Mathematik

From SkyWiki

Revision as of 10:07, 16 September 2008 (view source) Martin (Talk | contribs) ← Previous diff		Current revision (09:39, 18 September 2008) (view source) Martin (Talk | contribs) (→Geometrie)
(25 intermediate revisions not shown.)
Line 1:		Line 1:
-	[[category:remember]]
-
	== Analysis ==		== Analysis ==
-	<math>a x^2 + b x + c = 0</math>	+	<math>a \cdot x^2 + b \cdot x + c = 0 \quad \Rightarrow \quad x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4\cdot a\cdot c}} {2\cdot a}</math>
-	=> <math>x_{1/2} = \frac{(-b +/- sqrt(b^2 - 4ac))} {2a}</math>	+	'''Binome, Trinome'''
		+	{|
		+	|<math>(a+b)^2=a^2+2ab+b^2</math>
		+	|<math>(a+b)^3=a^3+3a^2b+3ab^2+b^3</math>
		+	|-
		+	|<math>(a-b)^2=a^2-2ab+b^2</math>
		+	|<math>(a-b)^3=a^3-3a^2b+3ab^2-b^3</math>
		+	|-
		+	|<math>(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc</math>
		+	|<math>a^3+b^3=(a+b)(a^2-ab+b^2)</math>
		+	|-
		+	|<math>a^2-b^2=(a+b)(a-b)</math>
		+	|<math>a^3-b^3=(a-b)(a^2+ab+b^2)</math>
		+	|}
		+	<math>a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^2+\cdots+ab^{n-2}+b^{n-1})</math>
		+
		+	<math>n!=1\cdot2\cdot3\cdot...\cdot(n-1)\cdot n,\ n\in N\setminus\{1\}</math>
	== Geometrie ==		== Geometrie ==
-	<math>U = \pi*D</math>	+	<math>U = \pi \cdot D = 2 \cdot \pi \cdot r</math>
-	<math>U = 2 r \pi</math>	+
		+	Satz des Pythagoras:
		+
		+	<math>a^2+b^2=c^2</math>
		+
		+	== komplexe Zahlen ==
		+
		+	'''Normalform'''
		+	{|
		+	|<math>z = x\, +\, \mathrm i y \quad</math>
		+	|<math>\mathrm i</math>:=<math>\sqrt{-1} \qquad i^2 = -1</math>
		+	|-
		+	|<math>x = \operatorname{Re}\, z</math>
		+	|<math>y = \operatorname{Im}\, z</math>
		+	|-
		+	|Summe
		+	|<math>z_1 + z_2 = (x_1 + x_2) + \mathrm i(y_1 + y_2)</math>
		+	|-
		+	|Produkt
		+	|<math>z_1 \cdot z_2 = (x_1x_2-y_1y_2) + \mathrm i(x_1y_2 + x_2_y_1)</math>
		+	|}
		+
		+	'''Polarform''':
		+	{|
		+	|<math>z = |z| \cdot E(\varphi)</math>
		+	|<math>E(\varphi) = |z| \cdot(\cos\varphi + \mathrm i\,\sin\varphi)</math>
		+	|-
		+	|<math>\varphi = arc(z)</math>
		+	|<math>\tan\varphi = \frac y x</math>
		+	|-
		+	|Produkt:
		+	|<math>z_1 \cdot z_2 = |z_1||z_2| \cdot E(\varphi_1+\varphi_2) = |z_1||z_2| (\cos(\varphi_1+\varphi2) + \mathrm i\, \sin(\varphi_1+\varphi_2))</math>
		+	|-
		+	|Quotient:
		+	|<math>\frac{z_1}{z_2} = \frac{|z_1|}{|z_2|} E(\varphi_1-\varphi_2) = \frac{|z_1|}{|z_2|} (\cos(\varphi_1-\varphi2) + \mathrm i\, \sin(\varphi_1-\varphi_2))</math>
		+	|-
		+	|Potenz:
		+	|<math>{z}^n = |z|^n E(n\, \varphi) = |z|^n (\cos\, n \varphi + \mathrm i\,\sin\, n \varphi)</math>
		+	|}
		+
		+	'''<math>E(\varphi)</math>-Rechenregeln:'''
		+	{|
		+	|<math>E(\varphi+k\cdot2\phi) = E(\varphi)</math>
		+	|<math>|E(\varphi)| = 1</math>
		+	|-
		+	|<math>E(\varphi_1)\cdot E(\varphi_2) = E(\varphi_1 + \varphi_2)</math>
		+	|<math>E(\varphi)\cdot E(-\varphi) = 1</math>
		+	|-
		+	|<math>[E(\varphi)]^n = E(N\, \varphi)</math>
		+	|-
		+	|<math>E(\varphi) = e^{\mathrm i\varphi}</math>
		+	|<math>e^{2\pi\mathrm i} = 1</math>
		+	|}
		+
		+	'''konjungierte komplexe Zahlen''':
		+
		+	{|
		+	|<math>\bar z = \bar{x + \mathrm i y} = x - \mathrm i y</math>
		+	|-
		+	|<math>\bar{z_1+z_2} = \bar{z_1} + \bar{z_2}</math>
		+	|<math>\bar{z_1-z_2} = \bar{z_1} - \bar{z_2}</math>
		+	|-
		+	|<math>\bar{z_1 \cdot z_2} = \bar{z_1} \cdot \bar{z_2}</math>
		+	|<math>\bar{\frac{z_1}{z_2}} = \frac{\bar{z_1}}{\bar{z_2}}</math>
		+	|-
		+	|<math>\bar{z^n} = (\bar z)^n</math>
		+	|<math>z \cdot \bar z = |z|^2</math>
		+	|<math>\bar{(\bar z)} = z</math>
		+	|-
		+	|<math>z + \bar z = 2 \cdot \operatorname Re(z)</math>
		+	|<math>z - \bar z = 2\mathrm i \cdot \operatorname Im(z)</math>
		+	|}
		+
		+
		+	[[category:Mathematik]]
		+	[[category:remember]]

---

Current revision

Analysis

Binome, Trinome

Geometrie

Satz des Pythagoras:

komplexe Zahlen

Normalform

	:=
Summe
Produkt

Polarform:

Produkt:
Quotient:
Potenz:

-Rechenregeln:

konjungierte komplexe Zahlen: