{\begin{array}{cccc} |
| - | \end{array}\right\} | + | \Gamma_{11} & \Gamma_{12} & \cdots & \Gamma_{1n}\\\\ |
| - | \end{displaymath}$</ | + | \Gamma_{21} & \Gamma_{22} & \cdots & \Gamma_{2n}\\\\ |
| - | $\begin{displaymath} | + | \vdots & \vdots & \ddots & \vdots\\\\ |
| - | \left\{\begin{array}{cccc} | + | \Gamma_{m1} & \Gamma_{m2} & \cdots & \Gamma_{mn} |
| - | \Gamma_{11} & \Gamma_{12} & \cdots & | + | \end{array}\right\}$ |
| - | \Gamma_{1n}\\ | + | |
| - | \Gamma_{21} & \Gamma_{22} & \cdots & | + | < |
| - | \Gamma_{2n}\\ | + | x & \textrm{fuer } x \ge 0\\\\ |
| - | \vdots & \vdots & \ddots & | + | -x & \textrm{fuer } x < 0\\\\ |
| - | \vdots\\ | + | \end{array}\right\}$ |
| - | \Gamma_{m1} & \Gamma_{m2} & \cdots & | + | </ |
| - | \Gamma_{mn} | + | |
| - | \end{array}\right\} | + | $ |x|= \left\{ \begin{array}{ll} |
| - | \end{displaymath}$ | + | x & \textrm{fuer } x \ge 0\\\\ |
| - | < | + | -x & \textrm{fuer } x < 0\\\\ |
| - | x & \textrm{f¨ur } x \ge 0\\ | + | \end{array}\right\}$ |
| - | -x & \textrm{f¨ur } x < 0\\ | + | |
| - | \end{array}\right.$</ | + | < |
| - | $|x|= \left\{ \begin{array}{ll} | + | |
| - | x & \textrm{f¨ur } x \ge 0\\ | + | |
| - | -x & \textrm{f¨ur } x < 0\\ | + | |
| - | \end{array}\right.$ | + | |
| - | < | + | |
| \begin{array}{c@{}c@{}c} | \begin{array}{c@{}c@{}c} | ||
| \begin{array}{|cc|} | \begin{array}{|cc|} | ||
| \hline | \hline | ||
| - | a_{11} & a_{12} \\ | + | a_{11} & a_{12} |
| - | a_{21} & a_{22} \\ | + | a_{21} & a_{22} |
| \hline | \hline | ||
| - | \end{array} & 0 & 0 \\ | + | \end{array} & 0 & 0 \\\\ |
| 0 & \begin{array}{|ccc|} | 0 & \begin{array}{|ccc|} | ||
| \hline | \hline | ||
| - | b_{11} & b_{12} & b_{13}\\ | + | b_{11} & b_{12} & b_{13}\\\\ |
| - | b_{21} & b_{22} & b_{23}\\ | + | b_{21} & b_{22} & b_{23}\\\\ |
| - | b_{31} & b_{32} & b_{33}\\ | + | b_{31} & b_{32} & b_{33}\\\\ |
| \hline | \hline | ||
| - | \end{array} & 0 \\ | + | \end{array} & 0 \\\\ |
| 0 & 0 & \begin{array}{|cc|} | 0 & 0 & \begin{array}{|cc|} | ||
| \hline | \hline | ||
| - | c_{11} & c_{12} \\ | + | c_{11} & c_{12} |
| - | c_{21} & c_{22} \\ | + | c_{21} & c_{22} |
| \hline | \hline | ||
| - | \end{array} \\ | + | \end{array} |
| \end{array} | \end{array} | ||
| - | \right)$</ | + | \right)$ |
| - | $\left( | + | </ |
| + | |||
| + | $ \left( | ||
| \begin{array}{c@{}c@{}c} | \begin{array}{c@{}c@{}c} | ||
| \begin{array}{|cc|} | \begin{array}{|cc|} | ||
| \hline | \hline | ||
| - | a_{11} & a_{12} \\ | + | a_{11} & a_{12} |
| - | a_{21} & a_{22} \\ | + | a_{21} & a_{22} |
| \hline | \hline | ||
| - | \end{array} & 0 & 0 \\ | + | \end{array} & 0 & 0 \\\\ |
| 0 & \begin{array}{|ccc|} | 0 & \begin{array}{|ccc|} | ||
| \hline | \hline | ||
| - | b_{11} & b_{12} & b_{13}\\ | + | b_{11} & b_{12} & b_{13}\\\\ |
| - | b_{21} & b_{22} & b_{23}\\ | + | b_{21} & b_{22} & b_{23}\\\\ |
| - | b_{31} & b_{32} & b_{33}\\ | + | b_{31} & b_{32} & b_{33}\\\\ |
| \hline | \hline | ||
| - | \end{array} & 0 \\ | + | \end{array} & 0 \\\\ |
| 0 & 0 & \begin{array}{|cc|} | 0 & 0 & \begin{array}{|cc|} | ||
| \hline | \hline | ||
| - | c_{11} & c_{12} \\ | + | c_{11} & c_{12} |
| - | c_{21} & c_{22} \\ | + | c_{21} & c_{22} |
| \hline | \hline | ||
| - | \end{array} \\ | + | \end{array} |
| \end{array} | \end{array} | ||
| \right)$ | \right)$ | ||
| - | + | < | |
| + | </ | ||
