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<= n$ ist $tau_(i j)$ durch $tau_i j(l) = l$, falls $l in.not {i, j}$ sowie $tau_(i j) (i) = j$ und $tau_(i j)(j) = i$ ein Element von $S_n$ definiert. Zeigen Sie: $ angle.l {tau_(i j) : 1 <= i < j <= n} angle.r = S_n $ +// +//Hinweis: Wie würden Sie ein Bücherregal sortieren. +// +//Da $S_n$ genau alle bijektiven Abbildungen der Menge $N = {1, 2, 3, ..., n}$ auf sich selbst enthaelt und $tau_(i, j) in S_n$ ist $tau_(i j)$ bijektiv. +// +//Fuer $G := angle.l{tau_(i j) : 1 <= i < j <= n}angle.r$ gelten die Gruppenaxiome. +//Also ist $i d_N$ Element der Gruppe als neutrales Element. $tau_(i j)^2$ ist das inverse Element der Gruppe. +// +// +//Jede Verknuepfung von bijektiven Abbildungen auf eine Gruppe $G$ ist wieder eine bijektive Abbildung auf $G$. Daher ist $angle.l {tau_(i j) : 1 <= i < j <= n} angle.r subset S_n$ +// +//Nun gilt es zu zeigen, dass $S_n subset angle.l { tau_(i j) : 1 <= i < j <= n}angle.r$. Also, dass $ forall f in S_n: f in angle.l{ tau_(i j) : 1 <= i < j <= n}angle.r $ +// +//Der Defintionsbereich von $f in S_n$ und $g in G$ ist identisch. Es bleibt also zu zeigen, dass $ forall f in S_n exists g in G: f(N) = g(N) $ +// +//Stets laesst sich $g$ wie folgt konstruieren: +// +//Sei $g_0 = id_N$. Fuer das kleinste $n_k$ mit $f(n_k) != n_k$ kann man stets $g_1 = tau_(g_0 (n_k) f (n_k)) compose g_0$ konstruieren, da $tau_(i j) = tau_(j i)$. +// +//Man erhaelt nun also $f(n) = g_1 (n)$ + += Aufgabe 2 + +Sei $G$ eine beliebige Untergruppe der ganzen Zahlen $(ZZ, +)$. Zeigen Sie, dass eine ganze Zahl $n >= 0$ mit $G = n Z$ existiert. + +Hinweis: Jede nicht-leere Teilmenge der natürlichen Zahlen hat ein kleinstes Element. + +Sei $G$ eine beliebige Untergruppe der ganzen Zahlen $(ZZ, +)$. + +Falls $G$ eine triviale Untergruppe ist, dann liefert $n = 0$ bzw. $n = 1$ die Loesung. + +Da $G$ nun eine nicht-leere Untergruppe ist, enthält $G$ mindestens ein Element ungleich Null. +Da $G$ eine Gruppe unter der Addition ist, enthält $G$ auch entsprechende inverse Elemente, wenn es positive enthält. +Wir betrachten die Menge $G sect NN = S$, also die positiven Elemente, die in $G$ liegen. +Nach dem Hinweis hat jede nicht-leere Teilmenge der natürlichen Zahlen ein kleinstes Element. +Sei $d$ das kleinste Element, das in $S$ enthalten ist. + +Jetzt zeigen wir, dass $G = d ZZ$. + +Wenn $s$ ein beliebiges Element in $S$ ist, können wir $s$ in der maximal gekuerzten Form $s = k d + r$ schreiben, wobei $0 <= r < d$ und $k in ZZ$. +Aus der Gruppeneigenschaft von $G$ ist $r in G$, da $r = s - k d$ und $s, k d in G$. +Da $d$ das kleinste Element in $S$ ist, muss $r$ gleich Null sein, sonst würde $0 < r < d$, was im Widerspruch zu unserer Annahme, dass $d = min(S)$ ist, wäre. +Somit haben wir $s = k d$, was zeigt, dass $S$ von $d$ erzeugt wird. + +Da $s$ beliebig war, folgt $S = d ZZ$. + +Daher existiert eine ganze Zahl $n >= 0$, sodass $G = n ZZ$, da G alle Elemente von S und deren Inverse enthaelt. + +$qed$ + += Aufgabe 3 + +Eine ausfuerliche Begruendung warum der angesprochene Isomorphismus existiert steht auf der Abgabe in Papierform. + + +1. Wie viele verschiedene Gruppen mit 4 Elementen gibt es? + +Es gibt zwei Gruppen. Namentlich die Klein-4-Gruppe und die zyklische $Z_4$ Gruppe. +Jede weitere Gruppe mit $|G| = 4$ ist isomorph zu einer dieser beiden Gruppen. + +#stack( + dir: ltr, + table( + columns: 5, + [*$Z_4$*], [*0*], [*1*], [*2*], [*3*], + [*0*], [0], [1], [2], [3], + [*1*], [1], [2], [3], [0], + [*2*], [2], [3], [0], [1], + [*3*], [3], [0], [1], [2], + ), + + h(10pt), + + table( + columns: 5, + [*$K_4$*], [*0*], [*1*], [*2*], [*3*], + [*0*], [0], [1], [2], [3], + [*1*], [1], [0], [3], [2], + [*2*], [2], [3], [0], [1], + [*3*], [3], [2], [1], [0], + ), + +) + + +2. Wie viele verschiedene Gruppen mit 6 Elementen gibt es? + +Es gibt zwei Gruppen. Namentlich die zyklische $Z_6$ Gruppe und die $S_3$ Gruppe. +Jede weitere Gruppe mit $|G| = 6$ ist isomorph zu einer dieser beiden Gruppen. + + +#stack( + dir: ltr, + table( + columns: 7, + [*$Z_6$*], [*0*], [*1*], [*2*], [*3*], [*4*], [*5*], + [*0*], [0], [1], [2], [3], [4], [5], + [*1*], [1], [2], [3], [4], [5], [0], + [*2*], [2], [3], [0], [1], [0], [1], + [*3*], [3], [4], [5], [0], [1], [2], + [*4*], [4], [5], [0], [1], [2], [3], + [*5*], [5], [0], [1], [2], [3], [4], + ), + h(10pt), + + table( + columns: 7, + [*$S_3$*], [*0*], [*1*], [*2*], [*3*], [*4*], [*5*], + [*0*], [0], [1], [2], [3], [4], [5], + [*1*], [1], [2], [0], [4], [5], [3], + [*2*], [2], [0], [1], [5], [3], [4], + [*3*], [3], [5], [4], [0], [2], [1], + [*4*], [4], [3], [5], [1], [0], [2], + [*5*], [3], [4], [5], [0], [1], [2], +)) + +3. Finden Sie alle Normalteiler der Permutationsgruppe $S_3$. + +Sei $S_3 = {e, (12), (13), (23), (123), (132)} = G$ + +Nach Lagrange gilt $|G| = |G slash H||H|$. Da $|G| = 6$ muss $|H| in {1, 2, 3, 6}$. +Kandidaten fuer Normalteiler sind somit die trivialen Untergruppen ${e}, G$ und ${(12), e}, {(13), e}, {(23), e}, {e, (123), (132)}$ + +Die trivialen Untergruppen sind immer ein Normalteiler. +Sei $H$ eine Untergruppe von $G$ mit $|H| = 2$. Dann kann H kein Normalteiler von $G$ sein, da eine Spiegelung nicht kommutativ ist mit einer anderen Spiegelung ist. + +$A_3$ ist ein Normalteiler, da eine Spiegelung und dann eine Drehung des Dreiecks gleich zu einer Drehung in die andere Richtung und dann der Spiegelung ist. + +Eigenschaft eines Normalteilers: $g H = H g$ + +Die Normalteiler sind somit: + +- ${e}$ +- $A_3 = {e, (123), (132)}$ +- $S_3 = G$ + += Aufgabe 4 + +// Schritte der Argumentation +// +// - Gitter begruenden +// - Jede Zelle ist zueneinander symetrisch +// - Falls ein Punkt in dieser Zelle liegt, dann auch in der am Urspung (durch Gruppeneigenschaft) +// - Die Zelle hat die Form eines Parralelogramms +// - Jeder Punkt in einem Parralelogramm hat einen kleineren Abstand zu einem der Eckpunkte als die groesste Seitenlaenge (Siehe Lemma L1) +// + +Laut Vorlesung ist $Z^2 = {(x, y) : x, y ∈ ZZ}$ eine Gruppe. Sei $U ⊂ Z^2$ eine nicht triviale Untergruppe. In dieser gibt es ein Element $v != 0$ mit minimalem Abstand zum Ursprung $(0, 0)$. Angenommen $⟨v⟩ != U$. Dann gibt es ein weiteres Element $0 != w in U$, dass nicht in $⟨v⟩$ enthalten ist und minimalen Abstand zum Ursprung hat. Zeigen Sie, dass $⟨{v, w}⟩= U$ ist. + +Hinweis: Der Abstand von $(x, y)$ zum Ursprung ist $ sqrt(x^2 + y^2 )$. + +Sei $a = vec(a, b)$ ein beliebiges Element von $U$. Nun gilt es zu zeigen, dass $a = z_1 v + z_2 w$ (Darstellung durch Linearkombination von $v, w$) gilt, wobei $z_1, z_2 in ZZ$. + +Zuerst laesst sich feststellen, dass $v$ und $w$ linearunabhaengig voneinander sein muessen, da sonst nach dem Argument aus Aufgabe 2 die Annahme, dass $v, w$ minimalen Abstand vom Ursprung haben und $w in.not angle.l v angle.r$, ein Widerspruch erzeugen wuerde. + +Angenommen $a in.not angle.l v, w angle.r$. Dann liegt $a$ in einem durch $v$ und $w$, ausgehend von einem Punkt $p in angle.l v, w angle.r$, aufgespanntem Parralelogramm. Dies folgt aus der Gitterstuktur von $U$, welche durch Kombination von $w$ und $v$ erzeugt wird. Da $a$ ein Element der Untergruppe $U$ ist, existiert $a'$ auch in dem Parralelogramm welches ausgehend vom Ursprung durch $v$ und $w$ aufgespannt wird. Die beiden Parralelogramme und die relativen Positionen von $a, a'$ sind hier, wider durch die Gruppeneigenschaft, identisch. + +/// Geometrisches Lemma fuer Punkte in einem Parralelogramm ////////////////////////// +Nun wird folgendes Lemma gezeigt: + +Jeder Punkt in einem Parralelogramm hat einen kuerzeren Abstand zu einem der vier Punkte, welche es aufspannen, als $max(a, b)$ wobei $a, b$ die Seitenlaengen sind. + +Fuer den Beweis laesst sich feststellen, dass um jeden Eckpunkt des Parralelogramms eine offene Kreisscheibe mit Radius $r = max(|v|, |w|)$ gelegt werden kann. Diese Kreisscheiben ueberdecken das Parralelogramm vollstaendig. Das kommt daher, dass ein Parralelogramm sich durch die verbindenden Geraden von jeweils gegenueberliegenden Eckpunkten in vier Dreieicke aufteilen laesst. +Nun liegt jedes dieser Dreiecke vollstaendig in der jeweiligen Kreisscheibe bis auf maximal zwei Eckpunkte, welche aber in einer anderen Scheibe liegen. + +Die Aussage, dass der Punkt in einer offenen Kreisscheibe um einen Eckpunkt liegt ist aequivalent zu der, welche wir zeigen wollten. +/////////////////////////////////////// + +Nach Lemma existiert ein Vektor $t$, sodass $|t| < max(|v|, |w|)$ und $a' = vec(t_x, t_y)$. Dies steht im Wiederspruch zu der Annahme, dass $v$ und $w$ die Elemente der Gruppe sind, welche den kleinsten Abstand zum Urspung haben. + +Also muss $t = 0$ gelten. 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2 + 9 ZZ, 4 + 9 ZZ, 5 + 9 ZZ, 7 + 9 ZZ, 8 + 9 ZZ} subset ZZ slash 9 ZZ $ + + jeweils eine Gruppe bezueglich Multiplikation sind. + + ////////////////////////////// + Geforderte Eigenschaften an eine Gruppe: + + - Abgeschlossenheit: Das Produkt zweier Elemente muss wieder in der Gruppe sein. + + - Inverses Element: Jedes Element muss ein multiplikatives Inverses in der Gruppe haben. + + - Assoziativität: Diese Eigenschaft wird von den ganzen Zahlen vererbt. $checkmark$ + + - Neutrales Element: Es muss ein neutrales Element in der Gruppe geben. Dieses ist hier die 1. $checkmark$ + + Für $(ZZ slash 5 ZZ)^*$ gibt es durch Modulo-Rechnen folgende Elemente in der Gruppe $E = {1, 2, 3, 4}$. + + Das Produkt zweier Elemente aus $E$ ergibt wieder ein Element in dieser Menge: + + $ 1 dot 1 = 1, 1 dot 2 = 2, 1 dot 3 = 3, 1 dot 4 = 4 $ + + $ 2 dot 2 = 4, 2 dot 3 = 1, 2 dot 4 = 3 $ + + $ 3 dot 3 = 4, 3 dot 4 = 2 $ + + $ 4 dot 4 = 1 $ + + Jedes Element hat ein Inverses: + + $ 1^(-1) = 1, 2^(-1) = 3, 3^(-1) = 2, 4^(-1) = 4 $ + + Für $(ZZ slash 9 ZZ)^*$ gibt es durch Modulo-Rechnen folgende Elemente in der Gruppe $E = {1, 2, 4, 5, 7, 8}$. + + Das Produkt zweier Elemente aus $E$ ergibt wieder ein Element in dieser Menge: + + $ 1 dot 1 = 1, 1 dot 2 = 2, 1 dot 4 = 4, 1 dot 5 = 5, 1 dot 7 = 7, 1 dot 8 = 8 $ + + $ 2 dot 2 = 4, 2 dot 4 = 8, 2 dot 5 = 1, 2 dot 7 = 5, 2 dot 8 = 7 $ + + $ 4 dot 4 = 7, 4 dot 5 = 2, 4 dot 7 = 1, 4 dot 8 = 5 $ + + $ 5 dot 5 = 7, 5 dot 7 = 8, 5 dot 8 = 4 $ + + $ 7 dot 7 = 4, 7 dot 8 = 2 $ + + $ 8 dot 8 = 1 $ + + Jedes Element hat ein Inverses: + + $ 1^(-1) = 1, 2^(-1) = 5, 4^(-1) = 7, 5^(-1) = 2, 7^(-1) = 4, 8^(-1) = 8 $ + + Damit ist alles gezeigt. $qed$ + ++ Finden Sie ein Element $x in (ZZ slash 9 ZZ)^*$, sodass $angle.l {x} angle.r = (ZZ slash 9 ZZ)^*$ gilt. + + Nach Aufgabe 1a ist hat die Gruppe $(ZZ slash 9 ZZ)^*$ folgende Elemente $1, 2, 4, 5, 7, 8$. Es gilt $abs((ZZ slash 9 ZZ)^*) = 6$. + + Durch Testen jedes Elements finden wir, dass $x = 2$ die + gesamte Gruppe erzeugt, da: + $ 2^1 = 2 , quad 2^2 = 4 , quad 2^3 = 8 , quad 2^4 = 7 , quad 2^5 = 5 , quad 2^6 = 1 . $ + + Somit ist $x = 2$ ein Generator von $(bb(Z) \/ 9 bb(Z))^(\*)$, und wir haben + $angle.l 2 angle.r = (bb(Z) \/ 9 bb(Z))^(\*)$. + ++ Geben Sie einen Isomorphismus zwischen $(ZZ slash 9 ZZ, dot.op)$ und $(ZZ slash 6 ZZ, +)$ an. + + Ein Isomorphismus zwischen den Gruppen $(bb(Z) \/ 9 bb(Z) , dot.op)$ und + $(bb(Z) \/ 6 bb(Z) , +)$ kann durch die Funktion + $ phi : (bb(Z) \/ 9 bb(Z))^(\*) arrow.r bb(Z) \/ 6 bb(Z) $ gegeben + werden, wobei $phi (x)$ die Ordnung von $x$ in $(bb(Z) \/ 9 bb(Z))^(\*)$ + modulo 6 ist. + + Wählen wir nach Aufgabe 1b $x = 2$ als Generator von $(bb(Z) \/ 9 bb(Z))^(\*)$, so + laesst sich $phi$ definieren durch $ phi (2^k) = k mod 6 . $ + + Diese Zuordnung ist ein Isomorphismus, da sie bijektiv ist und die + Gruppenoperationen respektiert: + + $ phi (2^k dot.op 2^m) = phi (2^(k + m)) = (k + m) mod med 6 = phi (2^k) + phi (2^m) . $ + + Ist $(ZZ slash 5 ZZ)^∗$ auch zu einer Gruppe der Form $ZZ slash n ZZ$ für ein $n ∈ N$ isomorph? Falls ja, geben Sie einen Isomorphismus an. + + Die Gruppe $(bb(Z) \/ 5 bb(Z))^(\*)$ ist also isomorph zu + $bb(Z) \/ 4 bb(Z)$, da beide Gruppen die gleiche Ordnung haben und + zyklisch sind. + + Ein Isomorphismus zwischen $(bb(Z) \/ 5 bb(Z))^(\*)$ und + $bb(Z) \/ 4 bb(Z)$ ist durch die Abbildung + + $ phi : (bb(Z) \/ 5 bb(Z))^(\*) arrow.r bb(Z) \/ 4 bb(Z) , quad phi (1) = 0 , quad phi (2) = 1 , quad phi (3) = 3 , quad phi (4) = 2 $ + + gegeben. + + +// Aufgabe 2 += Homomorphismen + +Seien $G$ und $H$ Gruppen. Angenommen $G$ ist endlich und ihre Ordnung eine Primzahl $p$. Zeigen Sie: + ++ Jeder Homomorphismus $phi: G arrow H$ ist entweder trivial oder injektiv. + + Sei $phi.alt : G arrow.r H$ ein Homomorphismus. Da $G$ endlich und + $lr(|G|) = p$ eine Primzahl ist, gilt nach Lagrange für jede Untergruppe von $G$ + entweder $lr(|H|) = 1$ oder $lr(|H|) = p$. + + Fuer den Kern gilt: + + $ phi (e_G) = e_H ==> ker (phi) != emptyset $ + + Sei $a, b in ker (phi)$. Dann: + + $ phi (a) = e_H "und" phi (b) = e_H $ + + Da $phi$ ein Homomorphismus ist, folgt: + + $ phi (a dot b) = phi (a) dot phi (b) = e_H dot e_H = e_H $ + + Also ist $a dot b in ker (phi)$ + + Sei $a in ker (phi)$. Dann: + + $ phi (a) = e_H $ + + Da $e_H = e^(-1)_H$ folgt: + + $ phi (a^(-1)) = phi (a)^(-1)) = e_H^(-1) = e_H $ + + Da alle Eigenschaften fuer eine Untergruppe, namentlich nicht Leerheit, Abgeschlossenheit und die Existenz eines Inversen gegeben sind, ist das Kernbild des Homomorphismus, $ker (phi)$, eine Untergruppe von $G$. + + Falls $abs(ker (phi)) = G$, ist $phi$ trivial. Andernfalls ist + $lr(|ker (phi)|) = 1$, was bedeutet, dass $ker (phi) = { e }$, + das neutrale Element. + + Angenommen $phi (a) = phi (b)$ fuer beliebige $a, b in G$, dann: + + Wir betrachten das Element $a dot b^(-1) in G$ fuer beliebige $a, b in G$. + + $ phi (a dot b^(−1))=phi (a) dot phi (b^(−1)) $ + + Da $phi (a) = phi (b)$ und $phi (b^(-1)) = phi (b)^(-1)$, folgt: + + $ phi (a dot b^(−1))=phi (a) dot phi (a)^(−1) = e_H $ + + Da $a dot b^(-1)$ nun im Kern ist, aber $ker (phi) = {e}$, folgt: + + $ a dot b^(-1) = e ==> a = b $ + + Da $phi (a) = phi (b) ==> a = b$ ist $phi$ injektiv. + ++ Jeder Homomorphismus $phi: H arrow G$ ist entweder trivial oder surjektiv. + + Sei $phi : H arrow.r G$ ein Homomorphismus. Da $G$ die + Ordnung $p$ hat, ist jede echte Untergruppe von $G$ trivial. Das + Bild $"im"(phi)$ ist nach dem selben Argument wie in 2a eine Untergruppe von $G$, daher gilt + $lr(|"im"(phi)|) = 1$ oder $lr(|"im"(phi)|) = p$. + + Falls $lr(|"im"(phi)|) = 1$, ist $phi$ trivial. Sonst ist + $lr(|"im"(phi)|) = p$, was bedeutet, dass $"im"(phi) = G$. + + Fuer surjektivitaet gilt es jetzt zu zeigen, dass + + $ ∀y∈G: ∃x∈G: ϕ(x)=y. $ + + Dabei gilt: + + $ im (phi) = G ==> forall y in G: y in im (phi) $ + + $ y in im (phi) ==> exists x in G: phi (x) = y $ + + also ist $phi$ surjektiv. + +// Aufgabe 3 += Vektorraeume I + +Wir betrachten den Vekorraum $V = RR^4$ mit den folgenden Unterraeumen: + +$ U_1 = angle.l {(0, 1, 0, 2), (1, 0, 1, 0), (-1, 0, 1, 0)} angle.r, U_2 = angle.l {(1, 0, 3, 0), (0, 1, 0, 0), (1, 3, 3, 3)} angle.r $ + ++ Bestimmen Sie $U_1 sect U_2$. + + *Loesung* in Papierform. + +// Pruefen auf lineare Unabhaengigkeit der beiden Unterraeume: +// +// $ U_1: a vec(0, 1, 0, 2) + b vec(1, 0, 1, 0) + c vec(-1, 0, 1, 0) =^! 0 $ +// +// $ U_2: a vec(1, 0, 3, 0) + b vec(0, 1, 0, 0) + c vec(1, 3, 3, 3) =^! 0 $ +// +// Das Loesen mithilfe eines CAS liefert nur die triviale Loesung. +// +// Da jeweils alle drei Vektoren voneinander linear unabhaengig sind, koennen wir keinen verwerfen. +// +// Aufstellen des Gleichungssystems: +// +// $ mat(0, 1, -1, -1, 0, -1, 0; +// 1, 0, 0, 0, -1, -3, 0; +// 0, 1, 1, -3, 0, -3, 0; +// 2, 0, 0, 0, 0, -3, 0) $ +// +// Loesen durch ein CAS ergibt folgenden Loesungsvektor: +// +// $ X = vec(3/2 b, 2 a+2 b, a+b, a, -3/2 b, b) $ +// +// Einsetzen in die Gleichung von $U_2$ oder $U_1$ gibt: +// +// $ a vec(1, 0, 3, 0) + b vec(2, 3, 6, 6) $ +// +// Also gilt: +// +// $ U_1 sect U_2 = angle.l {(1, 0, 3, 0), (2, 3, 6, 6)} angle.r $ + ++ Ist $U_1 union U_2 = V$? + + //TODO + *Loesung* in Papierform. + +// Aufgabe 4 += Vektorraeume II + ++ Zeigen Sie, dass die Menge + $ V = {(x_1, x_2, x_3) in QQ^3: 7x_1 + 2x_2 - 3x_3 = 0} $ + ein Untervektorraum von $QQ^3$ ist. + + Sei $V = { (x_1 , x_2 , x_3) in bb(Q)^3 : 7 x_1 + 2 x_2 - 3 x_3 = 0 }$. + Um zu zeigen, dass $V$ ein Untervektorraum ist, prüfen wir die drei + Bedingungen: + + $V$ ist nicht leer: Es gilt $vec(0, 0, 0) in V ==> V != emptyset$. + + Abgeschlossenheit unter Addition: + Fuer $arrow(v) = (x_1 , x_2 , x_3) in V$ und + $arrow(w) = (y_1 , y_2 , y_3) in V$ gilt: + $ 7 (x_1 + y_1) + 2 (x_2 + y_2) - 3 (x_3 + y_3) = (7 x_1 + 2 x_2 - 3 x_3) + (7 y_1 + 2 y_2 - 3 y_3) = 0 , $ + also ist $arrow(v) + arrow(w) in V$. + + Abgeschlossenheit unter skalarer Multiplikation: + Fuer $arrow(v) = (x_1 , x_2 , x_3) in V$ und $lambda in bb(Q)$ gilt: + $ 7 (lambda x_1) + 2 (lambda x_2) - 3 (lambda x_3) = lambda (7 x_1 + 2 x_2 - 3 x_3) = 0 , $ + also ist $lambda arrow(v) in V$. + + Restliche Eigenschaften, wie die Assoziativitaet oder Kommutativitaet werden vom Koerper der rationalen Zahlen $QQ$ vererbt. + + Da sonst alle drei Bedingungen erfüllt sind, ist $V$ ein Untervektorraum von + $bb(Q)^3$. + + ++ Zeigen Sie, dass $V = angle.l {(2, -7, 0), (0, 3, 2), (1, 1, 3)} angle.r$ ist. + + *Loesung* in Papierform. + +// Damit $V$ durch die drei Vektoren $r = vec(2, -7, 0), v = vec(0, 3, 2), w = vec(1, 1, 3)$ vollstaendig aufgespannt wird muessen die Vektoren Elemente von V sein. +// +// // Was wenn V nur 2dimensional ist? +// +// Fuer $r: 7 dot 2 + 2 dot -7 - 3 dot 0 = 14 - 14 = 0$ +// +// Fuer $v: 7 dot 0 + 2 dot 3 - 3 dot 2 = 0 + 6 - 6 = 0$ +// +// Fuer $w: 7 dot 1 + 2 dot 1 - 3 dot 3 = 7 + 2 - 9 = 0$ +// +// Es gilt also $r, v, w, in V$. +// +// Da $w$ linear abhaengig von $v$ und $r$ ist muessen wir diesen nicht betrachten. Berechnen des Kreuzproduktes von $v$ und $r$ liefert: +// +// $ v times r = vec(7, 2, -3) $ +// +// Deshalb wird die Gesamte Ebene $V$ durch die Vektoren aufgespannt. + + + ++ Ist die Menge + + $ W = {(x_1, x_2, x_3) in QQ^3: 7x_1 + 2x_2 - 3x_3 = 1} $ + + ein Untervektorraum von $QQ^3$? + + Zuerst pruefe ich ob das neutrale Element in $W$ enthalten ist durch einsezten von $vec(0, 0, 0)$ in das Aussonderungsaxiom der Menge: + + $ 7 dot 0 + 2 dot 0 - 3 dot 0 = 0 != 1 $ + + So kann $W$ kein Untervektorraum von $QQ^3$ sein, da diese Menge kein neutrales Element enthaelt. + \ No newline at end of file diff --git a/S1/AGLA/Hausaufgaben/Zettel04.typ b/S1/AGLA/Hausaufgaben/Zettel04.typ new file mode 100644 index 0000000..8f943e1 --- /dev/null +++ b/S1/AGLA/Hausaufgaben/Zettel04.typ @@ -0,0 +1,226 @@ +// Load the preamble +#import "../conf.typ": conf +#show: conf.with(week: "4") + += Basis von Vektorraeumen + +Sei $V$ ein $RR$-Vektorraum und $b_1, b_2, b_3, b_4$ eine Basis von $V$. Definiere + +$ v_1 = b_1 + 2b_2 + b, $ +$ v_2 = 2b_1 + b_3 + b_4 , $ +$ v_3 = b_1 + b_2 + b_3 + b_4 , $ +$ v_4 = b_2 + b_3 + b_4 , $ +$ w = 2b_1 − b_3 . $ + ++ Zeigen Sie, dass $v_1, v_2, v_3, v_4$ eine Basis von $V$ ist. + + *Abgabe in Papierform.* + +// Jede Basis von $V$ kann durch Kombination von $v_i$ dargestellt werden. +// +// $ b_1 = v_3 - v_4 \ +// b_2 = v_4 - v_2 + 2 b_1 \ +// b_3 = v_4 - b_2 - b_4 \ +// b_4 = v_1 - b_1 - 2 b_2 $ +// +// Es gilt $dim(V) = 4$. Angenommen, $dim(angle.l v_1, v_2, v_3, v_4 angle.r) = n != dim(V)$. Falls $n > dim(V)$, dann steht dies im Widerspurch zu der Annahme, dass $v_i$ nur durch 4 Basisvektoren konstruiert werden koennen. +// +//TODO Zweiter FALL +// + + + ++ Definiere die beiden Unterräume $U_1 = ⟨{v_1, v_2, v_3}⟩$ und $U_2 = ⟨{v_4, w}⟩$. + + Bestimmen Sie jeweils eine Basis von $U_1 ∩ U_2$ und $U_1 + U_2$. + + *Abgabe in Papierform.* + //$A = U_1 + U_2 := {a, b: a in U_1, b in U_2}$ + + //$BB_A = {v_1, v_2, v_3, v_4, w}$, da alle 5 Vektoren linear unabhaengig sind. + + //$B= U_1 sect U_2$ + + //Aufstellen einer Matrix: + + //$ mat(2, 0, 1, 2, 1, 0; 0, 1, 1, 0, 2, 0; -1, 1, 1, 1, 0, 0; 0, 1, 1, 1, 1, 0) $ + + //Dies ergibt folgenden Loesungsvektor: + + //$ X = vec(1, 1, -1, 1, 1) $ + + //Also folgt: $BB_B = {(2, 1, 0, 1)}$ + + += Lineare Unabhaengigkeit + +Im $QQ^4$ definieren wir die Vektoren + +$ v_1 = (1, 0, 0, 1), v_2 = (1, 2, 3, 4), v_3 = (4, 3, 2, 1), v_4 = (0, 1, 1, 0), $ + +$ w_1 = (1, 2, 1, 2), w_2 = (−1, 0, 1, 0), w_3 = (0, 1, 0, 1) "und" w_4 = (1, 1, 1, −1). $ + +Entscheiden Sie jeweils, ob die $v_i$ bzw. $w_i$ linear unabhängig sind. + +*Abgabe in Papierform.* + +//Aufstellen einer Matrix: +// +//$ M_v = mat(1, 0, 0, 1; 1, 2, 3, 4; 4, 3, 2, 1; 0, 1, 1, 0) M_w = mat(1, 2, 1, 2; -1, 0, 1, 0; 0, 1, 0, 1; 1, 1, 1, -1) $ +// +//Ermitteln der Determinanten: +// +//#image("Media/mat_v.png") +//#image("Media/mat_w.png") +// +//$therefore$ Die $v_i$ sind nicht linear unabhaengig, wohingegen die $w_i$ linear unabhaengig sind. +// += Vektorraum von Funktionen + +Sei $KK$ ein Körper und $V$ der Vektorraum aller Funktionen $f : KK → KK$. + +Definiere $U_1 = {f ∈ V | f (x) = f (−x)}$ und $U_2 = {f ∈ V | f (x) = −f (−x)}$. + ++ Zeigen Sie, dass $U_1$ und $U_2$ Unterräume von $V$ sind. + + Ein Unterraum $U subset.eq V$ braucht drei Eigenschaften: + + 1. $f (x) = 0 in U$ . + 2. $f , g in U: f + g in U$. + 3. $f in U, lambda in K: lambda f in U$. + + == $U_1 = { f in V divides f (x) = f (- x) }$ + + Nullfunktion: $f (x) = 0$ gehört zu $U_1$, da $0 = 0, quad forall x in K$ gilt. + + Abg. unter Addition: Für $f , g in U_1$ gilt: + + $ (f + g) (x) = f (x) + g (x) $ + + $ (f + g) (- x) = f (- x) + g (- x) $ + + $ therefore f (x) = f (- x) and g (x) = g (- x) &==>^(#[@sum1, @sum2]) (f + g) (x) = (f + g) (- x) \ + &<==>^("Def.") f + g in U_1 $ + + Abg. unter sk. Mult. : Für $f in U_1$ und $lambda in K$ gilt: + + $ (lambda f) (x) = lambda f (x) $ + + $ (lambda f) (- x) = lambda f (- x) $ + + $ therefore f(x ) = f(-x) ==> (lambda f) (x) = (lambda f) (- x) <==> lambda f in U_1 $ + + Damit ist $U_1$ ein Unterraum. + + == $U_2 = { f in V divides f (x) = - f (- x) }$ + + Nullfunktion: $f (x) = 0$ gehört zu $U_2$, da $0 = - 0, quad forall x in K$ gilt. + + Abg. unter Addition: Für $f , g in U_2$ gilt: + + $ (f + g) (x) = f (x) + g (x) $ + + $ (f + g) (- x) = f (- x) + g (- x) $ + + $ f (x) = - f (- x) and g (x) = - g (- x) $ + + $ therefore (f + g) (- x) = - f (x) - g (x) = - (f (x) + g (x)) = - (f + g) (x) <==> f + g in U_2 $ + + Abg. unter sk. Mult.: Für $f in U_2$ und $lambda in K$ gilt: + + $ (lambda f) (x) = lambda f (x) $ + + $ (lambda f) (- x) = lambda f (- x) $ + + $ f (x) = - f (- x) $ + + $ therefore (lambda f) (- x) = - lambda f (x) = - (lambda f) (x) <==> lambda f in U_2 $ + + Damit ist $U_2$ ein Unterraum. + +//////////////////////////////////////////////////// + ++ Angenommen in $KK$ gilt $-1 != 1$. Zeigen Sie, dass $V = U_1 ⊕ U_2$ ist. + + Die Bedingung $V = U_1 xor U_2$ bedeutet: + + $V = U_1 + U_2 <==> f = f_1 + f_2, f_1 in U_1 and f_2 in U_2$. + + + + $U_1 sect U_2 = { 0 }$ + + Für $f in V$, definiere: + + $ f_1 (x) = frac(f (x) + f (- x), 2) , quad f_2 (x) = frac(f (x) - f (- x), 2) . $ + + Es gilt: + + $ f_1 (- x) = frac(f (- x) + f (x), 2) = frac(f (x) + f (- x), 2) = f_1 (x) ==> f_1 in U_1 $ + + $ f_2 (- x) = frac(f (- x) - f (x), 2) = - frac(f (x) - f (- x), 2) = - f_2 (x) ==> f_2 in U_2 $ + + $ f_1 (x) + f_2 (x) = frac(f (x) + f (- x), 2) + frac(f (x) - f (- x), 2) = f (x) $ + + Damit ist $V = U_1 + U_2$. + + Sei $f in U_1 sect U_2$. + + Dann gilt: + + $ f (x) = f (- x) quad upright("und") quad f (x) = - f (- x) . $ + + $ f (x) = - f (x) arrow.r.double.long 2 f (x) = 0 . $ + + $ - 1 eq.not 1 <==> 0 eq.not 2 ==> f (x) = 0, forall x in K <==> f = 0 $ + + Da beide Bedingungen erfüllt sind, folgt: $V = U_1 xor U_2$. + + + + + += Polynome +Sei $KK$ ein Körper und $f ∈ K[X]$ gegeben durch + +$ f(X) = sum_(j=0)^n a_j X_j $ + +mit $n >= 1 "und" a_n != 0$. + +Zeigen Sie: Zu einem $c ∈ KK$ gibt es genau dann ein Polynom $g ∈ K[X]$ mit + +$ f = (X − c)g , $ + +wenn gilt + +$ f(c) = sum_(j=0)^n a_j c_j = 0 $ + + +// [=>] ++ Angenommen $f = (X -c)g$. Dann gilt: + + $ f(c) = (c-c)g = 0 quad checkmark $ + ++ Angenommen $f(c) = 0$. + + Sei $X = Y + c <==> Y = X - c$. + + $ f &= sum_(j=0)^n a_j (Y+c)^j $ + + Falls $Y=0$, dann gilt: + + $ f &= sum_(j=0)^n a_j c^j = 0 = 0g $ + + nach Vorraussetzung. + + Sonst gilt: + + $ f &= sum_(j=0)^n a_j (Y+c)^j = sum_(j=0)^n a_j (Y+c)^j - a_j c^j + a_j c^j \ + &= sum_(j=0)^n a_j (Y+c)^j - a_j c^j + underbrace(sum_(j=0)^n a_j c^j, f(c) = 0) $ + + Wobei in der expandierten Form von $(Y+c)^j$ nach dem Binomial Satz, jeder Summand den Faktor $Y$ entaehlt, ausser $c^j$, was sich hier aber kuerzt. + + Da jeder Summand von $f$ den Faktor $Y = X - c$ mindestens einmal enthaelt ist $f$ restlos durch $X - c$ teilbar. Damit laesst sich $f$ wie folgt schreiben: + + $ f = (X-c)g <==> g = f/(X-c) quad checkmark $ + +$qed$ + diff --git a/S1/AGLA/Hausaufgaben/Zettel05.typ b/S1/AGLA/Hausaufgaben/Zettel05.typ new file mode 100644 index 0000000..1da1c0d --- /dev/null +++ b/S1/AGLA/Hausaufgaben/Zettel05.typ @@ -0,0 +1,232 @@ +// Load the preamble +#import "../conf.typ": conf +#show: conf.with(week: "5") + += $CC$-Vektorraum + +Betrachte den $CC$-Vektorraum $V = CC^4$ und + +$ w_1 = (1, i, 0, −i), w_2 = (i, 1, 0, i), w_3 = (0, −2i, 0, i + 1) in CC^4 . $ + +Sei $W = ⟨{w_1, w_2, w_3}⟩$. Bestimmen Sie einen zu $W$ komplementären Unterraum vom $V$ und geben Sie jeweils eine Basis von $U$ und $W$ an. + +*Abgabe in Papierform*. + +//$ W_B = angle.l {w_1, w_2} angle.r $ +// +//$ U_B = angle.l {(0, 0, 1 ,0), (i - 1, 1 - i, 0, 2)} angle.r $ +// +//Es gilt: +// +//$ V = W plus.circle U $ +// +//was aus der linearen Unabhaengigkeit von den Basisvektoren von $W$ und $U$ folgt und da +// +//$ dim(V) = dim(W) + dim(U) = 4 . $ + += Komplementaere Unterraeume + +Sei $K$ ein Körper und $V$ ein K-Vektorraum mit $m$ Unterräumen $W_1, . . . , W_m ⊂V$. + ++ Zeigen Sie, dass die folgenden zwei Aussagen äquivalent sind: + + #enum(numbering: "(i)", + enum.item(1)[Es gilt $V = W_1 ⊕. . . ⊕W_m$.], + enum.item(2)[Für jedes $v ∈V$ gibt es eindeutige $w_1 ∈W_1, . . . , w_m ∈W_m "mit" v = w_1 + . . . + w_m$. ]) + + + Angenommen (i). Dann folgt: + + Also ist hier die Null auf genau eine Weise darstellbar: + + $ 0 = w_1 + w_2 + ... <==> 0 = 0 + 0 ... $ + + + Sei $v in V$. + + $ v = w_1 + w_2 + ... = w'_1 + w'_2 + ... \ + <==> 0 = (w_1 - w'_1) + (w_2 - w'_2) + ... $ + + Nach @null muss also $w_i = w'_i$, wodurch die Darstellung von $v$ eindeutig ist. + + Angenommen (ii). Dann gilt: + + $ v = w_1 + . . . + w_m <==> V = W_1 plus ... plus W_m . $ + + Sei $A(n) :<==> U = W_1 + ... + W_(n), O = W_(n+1) + ... + W_m: U plus.circle O$ + + Induktionsanfang (IA): Sei $V$ wie hier mit $n=1$ folgt: + + $ underbrace(W_1, U) plus underbrace(W_2 plus ... plus W_(m), O) = V $ + + + + + Sei nun $v in U sect O$, dann folgt: + + $ v = u_1 = o_m $ + + Da $v$ jedoch eindeutig dargestellt werden muss, folgt $u_1 = o_m = 0$. + + Also ist: + + $ U_1 sect W_m = 0 ==>^("mit" #[@comp]) U_1 plus.circle W_m $ + + + Induktionshypothese (IH): Angenommen A(n), fuer $n in N$. + + Induktionsschritt (IS): Nun gilt es zu zeigen, dass A(n+1) auch gilt. + + ... + + *Der Rest ist in Papierform* + + ++ Ist die folgende Aussage ebenfalls äquivalent zu den ersten beiden? Geben Sie einen Beweis oder ein Gegenbeispiel an. + + (iii) Es gilt $V = W_1 + . . . + W_m$ und $W_i ∩ W_j = {0}$ für alle $1 ≤i < j ≤m$. + + Gegenbeispiel: + + $ V = RR^2, quad W_1 = angle.l (0, 1) angle.r , angle.l (1, 0) angle.r, angle.l (1, 1) angle.r $ + + Dann gilt: + + $ V = W_1 + W_2 \ + sect.big W_i = 0 $ + + da $W_1$ und $W_2$ linear unabhaengig sind und sich die drei Geraden im Ursprung schneiden. + + Sei $v = (1, 1)$, dann folgt: + + $ v = w_1 + w_2 = w_3 = vec(1, 0) + vec(0, 1) = vec(1, 1) $ + + Also ist die Darstellung von $v in V$ nicht eindeutig. + += Projektionen + +Sei $K$ ein Körper und $V$ ein K-Vektorraum. +Angenommen $ϕ: V →V$ ist eine lineare Abbildung mit der Eigenschaft $ϕ ◦ϕ = ϕ$ <2b1> . + +Zeigen Sie, dass es zwei Untervektorräume $W_1, W_2 ⊂V$ mit $V = W_1 ⊕W_2$ gibt, sodass die Gleichungen $ϕ(w_1) = w_1$ für alle $w_1 ∈W_1$ und $ϕ(w_2) = 0$ für alle $w_2 ∈W_2$ gelten. + +Setze folgenden UVR von $V$: + +$ W_1 = im(phi.alt), quad W_2 = ker(phi.alt) $ + +Nun gilt fuer $v in V$: + + +$ phi.alt(phi.alt(v)) &=^(#[@2b1]) phi.alt(v) \ +<==> phi.alt(phi.alt(v)) - phi.alt(v) &= 0 \ +<==>^("lin.") phi.alt(phi.alt(v) - v) &= 0 $ + +Also folgt: + +$ k &= phi.alt(v) - v, quad k in ker(phi.alt) \ +<==> v &= phi.alt(v) - k, quad k in W_2, quad phi.alt(v) in W_1 $ + +Da $v$ beliebig war, gilt: $V = W_1 + W_2$. + +Sei $x in W_1 sect W_2$. + +Dann gilt: + +$ x in W_1 &==> phi.alt(x) = x \ + x in W_2 &==> phi.alt(x) = 0 \ + therefore x &= 0 $ + +Da $x$ beliebig war, folgt: $W_1 sect W_2 = 0$. + +Also ist $V = W_1 plus.circle W_2$. Die Eigenschaften von $phi.alt$ gelten nach Konstruktion. + +$qed$ + += Fibonacci + +Sei $V := {(a_i)_(i∈N) : a_i ∈R}$ der R-Vektorraum aller reellen Folgen. + +Wir definieren die lineare Abbildung $ϕ$ durch + +$ ϕ: V →V, (a_1, a_2, a_3, . . .) |-> (a_2, a_3, . . .) . $ + +Definiere nun + +$ psi = ϕ ◦ϕ −ϕ − id(V) . $ + ++ Bestimmen Sie die Dimension von $ker(ψ)$. + + Zuerst bestimme ich den Kern von $psi$: + + $ ker(psi) = {v in V: psi(v) = 0 <==> phi(phi(v)) - phi(v) - v = 0} $ + + Die Funktion $phi$ verschiebt alle Elemente von $v in V$ um einen Index nach links. + + Aus der Definition von $phi$ folgt, dass folgendes gelten muss: + + $ forall k in ker(psi): k_n = k_(n-1) + k_(n-2), quad n > 2 $ + + Also gilt: Damit ein $v in V$ im Kern von $psi$ liegt muessen alle Elemente durch die ersten Beiden eindeutig bestimmt sein. + + Eine moegliche Basis fuer den Kern ist: + + $ bold(b'_1) = (1, 0, a_3, a_4, ...) \ + bold(b'_2) = (0, 1, a_3, a_4, ...) $ + + Diese Basisvektoren sind linear unabhaengig und jedes Element laesst sich durch Linearkombination dieser beiden Vektoren darstellen. + + Da sich der Kern durch zwei Basisvektoren darstellen laesst, folgt: + + $ dim(ker(psi)) = 2 $ + ++ Finden Sei eine Basis $b_1, . . . , b_m$ von ker(ψ), die die Form $b_i = (α_i, α_i^2 , α_i^3 , . . .)$ für reelle Zahlen $α_1, . . . , α_m$ hat. + + Jeder Basisvektor vom Kern muss im Kern liegen, also: + + $ alpha^n = alpha^(n-1) + alpha^(n-2), quad forall n > 2 $ + + Hier waehlen wir $n=2$ o.E.d.A, da wir die Gleichung immer durch $alpha^t$ teilen koennen, da $alpha != 0$, da der Nullvektor kein Basisvektor vom Kern sein kann. + + Wir wollen also die folgende Gleichung loesen: + + $ alpha^2 - alpha - 1 = 0 $ + + Nach der Lösungsformel für die allgemeine quadratische Gleichung folgt fuer $alpha$: + + $ alpha = (1+ sqrt(5))/2, quad dash(alpha) = (1- sqrt(5))/2 $ + + Da der Kern zweidimensional ist und $bold(b_1)$ und $bold(b_2)$ linear unabhaengig sind, folgt: + + Also folgt fuer diese spezifische Basis von $ker(psi)$: + + $ bold(b)_1 = (alpha, alpha^2, alpha^3, ...), quad bold(b)_2 = (dash(alpha), dash(alpha)^2, dash(alpha)^3, ...) $ + ++ Sei die Fibonacci-Folge $(F_n)_(n∈N)$ definiert durch: $F_1 = 1, F_2 = 1, F_(n+2) = F_(n+1) + F_n$ für alle $n ∈N$. Zeigen Sie, dass $F_n ∈ker(ψ)$ liegt und schreiben Sie $(F_n)$ als Linearkombination ihrer Basis $b_1, . . . , b_m$. + + $ F_(n+2) &= F_(n+1) + F_n \ + <==> F_(n) &= F_(n-1) + F_(n-2), forall n > 2 $ + + Damit liegt $F_n$ nach @ker im Kern von $psi$. + + Wir wissen, dass $F_1 = 1$ und $F_2 = 1$, wodurch $F_((0)) = 0$ sein muesste. + Ermitteln von $c_1, c_2$, durch einsetzen in die bekannte Gleichung fuer ein Element des Kerns: + + $ 0 &= c_1 ((1+ sqrt(5))/2)^0 + c_2 ((1- sqrt(5))/2)^0 <==> c_1 = - c_2 \ + 1 &= c_1 (1+ sqrt(5))/2 + c_2 (1- sqrt(5))/2 \ + <==> 1 &= c_2 (1- sqrt(5))/2 - c_2 (1+ sqrt(5))/2 <==> 1 = (-2 sqrt(5))/2 c_2 ==> c_2 = -1/sqrt(5) \ + therefore &underline(c_2 &= -1/sqrt(5)) = - c_1 ==> underline(c_1 = 1/sqrt(5)) $ + + Also: + + $ (F_n) = 1/sqrt(5) bold(b)_1 - 1/sqrt(5) bold(b)_2 $ + ++ Bestimmen Sie eine Formel $F_n = sum^m_(i=1) c_i alpha_i^n$ , mit der die Fibonacci-Folge direkt berechnet werden kann. + + Wir wissen aus den Vorherigen Aufgabenteilen alles um eine geschlossene Form fuer die Fibonaccifolge aufzuschreiben. + + Wir muessen nur das Element der beiden Basisvektoren an $n$ Stelle ermitteln, was wir durch die Eigenschaft der Basisvektoren von Aufgabenteil b) tun koennen. + + + Also ergibt sich die geschlossene Formel als: + + #rect($ F_n = 1/sqrt(5) ((1+ sqrt(5))/2)^n - 1/sqrt(5) ((1- sqrt(5))/2)^n $) diff --git a/S1/AGLA/Hausaufgaben/Zettel06.typ b/S1/AGLA/Hausaufgaben/Zettel06.typ new file mode 100644 index 0000000..5afbcbc --- /dev/null +++ b/S1/AGLA/Hausaufgaben/Zettel06.typ @@ -0,0 +1,361 @@ +// Load the preamble +#import "../conf.typ": conf +#show: conf.with(week: "6") + +#import "@preview/pavemat:0.1.0": pavemat + += Vektorraeume + ++ Zeigen Sie, dass die Menge + + $ V := {x+y sqrt(2) | x, y in QQ} $ + + mit der Addition auf $RR$ und der durch + + $ lambda dot (x + y sqrt(2)) := lambda x + lambda y sqrt(2) $ + + definierten Multiplikation mit Skalaren zu einem Vektorraum ueber $QQ$ wird und bestimmen Sie die Dimension von $V$. + + Eigenschaften wie Distributivitaet und Assoziativitaet werden von $QQ$ vererbt. + + Seien $v, w in VV$ und $lambda in QQ$. + + $ v + w &= (a + b sqrt(2)) + (c + d sqrt(2)) \ + &= a + c + b sqrt(2) + d sqrt(2) \ + &= x + (b + d) sqrt(2) \ + &= x + y sqrt(2) in V $ + + da $a+c in QQ$ ud $b+d in QQ$. + + $ lambda dot v &= lambda dot (x + y sqrt(2)) \ + &= lambda x + lambda y sqrt(2) in QQ $ + + da $lambda x in QQ$ und $lambda y in QQ$ + + Da $1$ und $sqrt(2)$ linear unabhaengig sind (ueber $QQ$) folgt: + + $ dim(V) = 2 $ + ++ Zeigen Sie, dass $V$ mit der Multiplikation auf $RR$ ein Koerper ist. + + Es gilt zu zeigen: + + *Addition*: + + - Kommutative Addition + + _Wird vererbt_ + + - Assoziative Addition + + _Wird vererbt_ + + - Inverses der Addition + + Sei $v = a + b sqrt(2)$. + + $ v + v' = 0 => v' &= -v \ + &= -(a + b sqrt(2)) \ + &=^(#[@dist]) underline(underline(-a - b sqrt(2))) $ + + - Neutrales der Addition + + Das Neutrale ist hier $0 + 0 sqrt(2)$. + + *Multiplikation*: + + - Kommutative Multiplikation + + _Wird vererbt_ + + - Assoziative Multiplikation + + _Wird vererbt_ + + - Inverses der Multiplikation + + Sei $v = a + b sqrt(2)$. + + $ v v' = 1 => v' &= 1/v \ + &= 1/(a + b sqrt(2)) = (a - b sqrt(2))/((a + b sqrt(2))(a-b sqrt(2)))\ + &= (a - b sqrt(2))/(a^2 + 2 b^2) \ + &= underline(underline(underbrace(a/(a^2 + 2 b^2), c in QQ) - (b )/underbrace(a^2 + 2 b^2, d in QQ)sqrt(2))) $ + + + + - Neutrales der Multiplikation + + Das Neutrale ist hier $1 + 0 sqrt(2)$. + + *Distributivitaet*: + + $ (x + y sqrt(2)) (z + u sqrt(2)) &= x z + 2y u + x u sqrt(2) + z y sqrt(2) \ + &= 2y u + x z + (x u + z y)sqrt(2) \ + &= a + b sqrt(2) + $ + + Wobei $a = 2y u + x z in QQ$ und $b = x u + z y in QQ$. + + $qed$ + ++ Sei $W$ der von $1, sqrt(2), sqrt(3)$ und $sqrt(6)$ in $RR$ erzeugte $QQ$-Vektorraum. Bestimmen Sie die Dimension. Zeigen Sie, dass $W$ auch ein Vektorraum ueber dem Koerper $V$ aus Teil (b) ist. Bestimmen Sie wieder die Dimensionen. + + Unabhaengigkeit der Basen von $W$ zeigen: + + Es muss gelten (mit $a, b, c, d in QQ$ und $a b c d != 0$): + + $ a + b sqrt(2) + c sqrt(3) + d sqrt(6) = 0 \ + b sqrt(2) + c sqrt(3) = -1 dot (d sqrt(6) + a) \ + 2 b^2 + 3 c^2 + 2 b c sqrt(6) = 6 d^2 + a^2 + 2 a d sqrt(6) \ + 2 b c sqrt(6) - 2 a d sqrt(6) = - (2 b^2 + 3 c^2) + 6 d^2 + a^2 \ + sqrt(6) (2 b c - 2 a d ) = 6 d^2 + a^2 - (2 b^2 + 3 c^2) \ + sqrt(6) = (6 d^2 + a^2 - (2 b^2 + 3 c^2))/(2 b c - 2 a d ) =: l \ + $ + + damit die Basen voneinander linear unabhaengig sind. + + Da $QQ$ ein Koerper ist, gilt $l in QQ$. Somit muss $sqrt(6) in QQ$ damit die Gleichung eine Loesung hat. + + Angenommen $sqrt(6) = p/q$, wobei $p, q$ teilerfremde positive natuerliche Zahlen sind. + + Die Quadratwurzel von 6 kann keine natuerliche Zahl sein, da $4 < 6 < 9 => 2 < sqrt(6) < 3$. + + $ sqrt(6) = p/q \ + 6 q = p^2/q $ + + wobei $6 q in NN$. Es gilt $p^2/q in.not NN$, da sie teilerfremd sind $arrow.zigzag$. + + Es folgt also $sqrt(6) in.not QQ$. Also kann @lind nicht geloest werden. + + Da so $1, sqrt(2), sqrt(3)$ und $sqrt(6)$ linear unabhaengig sind (ueber $QQ$) folgt: + + $ dim(W_QQ) = 4 $ + + Wenn wir nun $V$ als unseren Koerper betrachten, dann faellt direkt auf, dass $1 "und" sqrt(2)$ nicht mehr linear unabhaengig sind, da ($a + b sqrt(2), c + d sqrt(2) in V$): + + $ a + b sqrt(2) &= (c + d sqrt(2)) sqrt(2) \ + &= 2 d + c sqrt(2) $ + + Auch sind $sqrt(3), sqrt(6)$ voneinander linear abhaengig: + + $ sqrt(3)(a + b sqrt(2)) &= sqrt(6) (c + d sqrt(2)) \ + a sqrt(3) + b sqrt(6) &= 2 d sqrt(3) + c sqrt(6) $ + + Das gleiche laesst sich auch beim Einsetzen der Eigenschaften von $v in V$ und unter Betrachtung des ersten Teilergebnisses erkennen ($lambda_i in V$): + + $ lambda_1 + lambda_2 sqrt(2) + lambda_3 sqrt(3) + lambda_4 sqrt(6) \ + <==> undershell((a_1 + a_2 sqrt(2)) + (2 b_1 + b_2 sqrt(2)), #[lin. abh.]) + undershell((c_1 sqrt(3) + c_2 sqrt(6)) + ( 2 d_1 sqrt(3) + d_2 sqrt(6)), #[lin. abh.]) $ + + Nach dem Selben Argument wie oben sind die beiden Bloecke, durch betrachten der jeweiligen Bassen, voneinander linear unabhaengig. + + Es folgt also: + + $ dim(W_V) = 2 $ + += Gleichungssystem + +Loesen Sie das folgende lineare Gleichungssystem + +$ x+y &=-1 \ +-x + y &= -1 \ +x+y -z &= 0 $ + +sowohl ueber dem koerper $QQ$ als auch ueber dem Koerper $FF_3$. + +Zuerst wird als Koerper $QQ$ angenommen. + +Umformen in eine Matrix mit Loesungsspalte: + + +$ mat(1, 0, 1, -1; -1, 1, 0, -1; 1, 1, -1, 0; augment: #(-1)) $ + +Gaussverfahren anwenden: + +#grid(columns: 4, +$ mat(1, 0, 1, -1; 0, 1, 1, -2; 0, 1, -2, 1; augment: #(-1)) $, +$ mat(1, 0, 1, -1; 0, 1, 1, -2; 0, 0, -3, 3; augment: #(-1)) $, +$ mat(1, 0, 1, -1; 0, 1, 1, -2; 0, 0, 1, -1; augment: #(-1)) $, +$ mat(1, 0, 1, -1; 0, 1, 0, -1; 0, 0, 1, -1; augment: #(-1)) $, +) + +Loesungsmatrix: + +$ mat(1, 0, 0, 0; 0, 1, 0, -1; 0, 0, 1, -1; augment: #(-1)) $ + +Es folgt also: + +$ x &= 0 \ +y &= -1 \ +z &= -1 $ + +Definiere den Koerper $FF_3 = {0, 1, 2}$. + +Betrachten Matrix (Welche mit dem Gaussverfahren bistimmt wurde): + +$ mat(1, 0, 1, -1; 0, 1, 1, -2; 0, 0, -3, 3; augment: #(-1)) $, + +Hier duerfen wir nicht durch 3 teilen, da $3 = 0$ in $FF_3$ gilt. + +Also machen wir eine Fallunterscheidung fuer $z$: + +Auch kann $z$ jeden Wert annehmen, da + +$ -3 dot 0 = 0 \ + -3 dot 1 = 0 \ + - 3 dot 2 = 0 $ + +Fall 1: ($z = 0$) + +Es muss gelten: + +$ x = -1 = 2 \ + y = -2 = 1 $ + +Fall 2: ($z = 1$) + +$ x + 1 = -1 = 2 => x = 1 \ + y + 1 = -2 = 1 => y = 0 $ + +Fall 3: ($z = 2$) + +$ x + 2 = -1 = 2 => x = 0 \ + y + 2 = -2 = 1 => y = 2 $ + + +Die Loesungsmenge ist hier also: + +$ LL = {(0, 2, 2), (1, 0, 1), (2, 1, 0)} $ + += Matrix invertieren + +Ist die Matrix $A in QQ^(4 times 4)$ mit $A = mat(1, 1, 0, 1; 0, 1, 0, 3; -1, 0, 1, 2; 0, 1, 1, 4)$ invertierbar? + +Benutzen Sie nur die Hilfsmittel die einschliesslich der Vorlesung vom 29.1 zur Verfuegung stehen. + +Bijektive Umformungen mit dem Gaussverfahren: + +#grid( + columns: 2, + row-gutter: 10pt, +$ mat(1, 1, 0, 1; + 0, 1, 0, 3; + -1, 0, 1, 2; + 0, 1, 1, 4) $, +$ mat(1, 1, 0, 1; + 0, 1, 0, 3; + 0, 1, 1, 3; + 0, 1, 1, 4) $, +$ mat(1, 1, 0, 1; + 0, 1, 0, 3; + 0, 0, 1, 0; + 0, 0, 1, 1) $, +$ mat(1, 1, 0, 1; + 0, 1, 0, 3; + 0, 0, 1, 0; + 0, 0, 0, 1) $, +$ mat(1, 1, 0, 1; + 0, 1, 0, 0; + 0, 0, 1, 0; + 0, 0, 0, 1) $, +$ mat(1, 0, 0, 0; + 0, 1, 0, 0; + 0, 0, 1, 0; + 0, 0, 0, 1) $, + ) + +Da der $A$ eine $4 times 4$ Matrix ist und einen Rank von $4$ hat, ist $A$ invertierbar. + +$qed$ + += Polynome II + +Sei $V_n := {f in K[X]: deg(f) <= n}$ die Menge der Polynome von Grad hoechstens $n$ ueber einem Koerper $K$. + ++ Zeigen Sie, dass $V_n$ ein Untervektorraum der Dimension $n+1$ von $K[X]$ ist. + + Pruefen der Eigenschaften fuer einen Unterraum: + + Abgeschlossenheit unter Addition: + + Seien $f, g in V_n$, also $deg(f), deg(g) <= n$. + + Nun gilt: + + $ deg(f+g) = max({deg(f), deg(g)}) <= n ==> f+g in V_n $ + + Abgeschlossenheit unter skalarer Multiplikation: + + Fuer $c in K$ und $c != 0$, gilt: + + $ deg(c f ) = deg(f) ==> c f in V_n $ + + Das Nullpolynom ist in $V_n$ enthalten, da + + $ deg(0) = 0 <= n, quad forall n in NN $ + + Eine Basis von $V_n$ ist gegeben durch: + + $ B_(V_n) = {X^0, X^1, X^2, ... , X^n} $ + + Diese Basis hat $n+1$ Elemenente, weshalb die gilt: + + $ dim(V_n) = n+1 $ + ++ Zeigen Sie weiter, dass die Abbildung + + $ phi: V_n -> V_(n+2), quad f |-> (X^2 + 1)f $ + + eine lineare Abbildung ist. + + Zuerst pruefen der ersten Eigenschaft: + + Seien $f, g in V_n$ und $lambda in K$. + + $ phi(f + g) &= (X^2 + 1)(f + g) \ + &= X^2 dot f + f + X^2 dot g + g \ + &= (X^2 + 1)f + (X^2 + 1)g \ + &= phi(f) + phi(g) $ + + $ lambda dot phi(f) &= lambda dot (X^2 + 1) dot f \ + &= (X^2 + 1) dot lambda f \ + &= phi(lambda f) $ + + $qed$ + + Ein Polynom kann problemlos mit einem Skalar multipliziert werden. + Auch ist die Multiplikation mit einem Anderen Polynom anderer Ordnung moeglich. + + ++ In allen $V_n$ sei die Basis $X^0, X^1, ..., X^n$ gegeben. + Ist $f=a_n X^n + ... + a_0 X^0$ und $(X^2 + 1)f = b_(n+2) X^(n+2) + ... + b_0 X^0$, dann ist durch $phi_n$ + auch eine Abbildung $Phi_n: K^(n+1) -> K^(n+3)$ mit + + $ (a_0, a_1, ..., a_n) |-> (b_0, b_1, ..., b_(n+2)) $ + + gegeben. Zeigen Sie, dass $Phi_n$ linear ist und bestimmen Sie die zugehoerige Matrix. + + Sei $f,g in V^n$ und $lambda in K$. + + $ Phi(f+g) &= Phi(f) + Phi(g) $ + $ lambda dot Phi(f) &= Phi(lambda dot f) $ + + nach dem selben Argument wie in Teil (b). + + Die zugehoerige Matrix sieht wie folgt aus: + + #{ + set text(size: 1.1em) + + pavemat( + delim: "[", + )[$ mat(1, 0, 0, 0, 0, ..., 0; + 0, 1, 0, 0, 0, ..., 0; + 1, 0, 1, 0, 0, ..., 0; + 0, 1, 0, 1, 0, ..., 0; + 0, 0, 1, 0, 1, ..., 0; + dots.v, dots.v, dots.v, dots.down, dots.down, dots.down, 0; + 0, 0, 0, 0, 1, 0, 1;) $] + } + + wobei sie $n+1$ Spalten und $n+3$ Reihen hat. + \ No newline at end of file diff --git a/S1/AGLA/Hausaufgaben/Zettel07.typ b/S1/AGLA/Hausaufgaben/Zettel07.typ new file mode 100644 index 0000000..4910fa3 --- /dev/null +++ b/S1/AGLA/Hausaufgaben/Zettel07.typ @@ -0,0 +1,81 @@ +// Load the preamble +#import "../conf.typ": conf +#show: conf.with(week: "7") + += Vektorraeume + +Fuer eine Matrix $A = (a_(i j)) in K^(n times n)$ definieren wir die Reiehen- und Spaltensummen durch + +$ R_r (A) := sum^n_(i = 1) a_(r i) quad "und" quad S_s (A) := sum^n_(i = 1) a_(i s) . $ + +Sei $V$ die Menge aller $n times n$- Matrizen fuer die alle Reiehen- und Spaltensummen gleich sind. +Also + +$ V:= {A in K^(n times n):R_r (A) = S_s (A) "fuer alle" r, s = 1, ..., n} $ + ++ Zeigen Sie, dass $V$ ein Untervektorraum von $K^(n times n)$ ist. + + Seien $v, w in V$. + + Bei der Matrixaddition werden alle Reiehen und Spalten addiert. Dabei gilt: + + $ v_(i j) = c_1, quad w_(i j) = c_2 $ + + Also + + $ v_(i j) + w_(i j) = c_1 + c_2 = c => v + w in V $ + + $ lambda v = lambda v_(i j) = lambda c_1 = c => lambda v in V $ + + + ++ Bestimmen Sie die Dimension und eine Basis von $V$. + + Wir koennen alle Eintraege von $v in V$ beliebig waehlen, bis auf jeweils die letzen innerhalb einer Zeile oder Spalte. Daraus muss folgen: + + $ dim(V) = (n-1)(n-1) = (n-1)^2 $ + += Einheitsmatrix + +Sei $A in K^(n times n)$ mit der Eigenschaft, dass $A B = B A, quad forall B in K^(n times n)$ gilt. + +Zeigen Sie, dass $A = lambda E_n$ fuer ein $lambda in K$ ist. + +Hier gilt es zu zeigen, dass nur ein Vielfaches der Einheitsmatrix die Kommutativit mit einer beliebigen Matrix $B$ erfuellt. + +Sei $B = bb(1)$. Falls $A = bb(1)$ dann waehle $B != bb(1) and B != bb(0)$. + +Angenommen fuer $A$ gilt, dass $exists a_(i j): i != j: a = lambda != 0$. + +Nach Matrixmultiplikationsvorschrift ist diese nicht kommutativ $arrow.zigzag$. + +Wir wissen also, dass $A = mat(a, 0, 0, 0, ...;0, b, 0, 0, ...; 0, 0, c, 0, ...; 0, 0, 0, d, ...; dots.v, dots.v, dots.v, dots.v, dots.down)$ sein muss. + +Angenommen $a, b, c, ...$ sind ungleich. Dann laesst sich wieder durch Matrixmultiplikation ein Widerspruch erzeugen. + +Falls nun aber $a = b = c = ...$, dann ist $A = lambda E_n$. + +$qed$ + += Inverse einer Matrix + +Bestimmen Sie das Inverse der Matrix $A = mat(1, -1, 0, 1; 1, 0, -1, 1; 2, 3, -4, 1; 1, 0, 0, 1)$. + +Anwenden des Gaussverfahrens mit der Erweiterten Matrix: + +$ mat(1, -1, 0, 1, 1, 0, 0, 0; + 1, 0, -1, 1, 0, 1, 0, 0; + 2, 3, -4, 1, 0, 0, 1, 0; + 1, 0, 0, 1, 0, 0, 0, 1; augment: #4) $ + += Abbildungsmatrix + + +Sei $frak(B)_1 = {(1, 0, 1), (1, 1, 0), (0, 1, 1)}$ eine Basis von $QQ^3$ und $frak(B)_2 = {(1,1), (-1, 1)}$ eine Basis von $QQ^2$. Sei $phi: QQ^3 -> QQ^2$ gegeben durch die Abbildungsmatrix + +$ A = mat(1, 2, 3; 4, 5, 6) $ + ++ Bestimmen Sie die Abbildungsmatrix $A'$ von $phi$ bezueglich der Basen $frak(B_1)$ und $frak(B_2)$. + ++ Nach der Vorlesung gibt es $X in Q^(3 times 3)$ und $Y in QQ^(2 times 2)$, sodass $A = X A' Y$ gilt. + Bestimmen Sie $X$ und $Y$. \ No newline at end of file diff --git a/S1/AGLA/Hausaufgaben/Zettel08.pdf b/S1/AGLA/Hausaufgaben/Zettel08.pdf new file mode 100644 index 0000000000000000000000000000000000000000..15eb62c27eee5d2d4b879038f2fb7aed3b1fd08f GIT binary patch literal 36450 zcmce;1z256nl+r@7Tn$4-3cBnxVyW%1cC;4cXxujJHg%E2~KeR52Sng-sw9%^Y#4C zQ1x)meoOXSwd?F%d#zPPA|oV1!$8XhMe-H!2L;89Pmgb{X9mU14TXKWQOm|8j7%NyF68qo+?TNu!Y z85mkQm^!%9e0nV-Y^85)U}|NIPxoiLkfFW4p_PHImBVX|_W0^709WKR-`ucPv@!*h zGQ?+lwL$UnLIHBVx(W4~>A&??P)z8x&i^;JUr!7uDDP_TU}!03Wn_*2)+Yd~oT0I) zy@Q=AJ~=wiL{x!02u;ViU4gmE7&hgqOjIa4L|2Q&Uhsr-jhNPjDv4hEL z<6nFK_qZ06(zSU#QkHrhT5khd#MHvU&<=2UEOZ?Vg$!TE+-seHt8J)j3FVS%86#!A 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f`CpxJET{kZ0!N{P)2U42;RD<{aTyt7jZj~;^ literal 0 HcmV?d00001 diff --git a/S1/AGLA/Hausaufgaben/Zettel08.typ b/S1/AGLA/Hausaufgaben/Zettel08.typ new file mode 100644 index 0000000..be8773c --- /dev/null +++ b/S1/AGLA/Hausaufgaben/Zettel08.typ @@ -0,0 +1,21 @@ +// Load the preamble +#import "../conf.typ": conf +#show: conf.with(week: "8") + += Determinanten I + +Sei $K$ ein Koerper mit $"char"(K) != 2$. + ++ Seien $A, B$ zwei $n times n$ Matrizen mit Eintraegen in $K$. Definiere $b_(i j) := (-1)^(i+j) a_(i j)$. + + Zeigen Sie, dass $A = det(B)$. + ++ Bestimmen Sie abhaengig von $n$ die Determinante der Matrix: + + $ C = mat(0, 1, ..., 1; 1, dots.down, dots.down, dots.v; dots.v, dots.down, dots.down, 1; 1, dots, 1, 0) $ + += Determinanten II + += Lineare Abbildung + += Dualraeume diff --git a/S1/AGLA/Hausaufgaben/Zettel09.typ b/S1/AGLA/Hausaufgaben/Zettel09.typ new file mode 100644 index 0000000..f0f6b21 --- /dev/null +++ b/S1/AGLA/Hausaufgaben/Zettel09.typ @@ -0,0 +1,11 @@ +// Load the preamble +#import "../conf.typ": conf +#show: conf.with(week: "9") + += + += + += + += diff --git a/S1/AGLA/Hausaufgaben/Zettel10.typ b/S1/AGLA/Hausaufgaben/Zettel10.typ new file mode 100644 index 0000000..3ac9546 --- /dev/null +++ b/S1/AGLA/Hausaufgaben/Zettel10.typ @@ -0,0 +1,11 @@ +// Load the preamble +#import "../conf.typ": conf +#show: conf.with(week: "10") + += + += + += + += diff --git a/S1/AGLA/Hausaufgaben/Zettel11.typ b/S1/AGLA/Hausaufgaben/Zettel11.typ new file mode 100644 index 0000000..948001e --- /dev/null +++ b/S1/AGLA/Hausaufgaben/Zettel11.typ @@ -0,0 +1,11 @@ +// Load the preamble +#import "../conf.typ": conf +#show: conf.with(week: "11") + += + += + += + += diff --git a/S1/AGLA/Hausaufgaben/Zettel12.typ b/S1/AGLA/Hausaufgaben/Zettel12.typ new file mode 100644 index 0000000..7497696 --- /dev/null +++ b/S1/AGLA/Hausaufgaben/Zettel12.typ @@ -0,0 +1,11 @@ +// Load the preamble +#import "../conf.typ": conf +#show: conf.with(week: "12") + += + += + += + += diff --git a/S1/AGLA/Vorlesungen/VL10.pdf b/S1/AGLA/Vorlesungen/VL10.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7318e52b4857a115da1a7d27353064882fd1b3e8 GIT binary patch literal 20579 zcmch91z1(v)~J*yrP2uKmJW%%VY310Qo2D}ViVGx(jg%oQlcOr-Q7qx(jq0@NQ1;% z8$IXfx#!;d-v58!+iUW%)|g|C*>jFD#~71ZPDG3e%nZYz_5^+y80;Wckfq*p3?3c~ z5Qs&|*$NI~5wo)Bla5MWjWL`c^bj)(=ZC`(BzE2%O{ndrf7>`W}|Z581* zCWcHxmSzS_;s$UFI}991YNZW 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Dann ist durch + +$ Phi : V slash K -> W, quad v + K |-> phi(v) $ + +ein Isomorphismus zwischen $V slash K$ und $phi(V)$ gegeben. +df + +Lemma: + +$phi: V -> W$ linear injektiv $<==>$ $ker(phi) = {0}$. + +Es kommt nicht drauf an, welches $v$ ich waehle um eine Nebenklasse zu definieren! + +Bew: Eigenschaft Kern, und Nebenklassen von v1 und v2 + +Noch zu zeigen ist die surjektivitaet und die injektivitaet von $Phi$. + +Mit dem Umwegsargument ist Jeder Vektorraum zu jedem isomorph?? Wo scheitert das Argument. + +Nutzen der Definition von liearitaet + +FRAGEN: + +Seien V,W K-VR und phi linear was genau ist der Kern von phi?? + +Was ist eine lineare Abbildung zwischen zwei VR + + diff --git a/S1/AGLA/Vorlesungen/VL11.pdf b/S1/AGLA/Vorlesungen/VL11.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ff9190382860be0a0fd76ce0faa3be488b15b0d9 GIT binary patch literal 16168 zcmch81z1$w*1sT)q@*B@bO;R04Ba4I5>kTH&@4Mgg{O6qKnVGfM-e>JR>$ieMQ%;_nk4KPz#Si!p5I{j-kdwI` zftVNp2*j)7=?n+)$~z&DsDIjS=14RFMFyiNGG?xDR4It}x{8vvhCY|7tvTEUX^U`k z)rPy+T5-!bIaqQlS;7%WTcjtqBC5_+goTr(Ey5ba`zu`z?rH%?SehY_s1{v8Mle7W zEn~D8PPzzNpcWh?h@uflNDu&=C@~38O#e|{StU7CoBunpQ4aEa1umj;A@a{!zGRl~&+=rYhR_P_e|y+GT|+1UZ^2#f&;HE4kF05?h= zRF7S3osmv1AOUp00J@wWnu%8rWCR9iFfbUz2V8<+0T4fM!C-tK2yh95VIU}Q!G!og zFyMml!+bBMSR-vf5Me=77rreHSv50f)IU8n)cl}Vg}ki;67B-@(ZLJ} zmxH4w4%H?wWN8K)tsnX^cBNKWC!zYd$=AXHeq9~ 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diff --git a/S1/AGLA/Vorlesungen/VL12.typ b/S1/AGLA/Vorlesungen/VL12.typ new file mode 100644 index 0000000..32b05da --- /dev/null +++ b/S1/AGLA/Vorlesungen/VL12.typ @@ -0,0 +1,21 @@ +- Das Objekt Matrix $A in K^(r times s)$ + +- Zeilen und Spaltenvektor sind Teil von jeder Matrix + +- Hauptdiagonale einer Matrix Nebendiagonale wenn quadratisch +- r = s <==> quadratische Matrix + += Saetze + + +Ezaine lineare Abbildung $Phi: K^s -> K^r$ ist gleichbedeutend zu einer Matrix mit s spalten und k reihen. + +bash + +$dim(L(K^s, K^r)) = rs$; und es gilt dass die Summe der linearen Abbildungen gleich des Summe der Matritzen ist. + +Es gelten folgende Rechenregeln: + +- Distributivitaet (Achtung; die Seiten sind unterschiedlich.) + + diff --git a/S1/AGLA/Vorlesungen/VL13.typ b/S1/AGLA/Vorlesungen/VL13.typ new file mode 100644 index 0000000..5f50897 --- /dev/null +++ b/S1/AGLA/Vorlesungen/VL13.typ @@ -0,0 +1,13 @@ += Matrizen + +Die Multiplikation ist im allgemeinen nicht kommutativ. + +Aufgrund der Gruppeneigenschaft, jedoch die Mult. mit der Inversen. + +Der Gauss Algorithmus ist vollstaendig reversibel. + - Rekursive Vorschrift fuer den Algo. + +$G L_n (K)$ general linear group over $K$ with $n$ rows. + +Wie schreibt man ein LGS formal richtig auf (mit der Summensschreibweisse.) + diff --git a/S1/AGLA/Vorlesungen/VL7.typ b/S1/AGLA/Vorlesungen/VL7.typ new file mode 100644 index 0000000..d458ed6 --- /dev/null +++ b/S1/AGLA/Vorlesungen/VL7.typ @@ -0,0 +1,115 @@ += Vektorraeume + +Ein Vektorraum ueber einem Koerper $K$ ist eine abelische Gruppe $(V, +)$ mit einer Abbildung $K times K arrow V$ (Mult. mit Skalaren), die die Bedingungen + +$ (lambda + mu)v = lambda v + mu v, lambda(v+w) = lambda v + lambda w | forall lambda, mu in K, v, w, in V $ +$ (lambda mu)v = lambda(mu v), 1 v = v $ + +erfuellt. + +== Bsp + +$K^n = K times ... times K = {(x_1, ..., x_j): x_j in K}, lambda (x_1, ..., x_n) = (lambda x_1, ..., lambda x_n)$ + +$RR$ ist ein $QQ$-Vektorraum + +$V_1, V_2$ seien K-Vektorraum. $V_1 times V_2 = {(v_1, v_2): v_j in V_j}$ + +$K times (V_1 times V_2) arrow V_1 times V_2, (lambda, (v_1, v)2)) arrow.flat (lambda v_1, lambda v_2). + +$(V_i)_(i in I)$ sumtlich K-Vektorraum. + +$V:= bigX_under(i in I) V_i = {f: I arrow bigU_under(i in I) V_i: f(i) in V_i} + +in V bildet + +$ W = {f: I arrow bigU_under(i in I) V_i, f(i) in V_i, f(i) != 0 nur fuer endlich viele i in I} $ + +einen Untervektorraum. Ist $I$ endlich, dann $W = V$. Sei $f, g in W$ und $I(f) = {i in I: f(i) !=0}$. + +//////////////// + +Lemma 1: Sei $V$ ein K-Vektorraum $lambda in K, v in K$. Dann ist $lambda v = 0 <==> lambda = 0 or v = 0$ + +Ist $lambda = 0$, dann benutze $0 v = (0+0)v = 0v + 0v <=> 0 = 0 v + 0v v + (-0 v) = 0 v$. Genauso: $V=0 => lambda v = 0$. + +Ist $lambda != 0$ aber $lambda v = 0$, dann $v = 11 v = lambda^(-1) (lambda v) = lambda(-1) 0 = 0 = 11 v = v$ + +$ v + (-w) = v -w $ + +/////////////// + +Lemma 2: + +$V$ ein K-Vektorraum und $lambda in K, v in V$. Kann ist $(- lambda) v = - (lambda) v = - (lambda v), wir schreiben deshalb kurz $- lambda v$ + +denn: $0 = (lambda - lambda) v = lambda v + (- lambda) v = lambda v - (lambda v)$ + +/////////// + +Mehr zu Untervektorraeumen. Gegeben $V$ ein K-Vektorraum. Hat die "tivialen Untervektorraeume" ${0}$ (Nullraum) und $V$. + + +Lemma 3: + +Ist $(W_i)_(i in I)$ eine Familie von Untervektorraeumen von $V$, dann ist $sect.bigU_under(i in I) W_i$ ein Untervektorraum. + +Idee: $M subset V$ eine beliebige Teilmenge. Dann $GrosserSchnitt (W subset V UVR; M subset W) W = angle.l M angle.r$ die von $M$ erzeugte UVR von $V$. Es ist $M subset angle.l M angle.r$ und $angle.l M angle.r$ hat kienen nicht trivialen UVR, der $M$ enthaelt. + +$v in V, M = {v}, angle.l {w} angle.r in.reverse lambda v, lambda in K$. Da die ${lambda v, lambda in K}$ + +einen Untervektorraum bilden, ist $angle.l {v} angle.r = {lambda v: lambda in K}. + +$angle.l {0} angle.r = {0}, Ist v != 0, dann gibt es eine Bijektion (Ue) + +$ K arrow angle.l {u} angle.r; lambda arrow.flat lambda v$ + +$angle.l {v, w} angle.r = {lambda v + mu w: lambda, mu in K}$ + + +Seien $W_1, W_2$ Untervektorraeume von K-Vektorraum $V$. + +$ W_1 + W_2 := angle.l W_1 union W_2 angle.r$ heisst Summe von $W_1, W_2$. +$ = {w_1 + w_2: w_j in W_j}$ + +$ lambda (w_1 + w_2) = (lambda w_1) + (lambda w_2) $ + +$ w_1 + w_2 + w_1' + w_2' = (w_1 + w_1') + (w_2 + w_2') $ + +Frage: Sei $w in W_1 + W_2$. Wie viele Darstellungen $w = w_1 + w_2, w_j in W_j$ gibt es? + +Sei $w_1 + w_2 = w_1' + w_2' <==> w_1 = w_1' = w_2 - w_2' in W_1 sect W_2$ + +Antwort: Es gibt unendlich viele Optionen?. Aber wenn $W_1 sect W_2 = {0}$, dann $w_1 = w_1', w_2 = w_2'$, also sind $w_1$ und $w_2$ eindeutig bestimmt. In diesem Fall heisst $W_1 + W_2$ die innere dierkte Summe von $W_1$ und W_2, in Zeichen $W_1 plus W_2$. + + + +Def: $M subset V$ heisst Eruzeugendensystem, wenn $angle.l M angle.r = V$. +Ist M endlich, und Eruzeugendensystem, ann heisst V endlich erzeugt. + +Bsp: $e_1 = (1, 0, ..., 0), e_2 = (0, 1, 0, ... , 0), ... , e_n = (0, ... , 0, 1) in K^n$ + +ist ein Eruzeugendensystem fuer $K^n$. DAnn fuer jedes v = (v_1, ..., v_n) in K^n$ ist v = v_1 e_1 + ... + v_n e_n in angle.l e_1, ... , e_n angle.r$ + +2) $M subset V$. Dann ist $M$ Eruzeugendensystem $angle.l M angle.r$ + +Def: $V$ ein K-Vektorraum, $v_1 , ... , v_n in V$ heissen linear unabhaengig, wenn aus + +$ lambda_1 v_1 + ... + lambda_n v_n = 0 => lambda_1 = ... = lambda_n = 0 . $ + +Beispiele: $n =1$ Gegeben $v in V$. Test: $lambda v = 0 =>^? lambda = 0. Weil $1 0 = 0$, ist $v=0$ nicht linear unabhaengig. Aber: $v + v != 0, "dann" lambda = 0$. Also: jedes $v != 0 $ ist linear unabhaengig. + +$n = 2$ (Ue); $v_1, v_2 in V \ {0}$. ist $v_2 in angle.l v_1 angle.r, "dann"" v_2 = lambda v_1, "also" 0 = lambda v_1 - v_2 "also" v_1, v_2 nicht linear unabhaengig. _Sonst_ schon! + +Lemma 5: + +Sei $V$ ein K-Vektorraum, und $(v_i)_(i in I)$ eine Familie von $v_i in V$. + +$(v_i)_(i in I)$ ist linear unabhaengig genau dann, wenn jedes $v in V$ hoechstens eine Darstellungen $v = sum(lambda_i v_i)'_(i in I)$ hat. + +Hier heisst $sum()'_(i in I), dass nur endlich viele $lambda_i != 0$ sind. + +Allgemeiner nennen wir $(v_i)_(i in I)$ linear unabhaengig, wenn aus $sum(lambda_i v_i)'_(i in I) = 0 "stets folgt" lambda_i = 0 forall i in I$. + +$v_1, v_2 linear unabhaengig <==> angle.l v_1 angle.r sect$ + diff --git a/S1/AGLA/Zettel/Blatt_03.pdf b/S1/AGLA/Zettel/Blatt_03.pdf new file mode 100644 index 0000000000000000000000000000000000000000..13bf5be8968f5b75e89d6bcb707a316d0a813772 GIT binary patch literal 178267 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score boxes + let doc = wrapper(doc) + + + // Setting the references right + show ref: it => { + let eq = math.equation + let el = it.element + if el != none and el.func() == eq { + // Override equation references. + link(el.location(),numbering( + el.numbering, + ..counter(eq).at(el.location()) + )) + } else { + // Other references as usual. + it + } +} + + // Show the template for ease of use + // Do this at last + show: pset( + class: "AGLA", + student: "Max Offermann, Tom Witjes und Jonas Hahn", + title: [Woche #week], + date: datetime.today(), + doc + ) +} + +// Configuration for the Vorlesungen (vl) +#let vl(doc) = { + // Global settings + set page(numbering: "1") + set heading(numbering: "1.1") + + // Add some content to the docs + outline() + + // Load the document + doc +} diff --git a/S1/AGLA/lectures.txt b/S1/AGLA/lectures.txt new file mode 100644 index 0000000..6c8641a --- /dev/null +++ b/S1/AGLA/lectures.txt @@ -0,0 +1,31 @@ +Nummer Woche Starttag + +1 Woche 0 21 +2 + +3 Woche 1 28 +4 + +5 Woche 2 4 +6 + +7 Woche 3 11 +8 + +9 Woche 4 18 +10 + +11 Woche 5 25 +12 + +13 Woche 6 +14 + +15 Woche 7 +16 + +17 Woche 8 +18 + +19 Woche 9 +20 diff --git a/S1/AGLA/scores.txt b/S1/AGLA/scores.txt new file mode 100644 index 0000000..b094145 --- /dev/null +++ b/S1/AGLA/scores.txt @@ -0,0 +1,7 @@ +1. 48 +2. 43 +3. + +MAX: 578 +GRENZE: 288 + diff --git a/S1/ExPhyI/.ipynb_checkpoints/ha3-checkpoint.ipynb b/S1/ExPhyI/.ipynb_checkpoints/ha3-checkpoint.ipynb new file mode 100644 index 0000000..bf1da98 --- /dev/null +++ b/S1/ExPhyI/.ipynb_checkpoints/ha3-checkpoint.ipynb @@ -0,0 +1,311 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Hausaufgabe Blatt 3\n", + "## Gleichförmig beschleunigte, geradlinige Bewegung - Revisited\n", + "\n", + "In dieser Aufgabe werden wir die Bahnkurve eines gleichförmig beschleunigten Objektes in einer Dimension berechnen und dieses mal auch visualisieren. Die Position $x$ zum Zeitpunkt $t$ ist, wie auf dem Blatt 2, gegeben durch folgende Gleichung:\n", + "\\begin{equation*}\n", + "x\\!\\left( t \\right) = x_0 + v_0 t + \\frac{1}{2} a t^2 \n", + "\\end{equation*}\n", + "wobei $x_0$ und $v_0$ die Anfangsposition und -geschwindigkeit sind und $a$ die konstante Beschleunigung, die auf das Objekt wirkt. \n", + "\n", + "## 1. Numpy Arrays: Linspace\n", + "Anstelle, dass wir die Einträge in numpy arrays \"per Hand\" definieren, können wir eine nützliche Funktion verwenden. \n", + "\n", + "**a)** \n", + "Machen Sie sich mit der nachstehenden Zelle vertraut. Verstehen Sie die Syntax?" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "ExecuteTime": { + "end_time": "2019-11-01T10:22:27.227336Z", + "start_time": "2019-11-01T10:22:27.100666Z" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[0. 0.25 0.5 0.75 1. ]\n" + ] + } + ], + "source": [ + "import numpy as np # Laden der Numpy Bibliothek \n", + "\n", + "x = np.linspace(0, 1, 5) # Definieren von x\n", + "\n", + "print(x) # Ausgabe x" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**b)** Erstellen sie ein numpy array für die Zeit `t` indem sie `np.linspace()` korrekt verwenden. Dabei soll gelten $t_0 = 0$ und $t_N = 5$ mit der Anzahl der Einträge $N = 50$." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**c)** Benutzen Sie die in ha2 Aufgabe 2 definierte Funktion `printBahnkurve()` um sich nun die Bahnkurve für das gerade erstellte array `t` ausgeben zu lassen. Verwenden Sie die Werte $x_0=3$ und $v_0=10$ wie auf Blatt 2." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Return\n", + "\n", + "Bisher hat unsere definierte Funktion lediglich einen `print()` Befehl ausgeführt. Wir wollen nun, dass unsere Funktion einen Wert zurück gibt. Dadurch kann der Wert in einer Variablen gespeichert und somit weiterverarbeitet werden. Dazu verwenden wir das `return` Statement. \n", + "\n", + "**d)** Betrachten Sie die folgenden zwei Funktionen. Beschreiben Sie kurz (1-2 Sätze), was hier geschieht. " + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "ExecuteTime": { + "end_time": "2019-11-01T10:22:27.233313Z", + "start_time": "2019-11-01T10:22:27.230416Z" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "2 4\n" + ] + } + ], + "source": [ + "def identity(x): # definiere Funktion\n", + " return x # definiere Ausgabe\n", + "\n", + "def square(x):# definiere Funktion\n", + " return x**2 # definiere Ausgabe\n", + "\n", + "id2 = identity(2) # definiere id2 über Zugriff auf identity\n", + "square2 = square(2)# definiere id2 über Zugriff auf square\n", + "\n", + "print(id2, square2) # Ausgabe der Werte" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**e)** Schreiben Sie eine neue Funktion, indem Sie den `print()` Befehl in der Funktion `printBahnkurve()` durch das `return` Statement ersetzen. Wählen Sie einen geeigneten Namen für die neue Funktion. " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Visualisierung\n", + "Da Sie nun dazu in der Lage sind, viele Datenpunkte zu erzeugen, wollen wir als nächsten Schritt die berechnete Bahnkurve in einem plot mithilfe von `matplotlib.pyplot` visualisieren. `Matplotlib` ist eine beliebte und sehr vielseitige plot Bibliothek, die es uns ermöglicht Daten zu visualisieren. Wer einen Eindruck davon gewinnen möchte, was alles mit `matplotlib` möglich ist, kann ja mal [hier](https://matplotlib.org/3.1.1/gallery/index.html) vorbeischauen!\n", + "\n", + "Wir haben folgendes Grundgerüst vorbereitet, in dem die Funktion $f(x) = x^2$ beispielhaft geplottet wird." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "ExecuteTime": { + "end_time": "2019-11-01T10:22:42.277126Z", + "start_time": "2019-11-01T10:22:42.160402Z" + } + }, + "outputs": [ + { + "data": { + "image/png": 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", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "import matplotlib.pyplot as plt # lade matplotlib als Bibliothek\n", + "import numpy as np\n", + "x = np.linspace(-2, 2, 170) # definiere x\n", + "xQuadrat = x**2 # berechen x^2\n", + "\n", + "# ### Anfang Grundgerüst ( mit # kann man Kommentare schreiben )\n", + "\n", + "fig, ax = plt.subplots()\n", + "\n", + "ax.set_title(\"Parabel\") # Titel\n", + "ax.set_xlabel(\"X-Werte\") # x-Achsenbeschriftrung\n", + "ax.set_ylabel(\"y-Werte\") # y-Achsenbeschriftung\n", + "\n", + "ax.plot(x, xQuadrat) # x-Wert hier: x, y Wert hier: xQuadrat\n", + "\n", + "plt.show()\n", + "\n", + "# ### Ende Grundgerüst" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**a)** Machen Sie sich mit dem Grundgerüst vertraut, indem Sie \n", + " - `x` mit Werten Ihrer Wahl erweitern\n", + " - einen geeigneten Titel\n", + " - geeignete x- und y- Achsenbeschriftung wählen.\n", + " \n", + "Möchte man mehrere Kurven in einem Diagramm darstellen, so muss `ax.plot()` lediglich erneut aufgerufen werden. \n", + "Dabei ist es nützlich diese Kurven in einer Legende zu unterscheiden:" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "ExecuteTime": { + "end_time": "2019-11-01T10:22:27.961375Z", + "start_time": "2019-11-01T10:22:27.514251Z" + } + }, + "outputs": [ + { + "data": { + "image/png": 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", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "fig, ax = plt.subplots()\n", + "\n", + "ax.set_title(\"Mehrfachplot\") # Titel\n", + "ax.set_xlabel(\"X\") #x-Achsenbeschriftung\n", + "ax.set_ylabel(\"Y\") # y-Achsenbeschriftung\n", + "\n", + "ax.plot(x, xQuadrat, label=\"$x^2$\") # label: Eintrag Legende, versteht auch LaTex!\n", + "ax.plot(x, x**4, label=\"$x^4$\") # label\n", + "\n", + "ax.legend() # Zeige Legende\n", + "plt.show() " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**b)** Kopieren Sie das Grundgerüst und ersetzen sie die x-Werte durch das oben definierte array `t` und die y-Werte durch die errechnete Bahnkurve. Wählen Sie auch hier einen geeigneten Titel und Achsenbeschriftungen." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "ExecuteTime": { + "end_time": "2019-11-01T10:21:24.775267Z", + "start_time": "2019-11-01T10:21:24.689518Z" + } + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**c)** Variieren Sie nun die Anfangsgeschwindigkeit. Erstellen Sie zwei Kurven mit verschiedenen Bedingungen (z.B. $v_0 = 10$ und $v_0=20$). Vergleichen Sie die Kurven miteinander, indem Sie diese in einem Diagramm darstellen. Benutzen Sie angemessene Beschriftungen!" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "ExecuteTime": { + "end_time": "2019-11-01T10:21:24.776711Z", + "start_time": "2019-11-01T10:21:24.023Z" + } + }, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.9" + }, + "latex_envs": { + "LaTeX_envs_menu_present": true, + "autoclose": true, + "autocomplete": true, + "bibliofile": "biblio.bib", + "cite_by": "apalike", + "current_citInitial": 1, + "eqLabelWithNumbers": true, + "eqNumInitial": 1, + "hotkeys": { + "equation": "Ctrl-E", + "itemize": "Ctrl-I" + }, + "labels_anchors": false, + 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arrow(u) = arrow(u) (t) quad arrow(a) != arrow(a)' $ + +Beobachter im beschleunigten BS kann dies feststellen und miteinbeziehen -> die gleichen physikalischen Gesetze gelten. + +$S'$ ist jetzt kein Interalsystem! + +$ x = x' + u_0 t + 1/2 a t^2 $ + +BS eines fallenden Koerpers + +$ y = y' \ +t = t' $ + +=== Gedanken-Experiemente + ++ Beispiel: + diff --git a/S1/ExPhyI/VL11.pdf b/S1/ExPhyI/VL11.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c7356ea857e323e8bf11aed693c2e5a5ff7c74e2 GIT binary patch literal 18619 zcmch91z1$w)<2~PNVg)4gft8c-QAs1>HtHR!_a~>0+LE7(jwg|(v5_aG!hb$(jW*5 z|1+b$>ic@{{l5D>&;Ojm!<@a>-h1s@d+py@YcpxdDRA;|3F0#O0~anXKL`wRHn+nS z6T=08xb?hTU?6S7)Yq=<%)ngDmz&@gHUY( zgzCVoZCw#?FA$3~zyypHpaeoS;g&#^fqJq0sMq%hT{jmO2bd$!1|Vdk0n7u`NPdtt zhTFOzoZ%orRK0+3IYSf?w;{*`421E6!5|*s7U1Ut@d7uWkT3`W+yXqpAU@#c2lI0Y z1D7C(AGjgB{2&3~hVTmkISPXKc!hu#69Vz_gF(W`8_W;l0Wt&wl)NAwAYYrNq6UAGi;KHOrG-8O;~xOuY22 zB2|8t)~AE@0+cn$dCBASQTXe-H(m~j+PVv+o^j5R*RQ5F*?t|L9Q3T?$=cEiW0nht 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physicspackage + $ tan(alpha) = (omega^2 r)/2#pin(4) = (d z (r))/(d r) $ + +A simple #pin(1)highlighted text#pin(2). + +#pinit-highlight(1, 2) + +#pinit-point-from((1, 2))[It is simple.] +#pinit-point-from(5)[It is simple.] +#pinit-point-from(4)[It is simple.] + +he: + diff --git a/S1/ExPhyI/VL12.pdf b/S1/ExPhyI/VL12.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b8f7c798e5106d8777ebfe4c985e0c50d56d3a78 GIT binary patch literal 12082 zcmch7c|4Te+jvBlvSq0##+oI}jF~YbWbFICZy5~6GQ-SZ?3FD`BBd+|C0mrDWJ}Sa zpp3#IhC?@>>b=lQXL|5@XIRLok|CQU?>RD#8L2ID@4)W#NNWlLC%=q zjwe$I#9)dQo=otP*ChFS${Tp%iBtkLOkN-CqeJu{c@l`;5XH4{Z9K&TPxQnQsUSrP z#0d$oV(CI>gJeS_0KM=KWe~=$uFei%f^4#bNdJ*vEdy)T5*dP`&kKNS+tHB}?I2E20EUD@Au!-ULJ<%+@F*eC5GCM2B4H2&_@FTm zB=8{M$}X#y1ze|J%VLZtdQ*KMN+=jO^vb4cncxD!FFO-(UFo~2OYo=S$pCNuI4WKn z4=y-J6PPL<7r;)-3$Sb_-hy&B&GA3oxOJz`)+}31)-n#A`v-NHYTvt=l}frhU8UT6 z{oJXb+*js7ZrIR**N4HU)p?B;K{wk!VB&dwJ7f=KM%uOC46Nfh7m%K-CZ_Yk`_tBQ 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zShE0oN|3jj8olpY)E}Jgl24MIr~_sOmVNc9GzLqXOHd2uwv) z@mG^7@T~%rkJ{qN6rfohs3KJbaaO}XO8&qF8w}{N6#>%3d9SstC#e@VfFs zZv=<<)dyGvU@KHi>6%a-MxW{`z>j%a0+n(EkV!3n9EGADNG6d0URM_Ix3*uqfCo>i zI(UlZVK8|(#u}=u0*9#pM|ss|tRT@l0cV@w74{!GtwLIv(<*`gF-zbK0dIdFL-k+B z7?_2c7#TRE>)o#F*#}y{1>sNf2JZ;^0=Y<0SN-Tb8?VlGb6C3YZ~lVNe86 zn)EviqXgI(Yw_T41fZ#YgCRggvKEF^hW-JAqk&?S-|&=_P=DYdF^E54C>ZEztQ`Xl z2<7iE6i|Nh2MhyNs;upcfdVeoZ!k1c`HwL$Fw`Gwg#qGTzxTz!fzau1^e{^3KVcZ4 zu;zC>^g1xiAANzsu|M_=27>{i*}v&g$vB|)i~OT5-HH&72c->=T1yfMQ1`&>R_siA tp*9@U=ydIGpbc@7{-$ #underline([nach]) dem Stoss) + +EES: ("innere Energie") + +Wir unterscheiden: + +$U = 0$: elastischer Stoss + +$U < 0, E_("kin")' < E_("kin"): inelastischer Stoss + +- zentraler Stoss + +- nicht zentraler Stoss + +Beispiel: + +- zentraler, elastischer Stoss + +- Impuls wird sukzessiv weitergegeben + +- Warum fliegt nicht 1 Kugel mit 2v weg, wenn vorne 2 Kugeln stossen? IES und EES muessen erfuellt sein (EES ist quadratisch mit der Geschwindigkeit) + +- Nur 2 Kugeln: elastischer vs. inelastischer Stoss + += Elastische Stoesse + +Geg: $m_1, m_2, arrow(v)_1, arrow(v)_2$ #h(10pt) (vor dem Stoss) + +Ges: $arrow(v)'_1, arrow(v)'_2$ (nach dem Stoss) + +Wahl des Koordinatensystems: mitbewegt, $v_2 = 0$ + +EES: $ 1/2 m_1 v_1^2 = 1/2 m_1 v'_1^2 + 1/2 m_2 v'_2^2 $ + +JES: $ m_1 arrow(v_1) = m_1 arrow(v') + m_2 arrow(v'_2) $ + +Man berechne: + +@jes quadrieren + +@ees ($dot 2 m_1$) + +Es folgt: + +$ 2 m_1 arrow(v_1)' dot arrow(v_2)' + m_2 v_2'^2 = m_1 v'_2^2 $ + +$ arrow(v_1)' dot arrow(v'_2) = (m_1 - m_2)/(2 m_1) v'_2^2 $ + ++ $m_1 = m_2; arrow(v')_1 dot arrow(v')_2 = 0 --> v'_1 = 0$ + + zentraler Stoss + ++ $m_1 = m_2$ $arrow(v')_1 perp arrow(v')_2$ + + nicht zentral + + Teile die Geschwindigkeiten der beiden Massen in seine Komponenten auf. + + Dabei behaelt $m_1$ die Tangentialkomponente, wohingegen $m_2$ Zentralkomponente bekommt. + ++ $m_1 != m_2$, zentraler Stoss + + $ v'_1 = (m_1 - m_2)/(2 m_1) v'_2 $ + ++ $m_1 < m_2 -> v'_1 < 0$ ($m_1$ wird reflektiret) + ++ + diff --git a/S1/ExPhyI/VL17.pdf b/S1/ExPhyI/VL17.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b345fffb9f4d2c4ee7861ce5d42a0f54d2901c8b GIT binary patch literal 33202 zcmce-1y~)?k}ev8yE}o6YjAgWmp~x6J3)d&aJS%2(BQ${AxLm{cZU#MZtvvGJu`F9 zn|W{E_uajUkKNU^s;g?PZs`ACi&9x!@+}J^7XsyH;Ddm`0b&N(8(AUn^CN&jOzQ3q zrXVIsdpl?Fk(!H<^B)CR%?#Fv8akPR&jR11WJHuD>E$hrOdXvq?OdGHOdT!F-iq4W zn7oxSF|~8HbasC$4eldhXKZg`X=e^%`j2{XQzv6nI}<}YXRt>nkPashimL9PFzhw# zEP-C8Aa1Y@K~N9@ump#R05;}w$;2mAd07B=|8Ku0xqCudVz89Ot3(4U$BT2)hX zODAVXcMy#T5QHf$US*) z#0fMuRt~@y7l@668>kgGh?RpG!~--AW_Az@;EEZ<&cO*{0bBz1EIc3dI1AWufQ5qR zR%Wnp^YDOxuHfFxtUxG0{J^EyfYWSXVF&Ag19lJ(Gl%ZKA5-uk|8q>`P3_E`Er7UL zz;*jOBgGU99l%GL3gAicXDUis+Blm!0)@3PbT$^GUw zg3lhq7a%bNzJGj8VXHx!_(~P!NxAsb7%B0*bZ>OVc4vqFQl*$?srKeps(Yq;f0n#& zE6MJdDv5jhE$w~)pFm3Z-kscnV&PPIMR2gj%Mio+=!Mfb?Yz=x&!Gt0uZOF$h#I;u 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Schallwellen, Wasserwellen + +Wir betrachten die Ausbreitung in #underline[einer Richtung]. + +== Darstellung harmonischer Wellen + +$ xi(z, t) = A sin(omega(t-z/v)) = A sin(omega t - k z), quad k = 2 pi/lambda, v eq "Phasengeschwindigkeit" $ + +Hier ist $xi$ i.A. nicht der Ort, kann z.B auch der Schallauch sein, el. Feldstaerke. + +Wellenlaenge $lambda$: Abstand $Delta z$ zweier Punkte, fuer die die Auslenkung $xi(z_1, t) = xi(z_2,t)$ zur gleichen Zeit $t$ + +Phasengeschwindigkeit: Geschwindigkeit, mit der sich die Phase ausbreitet $v = omega/k = 2 pi v lambda/(2 pi) = v lambda$ + +$ xi(z,t)&=c e^(i(omega z - k z)) + c^* e^(-i(omega t - k z)) \ &=A cos(omega t-k z)+B sin(omega t-k z) \ "mit" A&=c+c^* quad B=i(c-c^*) $ + +an jedem festen Ort $z=z_0$: zeitlich periodische harmoische Schwingung + +$ xi(t)=A sin(omega t-k z_0)=A sin(omega t-phi) $ + +zu jedem festen Zeitpunkt: diff --git a/S1/ExPhyI/VL6.pdf b/S1/ExPhyI/VL6.pdf new file mode 100644 index 0000000000000000000000000000000000000000..922adf6a74011ea4e844603d6e403b795ebba258 GIT binary patch literal 47456 zcmcee1y~%-y5|!DL4y;5L(t$hGZ5U}J-7vTfKRi1Uk#xftz+8CKxn~*a7d%Cc(gQ2mtk-oJfv_%I}O$gwMqV}U3 zHY(O;KrLfZPG}kuFE0{M66z)rw9tR*uaJZ=w9WrFx1l!%Dk`};I2v0?SiiR+ee4q; zRngeQ%)!y#m6S#RaKiWvP!dR$vo``l2I8XmM_hkpD1ETCwKTQ@`hXPLX@K)UX{bNY 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(EES der Mechanik) + + Beschleunigungsarbiet (2.NG) wird als Zuwachs kinetischer Energie gespeichert. + +- Energie wird nicht "erzeugt" oder "verbraucht" +- Energieformen werden ineinander umgewandelt + +Beispiel: Pendel + +// Skizze einfuegen + +1. $v = 0 arrow E_("kin") = 0, h = h_("max") arrow E_("pot") = m g h_("max")$ + +2. $v = v_("max") arrow E_("kin") = 1/2 m v_("max")^2, h = 0 arrow E_("pot") = 0$ + +3. Siehe 1 + += Kraftfelder und Potential + +Wir nehmen ein konservatives Kraftfeld an und gehen von Punkt $P$ um $delta arrow(r)$ zu Punkt $P'$ + +// Skizze von verschiedenen Wegen von Punkt P zu P' in 2D und der Steitung dy +// Vektoren fuer die Kraefte sind eingezeichnet + +es aendert sich die potientielle Energie $E_p (x, y, z) = E_p (P)$ um $delta E_p = (delta E_p)/(delta x) delta x + (delta E_p)/(delta y) delta y + (delta E_p)/(delta z) delta z$ + +da wir eine Strecke in einem Kraftfeld zuruecklegen, verrichten wir Arbeit $delta W = arrow(F) dot d arrow(r) = - delta E_p$ $delta W = arrow(F) dot d arrow(r) = - delta E_p$ (gespeichert als potientielle Energie) + +--> $F_x delta x + F_y delta y + F_z delta z =^"von oben" - (delta E_p)/(delta x) delta x - (delta E_p)/(delta y) delta y - (delta E_p)/(delta z) delta z$ + +daher $F_x = - (delta E_p)/(delta x) delta x F_y = - (delta E_p)/(delta y) delta y F_z = - (delta E_p)/(delta z) delta z$ + +$arrow(F) = - "grad" E_p = - nabla E_p$ + +Nachtrag: Es existiert ein Skalarfeld DURCHSCHNITT $(arrow(r))$ (Potential), sodass $arrow(F) = - k nabla o (arrow(r))$ + +($<--> arrow(F)(arrow(r))$ ist ein konservatives Kraftfeld) + + + +== Drehimpuls und Drehmoment + +MP mit Impuls $arrow(p)$ auf Bahn $arrow(r) (t)$ + +Definition: $arrow(L) = arrow(r) times arrow(p) = m (arrow(r) times arrow(v))$ Drehimpuls + +wichtig: immer in Bezug auf den Koordinatenursprung $O$. + +wir zerlegen $arrow(v) "in" arrow(v)_r || arrow(r)$ und $arrow(v)_phi "senkrecht" r$ (Polarkoordinaten) + +$ --> arrow(L) = m [ arrow(r) times (arrow(v)_r + arrow(v)_phi)] = m arrow(r) times arrow(v)_phi$ + +$abs(arrow(L)) = m r omega r sin(90 degree) = m r^2 dot(phi)$ + +$dot(arrow(L)) = (d arrow(L))/(d t) = [(d arrow(r))/(d t) times arrow(p)] + [arrow(r) times (d arrow(p))/(d t)]$ + +$= arrow(v) times arrow(p) + arrow(r) times dot(arrow(p)) = arrow(r) times arrow(F)$ + + +Definition: $arrow(D) = dot(arrow(L)) = arrow(r) times arrow(F)$ (Drehmoment) + +=== Analogie + +#table( + columns: 2, + [Translation], [Rotation], + [Impuls $arrow(p)$], [Drehimpuls $arrow(L) = arrow(r) times arrow(p)$], + [Kraft $arrow(F) = dot(arrow(p))$], [Drehmoment $arrow(D) = dot(arrow(L)) = arrow(r) times arrow(F)$], + [1. NG $arrow(p) = "const" <--> arrow(F) = 0$], [$arrow(L) "const" <--> arrow(D) = 0$], + [Position $arrow(r)$], [Winkel $phi$], + [Geschwindigkeit $arrow(v) = dot(arrow(r))$], [Winkelgeschwindigkeit $arrow(omega) = dot(phi) arrow(e)_z$], +) + +Beispiel: Zentralkraftfelder $arrow(F) = f(r) arrow(e)_r$ + +$ arrow(D) = f(r) arrow(r) times arrow(e)_r = 0 = dot(arrow(L))$ + +$ --> arrow(L) = "const"$ + diff --git a/S1/ExPhyI/VL9.pdf b/S1/ExPhyI/VL9.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4a183bd46ccb161589e2211ab88c5817008bc45a GIT binary patch literal 30869 zcmce;1yml*mhT;i;1HbP65QS0Ex5b8ySuw4xI=J<-~_jz!QEYgJGYjiSMAzWUEQ7MzkUU=tdIybJq;@~u`lp}hGxR2!?)2lhvwpf#>b~sbh9v2F6A9k8%BVLcz(_*3!rdm;-#!qyfnTwL$rSMr?0p z>u6(-&+>a*fL}=EcO_aCd<{Cl$4p0uPY)CpW+r?FpfItq<1+$1{{na zV`T&!Odw-s0es9LV`5;#2f_fx#00cu1sMx7P!Di}dH|yQHdaM<6l-BghyTnDH5a1E41+;1qBGWX=dUSU|?e4m1Rs17l$X9Bd$CW~0~o z*Gmst3;(?IQbyJ$j;8obEbO4^{j+cdrS)ti^KBxD3y 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beschleunigten Objektes in einer Dimension berechnen und dieses mal auch visualisieren. Die Position $x$ zum Zeitpunkt $t$ ist, wie auf dem Blatt 2, gegeben durch folgende Gleichung:\n", + "\\begin{equation*}\n", + "x\\!\\left( t \\right) = x_0 + v_0 t + \\frac{1}{2} a t^2 \n", + "\\end{equation*}\n", + "wobei $x_0$ und $v_0$ die Anfangsposition und -geschwindigkeit sind und $a$ die konstante Beschleunigung, die auf das Objekt wirkt. \n", + "\n", + "## 1. Numpy Arrays: Linspace\n", + "Anstelle, dass wir die Einträge in numpy arrays \"per Hand\" definieren, können wir eine nützliche Funktion verwenden. \n", + "\n", + "**a)** \n", + "Machen Sie sich mit der nachstehenden Zelle vertraut. Verstehen Sie die Syntax?" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "ExecuteTime": { + "end_time": "2019-11-01T10:22:27.227336Z", + "start_time": "2019-11-01T10:22:27.100666Z" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[0. 0.25 0.5 0.75 1. ]\n" + ] + } + ], + "source": [ + "import numpy as np # Laden der Numpy Bibliothek \n", + "\n", + "x = np.linspace(0, 1, 5) # Definieren von x\n", + "\n", + "print(x) # Ausgabe x" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**b)** Erstellen sie ein numpy array für die Zeit `t` indem sie `np.linspace()` korrekt verwenden. Dabei soll gelten $t_0 = 0$ und $t_N = 5$ mit der Anzahl der Einträge $N = 50$." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**c)** Benutzen Sie die in ha2 Aufgabe 2 definierte Funktion `printBahnkurve()` um sich nun die Bahnkurve für das gerade erstellte array `t` ausgeben zu lassen. Verwenden Sie die Werte $x_0=3$ und $v_0=10$ wie auf Blatt 2." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Return\n", + "\n", + "Bisher hat unsere definierte Funktion lediglich einen `print()` Befehl ausgeführt. Wir wollen nun, dass unsere Funktion einen Wert zurück gibt. Dadurch kann der Wert in einer Variablen gespeichert und somit weiterverarbeitet werden. Dazu verwenden wir das `return` Statement. \n", + "\n", + "**d)** Betrachten Sie die folgenden zwei Funktionen. Beschreiben Sie kurz (1-2 Sätze), was hier geschieht. " + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "ExecuteTime": { + "end_time": "2019-11-01T10:22:27.233313Z", + "start_time": "2019-11-01T10:22:27.230416Z" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "2 4\n" + ] + } + ], + "source": [ + "def identity(x): # definiere Funktion\n", + " return x # definiere Ausgabe\n", + "\n", + "def square(x):# definiere Funktion\n", + " return x**2 # definiere Ausgabe\n", + "\n", + "id2 = identity(2) # definiere id2 über Zugriff auf identity\n", + "square2 = square(2)# definiere id2 über Zugriff auf square\n", + "\n", + "print(id2, square2) # Ausgabe der Werte" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**e)** Schreiben Sie eine neue Funktion, indem Sie den `print()` Befehl in der Funktion `printBahnkurve()` durch das `return` Statement ersetzen. Wählen Sie einen geeigneten Namen für die neue Funktion. " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Visualisierung\n", + "Da Sie nun dazu in der Lage sind, viele Datenpunkte zu erzeugen, wollen wir als nächsten Schritt die berechnete Bahnkurve in einem plot mithilfe von `matplotlib.pyplot` visualisieren. `Matplotlib` ist eine beliebte und sehr vielseitige plot Bibliothek, die es uns ermöglicht Daten zu visualisieren. Wer einen Eindruck davon gewinnen möchte, was alles mit `matplotlib` möglich ist, kann ja mal [hier](https://matplotlib.org/3.1.1/gallery/index.html) vorbeischauen!\n", + "\n", + "Wir haben folgendes Grundgerüst vorbereitet, in dem die Funktion $f(x) = x^2$ beispielhaft geplottet wird." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "ExecuteTime": { + "end_time": "2019-11-01T10:22:42.277126Z", + "start_time": "2019-11-01T10:22:42.160402Z" + } + }, + "outputs": [ + { + "data": { + "image/png": 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", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "import matplotlib.pyplot as plt # lade matplotlib als Bibliothek\n", + "import numpy as np\n", + "x = np.linspace(-2, 2, 170) # definiere x\n", + "xQuadrat = x**2 # berechen x^2\n", + "\n", + "# ### Anfang Grundgerüst ( mit # kann man Kommentare schreiben )\n", + "\n", + "fig, ax = plt.subplots()\n", + "\n", + "ax.set_title(\"Parabel\") # Titel\n", + "ax.set_xlabel(\"X-Werte\") # x-Achsenbeschriftrung\n", + "ax.set_ylabel(\"y-Werte\") # y-Achsenbeschriftung\n", + "\n", + "ax.plot(x, xQuadrat) # x-Wert hier: x, y Wert hier: xQuadrat\n", + "\n", + "plt.show()\n", + "\n", + "# ### Ende Grundgerüst" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**a)** Machen Sie sich mit dem Grundgerüst vertraut, indem Sie \n", + " - `x` mit Werten Ihrer Wahl erweitern\n", + " - einen geeigneten Titel\n", + " - geeignete x- und y- Achsenbeschriftung wählen.\n", + " \n", + "Möchte man mehrere Kurven in einem Diagramm darstellen, so muss `ax.plot()` lediglich erneut aufgerufen werden. \n", + "Dabei ist es nützlich diese Kurven in einer Legende zu unterscheiden:" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "ExecuteTime": { + "end_time": "2019-11-01T10:22:27.961375Z", + "start_time": "2019-11-01T10:22:27.514251Z" + } + }, + "outputs": [ + { + "data": { + "image/png": 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", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "fig, ax = plt.subplots()\n", + "\n", + "ax.set_title(\"Mehrfachplot\") # Titel\n", + "ax.set_xlabel(\"X\") #x-Achsenbeschriftung\n", + "ax.set_ylabel(\"Y\") # y-Achsenbeschriftung\n", + "\n", + "ax.plot(x, xQuadrat, label=\"$x^2$\") # label: Eintrag Legende, versteht auch LaTex!\n", + "ax.plot(x, x**4, label=\"$x^4$\") # label\n", + "\n", + "ax.legend() # Zeige Legende\n", + "plt.show() " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**b)** Kopieren Sie das Grundgerüst und ersetzen sie die x-Werte durch das oben definierte array `t` und die y-Werte durch die errechnete Bahnkurve. Wählen Sie auch hier einen geeigneten Titel und Achsenbeschriftungen." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "ExecuteTime": { + "end_time": "2019-11-01T10:21:24.775267Z", + "start_time": "2019-11-01T10:21:24.689518Z" + } + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**c)** Variieren Sie nun die Anfangsgeschwindigkeit. Erstellen Sie zwei Kurven mit verschiedenen Bedingungen (z.B. $v_0 = 10$ und $v_0=20$). Vergleichen Sie die Kurven miteinander, indem Sie diese in einem Diagramm darstellen. Benutzen Sie angemessene Beschriftungen!" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "ExecuteTime": { + "end_time": "2019-11-01T10:21:24.776711Z", + "start_time": "2019-11-01T10:21:24.023Z" + } + }, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.9" + }, + "latex_envs": { + "LaTeX_envs_menu_present": true, + "autoclose": true, + "autocomplete": true, + "bibliofile": "biblio.bib", + "cite_by": "apalike", + "current_citInitial": 1, + "eqLabelWithNumbers": true, + "eqNumInitial": 1, + "hotkeys": { + "equation": "Ctrl-E", + "itemize": "Ctrl-I" + }, + "labels_anchors": false, + "latex_user_defs": false, + "report_style_numbering": false, + "user_envs_cfg": false + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/S1/ExPhyI/ha3.py b/S1/ExPhyI/ha3.py new file mode 100644 index 0000000..d2cca62 --- /dev/null +++ b/S1/ExPhyI/ha3.py @@ -0,0 +1,98 @@ +""" + "# Hausaufgabe Blatt 3\n", + + "## Gleichförmig beschleunigte, geradlinige Bewegung - Revisited\n", + + "In dieser Aufgabe werden wir die Bahnkurve eines gleichförmig beschleunigten Objektes in einer Dimension berechnen und dieses mal auch visualisieren. Die Position $x$ zum Zeitpunkt $t$ ist, wie auf dem Blatt 2, gegeben durch folgende Gleichung:\n", + "\\begin{equation*}\n", + "x\\!\\left( t \\right) = x_0 + v_0 t + \\frac{1}{2} a t^2 \n", + "\\end{equation*}\n", + "wobei $x_0$ und $v_0$ die Anfangsposition und -geschwindigkeit sind und $a$ die konstante Beschleunigung, die auf das Objekt wirkt. \n", + "## 1. Numpy Arrays: Linspace\n", + "Anstelle, dass wir die Einträge in numpy arrays \"per Hand\" definieren, können wir eine nützliche Funktion verwenden. \n", + + "**a)**", + "Machen Sie sich mit der nachstehenden Zelle vertraut. Verstehen Sie die Syntax?" +""" + +import numpy as np + +x = np.linspace(0, 1, 5) + +print(x) + +""" + "**b)** Erstellen sie ein numpy array für die Zeit `t` + indem sie `np.linspace()` korrekt verwenden. + Dabei soll gelten $t_0 = 0$ + und $t_N = 5$ mit der Anzahl der Einträge $N = 50$." +""" + +t = np.linspace(0, 5, 50) + +""" + "**c)** Benutzen Sie die in ha2 Aufgabe 2 definierte Funktion `printBahnkurve()` + um sich nun die Bahnkurve für das gerade erstellte array `t` ausgeben zu lassen. + Verwenden Sie die Werte $x_0=3$ und $v_0=10$ wie auf Blatt 2." +""" + +def printBahnkurve(x0, v0, t): + g = -9.81 + x = x0 + v0 * t + 1/2 * g * t ** 2 + print(x) + +x0 = 3 +v0 = 10 + +printBahnkurve(x0, v0, t) +""" + + "## Return\n", + "Bisher hat unsere definierte Funktion lediglich einen `print()` Befehl ausgeführt. + Wir wollen nun, dass unsere Funktion einen Wert zurück gibt. + Dadurch kann der Wert in einer Variablen gespeichert und somit weiterverarbeitet werden. Dazu verwenden wir das `return` Statement. \n", + + "**d)** Betrachten Sie die folgenden zwei Funktionen. Beschreiben Sie kurz (1-2 Sätze), was hier geschieht. " + +""" + +# Funktion nimmt ein Argument und gibt es wieder zurueck +def identity(x): + return x + +# Funktion nimmt ein Argument und probiert das Quadrat dieses zurueckzugeben +def square(x): + return x**2 + +# Weise dem Namen 'id2' den Rueckgabewert der Identityfunktion mit dem Argument 2 zu. id2 = 2 +id2 = identity(2) # definiere id2 über Zugriff auf identity + +# Weise dem Namen 'square2' den Rueckgabewert der squarefunktion mit dem Argument 2 zu. square2 = 4 +square2 = square(2)# definiere square über Zugriff auf square + +print(id2, square2) # Ausgabe der Werte + +""" + "**e)** Schreiben Sie eine neue Funktion, + indem Sie den `print()` Befehl in der Funktion `printBahnkurve()` durch das `return` Statement ersetzen. + Wählen Sie einen geeigneten Namen für die neue Funktion. " +""" + +def returnBahnkurve(x0, v0, t): + g = -9.81 + x = x0 + v0 * t + 1/2 * g * t ** 2 + return x + + +""" + "## Visualisierung\n", + "Da Sie nun dazu in der Lage sind, + viele Datenpunkte zu erzeugen, + wollen wir als nächsten Schritt die berechnete Bahnkurve in einem plot mithilfe von `matplotlib.pyplot` visualisieren. `Matplotlib` ist eine beliebte und sehr vielseitige plot Bibliothek, + die es uns ermöglicht Daten zu visualisieren. Wer einen Eindruck davon gewinnen möchte, was alles mit `matplotlib` möglich ist, kann ja mal [hier](https://matplotlib.org/3.1.1/gallery/index.html) vorbeischauen!\n", + "Wir haben folgendes Grundgerüst vorbereitet, in dem die Funktion $f(x) = x^2$ beispielhaft geplottet wird." +""" + + + + diff --git a/S1/ExPhyI/scores.txt b/S1/ExPhyI/scores.txt new file mode 100644 index 0000000..d1c8797 --- /dev/null +++ b/S1/ExPhyI/scores.txt @@ -0,0 +1,8 @@ +1. 20 +2. 19,5 +3. +4. + +MAX. 240 +GRENZE. 120 + diff --git a/S1/ExPhyI/Übungsblatt_1.pdf b/S1/ExPhyI/Übungsblatt_1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..07133338a7aab52cbab5e01aa44f3c42495c0b73 GIT binary patch literal 1012651 zcmeFYcT`i~w=Wu`Ns|r&LX;vPO_44T5di@a5a}fX(gmbR3yDha0s;a;06|2Ai1Zri zqSAYj5>z@Q)D0xz?eBTx{_Z*Fp8MW?Z`}9SbC2u+*;#vM&9y#juKD@Q84;t~+EUk~ z73t~aE?&Lp?ch$Ys!A_&%gx`<$jMjJ+w+OHmy?(OMJ0L}P4CCvz9vuX9i1-9+;)2E z=IC@^=N7$;w%cQWCtvWA=7ZZ#j^2-)=wtV}O+*U8Jp-}R!RlJd2y 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zziG_S-N|2kEI>A|ziDjH9n@cZ>}>xU8^g_}`AM}4vpHcthV+0KP7tP*Q z56GBji*z&Z=oEmb3Fr&~W|}+%0z7>{=kv`*O$=-dfdbV1b16uyLG|BUJvH%vFS^+s X0{-6C{$@ThGO{qUB9W5|$q4^{#oBKd literal 0 HcmV?d00001 diff --git a/S1/ReMe/VL18.typ b/S1/ReMe/VL18.typ new file mode 100644 index 0000000..2741913 --- /dev/null +++ b/S1/ReMe/VL18.typ @@ -0,0 +1,57 @@ +// 2025-01-06 09:05 + += Lineare Abbildungen + +Eine Abb. $f: V -> W$, zwischen VR $V$ und $W$ ueber $K$ ist linear wenn + +$ forall x, y in V space forall lambda in K: f(x+y) = f(x) + f(y) and f(lambda x) = lambda f(x) $ + += Matrizen und lineare Gleichungssysteme + +Eine Matrix ist gleichbedeutend zu einer linearen Abbildung. + +== Inhomogene LGS + +Sei $arrow(x)_"hom"$ Lsg. des homogenen LGS $A arrow(x)_"hom" = bb(0)$. + +Sei $arrow(x)_"part"$ Lsg. des inhomogenen LGS $A arrow(x)_"part" = arrow(c)$. + +$ A(arrow(x)_"hom" + arrow(x)_"part") = arrow(c) $ + +$arrow.r.curve$ Ein inhomogenes LGS hat genau dann eine eindeutige Lsg., wenn hom. Lsg. und inhom. Lsg. existieren. + +== Inverse + +Eine Matrix $A$ heisst invertierbar, wenn die zug. Abb. ein Isomorphismus ist. Die Matrix der Umkehrabb. heisst dann inverse Matrix $A^(-1)$. + +=== Eigenschaften + ++ jede invertierbare Matrix ist quadratisch ++ sind $A, B in K^(n times n)$, dann ist $ B = A^(-1) <=> A B = B A = E_n, quad E_n = (delta_(i j)) $ ++ ist $A$ invertierbar, so auch $A^(-1)$ ++ sind $A,B in K^(n times n)$ invertierbar, so auch $A B$ + +$arrow.r.curve$ Existiert inverse Matrix zur Kopf-Matrix eines inhom. LGS, so ist diese geloest durch $arrow(x) = A^(-1) arrow(c)$. + +== Determinante + +Diese ist relevant fuer das Loesen von LGS, invertieren von Matrizen, Subst. von Funktionen mehrerer Veraenderlicher und Eigenwertproblemen. + +$ n = 2, quad det(A) = abs(mat(a, b;c, d)) = a d - b c $ +$ n = 3, quad det(A) = abs(mat(a, b, c;d, e, f; g, h, i)) $ + +=== Laplace'scher Entwicklungssatz + +Determinante kannn nach beliebiger Zeile oder Spalte entickelt werden. + +$ det(A) = abs(mat(a_11, a_12, a_13;a_21, a_22, a_23; a_31, a_32, a_33)) = a_11 abs(mat(a_22, a_23; a_32, a_33)) - a_21 abs(mat(a_12, a_13; a_32, a_33)) + a_31 abs(mat(a_12, a_13;, a_22, a_23)) $ + +Allgemein gilt fuer $n >= 2$, $A in K^(n times n)$ + +$ det(A) = sum^n_(i=1) (-1)^(i+1) a_(i 1) U_(i 1), quad U:="Unterdeterminante an der Stelle" (i 1) $ + +== Transponierte + +Ist $A = (a_(i j))$, so ist die transponierte Matrix gegeben durch $a_(i j)^T = a_(j i)$.T +Fuer quadrasiche Matrizen bleibt die Determinante beim Transponieren gleich. 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beschraenkt. +Ein Konvergenzkriterium ist bspw. das Wurzel oder Quotientenkriterium. +> BSP: e^x = Die Reihe ueber x^n/n! -> r = oo + +Innerhalb des Konvergenzbereichs definiert die Potenzreihe eindeutig eine Funktion. + +## Entwickeln von Funktionen + +Blueprint: + +- Betrachten der Funktion +- Ansatz fuer Potenzreihe + - f(x) = a0 + a1x + a2 x**2 + ... +- Einsetzen des Entwicklungspunkts +- Ableitungen bilden +- Zusammenfassung in eine Reihe +- Konvergenzradius pruefen + +Zum Ableiten koennen wieder die Allgemeinen Ableitungen von Potenzreihen gebildet werden und dann einfach den Entwicklungspunkt einsetzen. diff --git a/S2/AnaMech/AMechVL1.pdf b/S2/AnaMech/AMechVL1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c927465642288665026253b46118a2f64333d725 GIT binary patch literal 50853 zcmce;1yo$kwk;goH8=za?rEfP2rj|h-QAtw65Js;L4%VJAh<)20KwheJp{|!O}=x_ zIrn}4z2m=kjMue#?C#oCwbrU#dpBd&oKP!@Niegra3NC%054=@Fgb|a!Ppv^pC6f= 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zoTM@a11u$h7H@z5mqQz30AR^qw*cZK`+pjxuQ2`Rq^(yAz)+3FZTUwY0bVX%vP+kw H6s7+k&k)Iw literal 0 HcmV?d00001 diff --git a/S2/AnaMech/AMechVL1.typ b/S2/AnaMech/AMechVL1.typ new file mode 100644 index 0000000..385745c --- /dev/null +++ b/S2/AnaMech/AMechVL1.typ @@ -0,0 +1,113 @@ +#import "./preamble.typ": * + +#show: conf.with(num: 4) += Einleitung + +Newton < Lagrange < Hamilton + +== Ziel der Analytischen Mechanik + +Dynamik von mechanischen Systemen (N Massepunkte) + +Aufstellen von BWGL mit der Forminvarianz + +Loesen von BWGLn. Hier gibt es nur wenig loesbare Beispiele: Zentralpotential und gekoppelte Oszillatoren. + +Erhaltungsgroessen & Symetrien beim Loesen ausnutzen und neue Erhaltungsgroessen auffinden. + +Newton <= Lagrange = Hamilton < Hamilton'sches Prinzip + +Abstraktionsweg: + +$"Kraefte" -> "Potentiale" -> "Lagrange-/ Hamiltonfunktion" -> "Wirkung"$ + +== Vorlesung + +24VL + +Newton'sche Mechanik (6VL) + +1D Probleme, Numerik, Reduktion auf DGL 1. Ordnung, Zentralpotential, Streuprobleme + +#example[ + Ich meine das PDF gibts auch online +] + += Notes + +Es wird ein Skript geben, welches vom letzten Jahr etwas umgebaut werden wird. + +Hoersaaluebung sei das wichtigste (Fr 16-18). Vielleicht sogar wichtiger und ersetzend zu den kleinen Uebungen. + +Donnerstag nach Ostern Vorstellung der TeamCaptains. Das sind irgendwelche Studierende, welche freiwillig Ansprechpartner fuer Fragen sein wollen und einen Kommunikationskanal zum Dozenten herstellen. + += Tafelanschriebe + +- Alles vom Standpunkt Newton mit Erhaltungsgroessen (6-7VL) +- Loesen vom Zentralkraftproblem + +== Newton Mechanik + +=== 1D Systeme + +Grundlage fuer die ersten Wochen + +Newton II. BWGL + +// Arrow und dot in typstar +// TODO: make it work with subscripts + +$ +dot(arrow(p)) = dif / (dif t) arrow(p) = m dot.double(r) = sum_(i=0)^(n) arrow(F_i) (arrow(r) dot(arrow(r))) +$ + +Dabei ist $arrow(r)$ ein Vektor in Kartesichen Koordinaten. + +Ziel: $arrow(r) = arrow(r)(t)$ bzw $x(t), y(t) z(t)$ also eine Lsg von @bwg durch das Anfangswertproblem + +$ + arrow(r)(t) = arrow(r_0) \ + dot(arrow(r))(t_0 ) = dot(arrow(r_0)) +$ + +#example[ + 1D Oszillator im Gravitationsfeld. + + Das KS wird so gewaehlt, dass $z$ zu einem kraeftefreien Punkt wird. $arrow(F_g) + arrow(F_r)(z) = arrow(0)$ + + $ + z = z(t) \ + arrow(F_g) = -m g \ + arrow(F_k) = -k z + m g = -k Delta l + $ + + BWGL: + + $ + m dot.double(z) = arrow(F_g) + arrow(F_k) = - m g - k z + m g\ + m dot.double(z) + k z = 0 + $ + + Gewoehnliche DGL 2. Ordnung in Zeit $t$ fuer $z=z(t)$ BWGL fuer $z$. + + - linear homogen, Koeffizienten konstant + + Standardloesungsansatz: + + $ + z(t) = z_0 e^(lambda t), z_0 in CC, lambda in CC + $ + + Ableiten und Einsetzen + $ + dot.double(z(t)) = z_0 lambda^2 e^(lambda t) ==> z_0 e^(lambda t) [lambda^2 + k/m] = 0, forall t + $ + + Lsg. $z_0 = 0 or lambda = p_m i omega_0, omega_0 = sqrt(k/m)$ +] + +Allgemeine Loesung des Beispiels: + +// TODO: mit dem Skript weiterschreiben zuende machen + + diff --git a/S2/AnaMech/AnaMech_Hahn_Penning_Zettel_1.pdf b/S2/AnaMech/AnaMech_Hahn_Penning_Zettel_1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e514c1db3134d25fde3a792356ae12a70065b540 GIT binary patch literal 28616 zcmce+1y~);mhX*2a1R9AI0SchcXtTx?(V@QxJz&+xVt;SU4sXA3w)d8J#)^P`_9bs z&3&G`dy(v}uC7|Ey1I9-`u$1eg+*x?XxU&%{eT}>SQY?1z}CP5mYW+E0H9NLw=)LN ziQ3vYz3nME8#w(gcq^uVD-qOpG=4h@pi`HU5KxwBV`Zi8)jXDCfSb^Rs>iqV^ zR>j5~c$P7M{jCg^mlqaT^UYJ(w@Uw`%R&;uZ*Bhn_WZ4H;JHffj!wqb5;i8bfZrnn 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0000000..234f41b --- /dev/null +++ b/S2/AnaMech/AnaMech_Hahn_Penning_Zettel_1.typ @@ -0,0 +1,47 @@ +#import "./preamble.typ": *; #show: conf.with(num: 1, ueb: true) + += Aufgabe 1 + ++ Gegeben ist eine homogene gewoehnliche lineare Differentialgleichung zweiter Ordnung $dot.double(y) + a y = 0$. + Diese kann mittels eines Exponentialansatz $y = c e^(lambda t)$ geloest werden. + Umformen liefert fuer $lambda$ die beiden Loesungen $lambda_1$ und $lambda_2$ + + $ + (c_1 e^(lambda t))'' + (a c_2 e^(lambda t))' = 0\ + <==> c_1 lambda^2 e^(lambda t) + a c_2 lambda e^(lambda t) = 0\ + ==> lambda^2 + a lambda = 0\ + ==> lambda_1 = 0 and lambda_2 = -a. + $ + + Da die Nullfunktion nicht interessant ist folgt fuer die allgemeine Loesung: + + $ + y(t) = c_1 + c_2 e^(-a t) + $ + ++ Der Ansatz fuer das eindimentionsale Problem ist hier $x(t) = x_0 + v_0 t + 1/2 a t^2$. + + Einsetzen liefert die BWGL + + $ + r(t) = x(t) = x_0 + v_0 t - 1/2 g t^2. + $ + ++ Grundsaetzlich gilt, dass $nabla V(r) = arrow(F)(arrow(r))$. Es kann also das Kraftfeld integriert werden um das relative Potential zu erhalten. + $ + arrow(F)(arrow(r)) = (alpha) / (r^2 ) arrow(e)_(r) \ + + $ + + $ + arrow(F)(arrow(r)) = (beta) / (r^3 ) arrow(e)_(r) + $ + + #highlight[Wie das am besten rechen? Nutzen von Abkuerzungen.] + ++ Potentiale Ableiten. + + $ + dif / (dif x) kappa x^2 = 2 kappa x\ + dif / (dif x) V_0 sin^2 (kappa x) = + $ diff --git a/S2/AnaMech/UebungHA1.pdf b/S2/AnaMech/UebungHA1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..381cac55b13a19f185cdf670cea1bbcb468534a1 GIT binary patch literal 419403 zcmeFaWk8f$+cu1{6~#bBMB1WDNoltz9nuW~(j7y?ScrmjNh|5l(v5Ld%6Dm~vw{`h3r ziN{a;R{R2blJ%M;;$|l%%;#Hlz6)m{SDR*Ax(B|c&=8;f)_ON6E_;pUy>d{pOr&Mv zt$W-N-u$OCBo%y5;av-?wEFRCS*JfIP^&I-Wjh`v(LXX>*jhv;edn%bObU&0Z;AQ` z>&@P7p1Toe8gnj^eaqKt8=LOYZEjXbHNlBtWX!m8*K9zY!Mm4F#pm($KDISVnO8sN z^L~)c|F|pwtKmMM%;j3qhpf3l`zYUu{hG1OQuRnSclnKx(UDiR5v}(}T$|Tz;8NY? zyh}Ifl~lqUFKa6_2KjJb)~mSX&wUx{-s142-QB2Tw8GU@S-+|=ua%;qiiQ4DmabgX zx@)ao7ZD`l^B=al4#~V|<}4U((&LW2*Coix?&zS?DXvI>)|#y+gYh;X5HADPwwH%axn?$(Oa~ZuV$uoDk&v0bfyHu z@S3V&=R3?4%Kw~Str6m6wz;0jprH80w|u_kyn*F8IU|;$ybmeHLtjtE?oVx5cH=bGj!y4 z+k0~HYf1ME6By(*W`glNR#0(7`_|{i%g( z{xK}Den)ZVqB(l%H*=pvt$2L7VCo;=@Begz^r+Mp5?#KS_ETWaD z($lt4W1|g0nPlu-1s}ihu(Dq4F`NxafwW!vUTP)jY6N+Cm(}B52BN*_`ONgCq|;X} z`?3iMcLeu6>tI{Zuc(D#$Nmc875$obRat7%^a3jAVn1T=^7UUTV}#bg!FfIyS5q-1g*`Hepuh;y z(f3uCez@sRvbsf?uJCps)E7VALx=zR2Z+$=tD=_5>Wta0IM5&IxVpdD910tHI;jn2 zzUC0Ve_5BNZ+G#Yy-dXEnUE{j8t)Mcis2dBnIXb*PvjbR{vjv(tCIbq^-M9H_2(Ho zmY;ErOodPBEqc;ImcMnsX_gS9JVA-WLsQFmxo1I0iso_8?Q&`F^iy&%Co_>eOizZt znikzX$NE9{LAXdA+x>K=&X28b;~}#mZhki?+M8bINWP28^3>yBj9I5Y`=j6arQM?E zIHwJ3g3H*@&v5-oV-=gRgw6=2F8e%@Ub4UM(+`Ofexjn6x|Q?DLC8q+R`B>T6?2B< zfCCeg@QGEJG3$n@8UBFlDe8w+3~=15y1didFZe^vS+dtmm#kvtW^Q!|o$e=s_6GZ% z`76Gu+6?(pXE2I+p+YOCWJ&8QLv&N2tdm#78cv3Ms|oW} z#4UfHCt-aGT&Qn>`Jku&`Db5&oSO1aCsZu=h1 z1%sd0t`;oRjs$L83)XJUul5%B^~cp2CtvX4<4bDqqH0r~iLO<+-kO2l4f&!C&7+>;g|W96uo!%TW@H0cwLWHl|WUok^90n6W6*yj_m^x%Y$N9)ebN73jg(Jy&ka^<@3xW z=kXUF>b53t$o-dCQD%S3bTu0E4LXfq{YbM$^2o*+!U;uK4QttH*nRK5zVYE@lXy10 zm`9H-!DdB+p+r_uc)kWgRe!ne!WGUcs7(anT>}M9ytnINL9Vj7cg~E|dG=RN)^JDU z+sOHCYF~FD!RDM>;A3yR#~~S~GMzbJcAtgq46R4$1UF}I$f(<;?9VryxJU~=YoN{% z$)9?Mt7rafk1~RPL(@&pBeYZA(NgaQv$C{~v7W9iF0&j&$9CsWM9s7f^l+J#jGm!M zDf`{KxXdDYMh1qqdp{l;+1fnOvlg^4v$QbRGq=5QAD3Cs!qmcA&Qe;U+DFOmd3ADMzEJVAqKaqzg5r#x3%!05WX#7}*1@WoPUe!4F&|=j<85*O18kp+ zWg-~#A?ig?*p`!-JguzUvPWx;U^B&LHq@njDQwol4IAs<|NPqm|F*!tE%0v({M!Qm zw!r_s1-_y@>zRt>*Uh$mQ>Pcq^<~dE++7`$_sb9qxFfo55tQ;5ESkhCLocaQEgBUJ z{%ATuYu%SQ9;soeI;Z6;f3XnjIio_JylB9k+7LR?0LJzZT&ylxJ#p}EfGG_9r~mQq zXW*@)&;PUExd~{A&>NY;&|795{f(6;-z`fCWKc^+zk`LPmW~Hu~JQ)3^D0A2T9RUt?-WBLS_-*{ZrQ^R= z=N{P;m_|%yB)~9B0LJgAnq5Qf`(fX3|Ml;ks5vlno;>F7qmk4NL^6f_FZvzfckZ9} z?2%C1@%(@K8zb1dWGOLz|IdQ&|I!Bklk_x&9j4p=rH=nnosWlVR>JXUMu*ZKM6{xY z8FP`+P6L^HxxXJ`>kr;7Kv1iTn^SKV3wBUtA2s5mbP}Zb9qw}uYwW_!!2^wijpmX? zSgFT=xDR8opZE@#>@V1}_RQF%xpB0U$Ax7Vmi9g0Iul#fEU?n{(3N#agbAS5VX93$ ztDQMFLni8-@%;@UX)R`!xkE@-MbfP|zS+y-)BB8|@#;XY06PK6(l4vV=K7WX>N_BK zxSMpM=JF5qva*Ten_ z+ib|dG!&jGtcspIyspu_;2%Eex9`wseq_8@+d`RSy^rd?1-$x zlw*4P8Dbo2%QkD^cI*L@X4%O0J0Do?XOluqG5TX9N*5=7d|Dgn$~yim>dGiHp;rBR zm=>lC#a`D0kblrdQpNO$QBp^=uWL%kb(QzN4BQ_uEy2B33uj$u=ZSXiJZ?aDg43@; z(o@L8>YG+IkHyV&&xSPpd7+Dq`I4mejG-7s@hZQVx2@~SZ>MqpG!e6bTuf)m#x*?f zL-gDH()vHx%H3BsuBv_}=11eY?6Dm+d}Z5Qo#|>=+DzPjDPhEi>?#er;`(>jxNaif zz52HgIE@?$C3v-V;zWQg6j^ruS@;vuFRU@B^}Dp5Fbg6GnP>6?a+Eg4&7 zLLs2~p?4wSfH3TZ78#CI?fJg>sbVwyn*M6c5Q^HAMG)Xg>KSm79)IEzL>#|iA+Rb{ z12>Kz<2WV@@zvzNQyoT;NsG&pN>-SGgbwDZZR0_;?y%*zmB)|-UrWq8bo&>Npjlvz zZsoc3p<{Xo!`>}4_FIhTbKc5zJifB`l~J!5qKOZxi!{gZWCd?i5d<4a^roY>d41e* z8>I1#Z!Wie`CT>gC&2lbM48GHMKL9CuKWp^VqwUs#jpz)jby((`xmLIc81I$-15LSOhQ17_+~F(7FY4+h{Em*$CChLRbc

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Hier steht S fuer Statik und D fuer Dynamik. + + Zusaetlich wird die *Lorentzkraft* + + $ + arrow(F)=q(arrow(E)+arrow(v)times arrow(B)) + $ + + dafuer genutzt. + +In dieser Vorlesung naehern wir und dem Verstaendnis dieser Gleichungen langsam. +- Maxwell Gleichungen @max sind die #underline[grundlegenden Axiome der Elektrodynamik] +- Im statischen Fall ($arrow(B), arrow(E), rho, arrow(j)$ aendern sich nicht mit der Zeit) entkoppeln die Gleichungen die $arrow(E)$ und $arrow(B)$ +- Elektrizitaet und Magnetismus sind getrennt solagne Stroeme und Ladungen statisch sind + +Unterschied zwischen ruhenden und bewegten Ladungen wird anhand Lorentzkraft @lor klar. + +Rechte-Handregel:\ +$arrow(v): "Technische Stromrichtung (Daumen)"$\ +$arrow(B): "Magnetische Feldstaerke (Zeigefinger)"$\ +$arrow(F): "Kraftrichtung (Mittelfinger)"$ + +== 0.1 Vektroranalysis + +$arrow(E), arrow(B)$ sind Vektorfelder, d.h an jedem Raumpunkt ist ein Vektor spezifiziert.\ +#underline[Divergenz] eines Vektorfeldes ist Skalarprodukt vom Nabla-Operator $arrow(nabla)=(partial / (partial x), partial / (partial y), partial / (partial z) )$ mit dem Vektorfeld +$ +arrow(nabla) dot arrow(E) = (partial E_(x) ) / (partial x) + (partial E_(y) ) / (partial y) + (partial E_(z) ) / (partial z) +$ +diese trifft eine Aussage ueber das "Auseinanderdriften" oder die Quellstaerke an einem Punkt. + +Die #underline[Rotation] ist als Vektorprodukt des Nabla-Operators mit einem Vektorfeld definiert +$ + arrow(nabla)times arrow(e)=rot(arrow(E))=sum_(i,j,k) epsilon_(i,j,k) partial / (partial x_i) E_(j) hat(e)_(k) = ((partial E_(z) ) / (partial y) - (partial E_(y) ) / (partial z) , (partial E_(y) ) / (partial z) - (partial E_(z) ) / (partial x), (partial E_(y) ) / (partial x) - (partial E_(x) ) / (partial y) ,) +$ +sie stellt den Grad der "Verwirberlungen oder die Wirbelstaerke eines Feldes am einem Punkt dar. + +Der #underline[Gradient] einer skalaren Funktion $f(x, y, z)$ besteht aus den drei partiellen Ableitungen +$ + arrow(nabla)f = grad(f) = ((partial f) / (partial x), (partial f) / (partial y) ,(partial f) / (partial z) ). +$ +Weitere Deteils und Anwendnungen auf $arrow(E)$ & $arrow(B)$ Vektorfelder in Uebung & Vorlesung. + +== 0.2 Anfaenge der Elektrodynamik + +Geschicktlich fielen drei Phaenomene der Elektrodynamik auf, ohne dass Zusammenhaenge dazwischen erahnt wurden. ++ Licht ++ Elektrizitaet ++ Magnetismus +Diese wurden von verschiedenen Personen zu verschiedenen Zeiten entdeckt. +- Erste Gesetzmaessigketen des Lichts (Licht nimmt immer den kuerzesten Weg): Heron v. Alexandrea (ca. 60 n.Chr.) +- Elektrizitaet: Thales von Milet (600 n. Chr.), geriebener Bernstein (griechisch: "electron") zieht leichte Koerper an +- Magnetismus: Petrus Peregrinu (1269) fuehrte erste Beobachtungen zu magnetischen Feldlinien durch +- Gilbert (1544-1605) erkannte wichtigen Unterschied zu $arrow(E)$ & $arrow(B)$ Feldern: Magnete rufen Drehwirkung hervor, elektrische Kraft aeussert sich als Anziehungs-Kraft +Fuer weitere geschichtliche Entwicklung z.B. siehe Geschichte der Elektrizitaet, H. Bortias. + += 1. Elektrostatik + +== 1.1 Ladung und Coulomb Gesetz + +=== 1.1.1 Zusammenfassung historischer Beobachtungen + ++ Es existieren zwei verschiedene Ladungen (+,-), diese koennen durch ihre kraftwirkung aufeinander und Ablenkung in elektischen Feldern unterschieden werden. ++ Ladungen gleichen Vorzeichens stossen sich ab. Ladungen mit unterschiedlichen Vorzeichen ziehen sich an (*Unterschied* zur immer attraktiven Gravitation) ++ Ladungen sind an Teilchen gebunden, insbesondere Elektronen ($e^(-)$) und Protonen ($p^(+)$) dessen Ladung sich nicht mit der Geschwindigkeit aendert ++ Ladung der Elektronen und Protonen stellt die kleinste #underline[frei] beobachtete Ladungen dar (Ausnahmen stellen kurzlebige Teilchen dar) diff --git a/S2/ExPhyII/preamble.typ b/S2/ExPhyII/preamble.typ new file mode 100644 index 0000000..fc568d5 --- /dev/null +++ b/S2/ExPhyII/preamble.typ @@ -0,0 +1,18 @@ +#import "../../default.typ": * + +#let rot = math.op("rot") +#let grad = math.op("grad") + +#let conf(num: none, date: "", body) = { + // Global settings + show: default + + // Set the header + [ExPhy II \ Vorlesung #(num) \ #(date) \ Jonas Hahn] + + // Make the outline + outline() + + // load the document + body +} diff --git a/S2/Praktikum/Vorbesprechung.pdf b/S2/Praktikum/Vorbesprechung.pdf new file mode 100644 index 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@@ +# Links +ExPhy Uebung Di 16-18 NR 7: +https://ecampus.uni-goettingen.de/h1/pages/cs/sys/portal/hisinoneIframePage.faces?id=studip&navigationPosition=link_studip&url=https%3A%2F%2Fstudip-ecampus.uni-goettingen.de%2Findex.php%3Fsso%3Dcasgoe%26cancel_login%3D1%26again%3Dyes%26redirect_to%3Dhttps%253A%252F%252Fstudip-ecampus.uni-goettingen.de%252Fdispatch.php%252Fmy_courses%26redirect_token%3D947a61f0a9df959879bce689abac0ffa + + +# Other modules +Zusammenhang Mensch und Natur; Modul B.Phy.1609 + diff --git a/courses.txt b/courses.txt new file mode 100644 index 0000000..aaaa2d0 --- /dev/null +++ b/courses.txt @@ -0,0 +1,37 @@ +# Physikbachelor GAU Goettingen + +## S1 + +AGLA Analytische Geometrie und Lineare Algebra 9CP +DiffI Differential und Integralrechnung I 9CP +ExPhyI Experimentalphysik I 6C +ReMe Rechenmethoden 6CP +GdCP Grundlagen der C-Programmierung 6CP +GdE Grundlagen des Experimentierens 0CP +PraktikumI Physikalisches Praktikum I 3CP + +## S2 + +ExPhyII Experimentalphysik II 6C +PraktikumII Praktikum zur Experimentalphysik II 3C +DiffII Differential und Integralrechnung II 9C +AnaMech Analytische Mechanik 8C +CWR Computergestuetztes Wissenschaftliches Rechnen 6C +CompNeu Computational Neuroscience Basics 3C + +## S3 + +ExPhyIII Experimentalphysik III +MaPhyIII Mathematik fuer Physikstudierende III +QP Quantenphysik + +## S4 + +ExPhyIV Experimentalphysik IV + +## S5 + +## S6 + +BaAr Bachelorarbeit 30CP + diff --git a/default.typ b/default.typ new file mode 100644 index 0000000..b85ae35 --- /dev/null +++ b/default.typ @@ -0,0 +1,42 @@ +#import "@preview/equate:0.2.1": equate +#import "@preview/quick-maths:0.2.1": shorthands +#import "./theorems.typ": * + +#let default(body) = { + // page setup + set page(margin: 2cm, numbering: "1") + set text(lang: "de", hyphenate: false) + // set par(justify: true) + + // equation setup + show ref: equate + show: equate.with(number-mode: "label", breakable: false) + set math.equation(numbering: "(1)", supplement: "Gl.", number-align: bottom) + show math.equation.where(block: false): it => box(it) + + // shorthands setup + show: shorthands.with( + ($*$, $dot.op$), + ($\\$, $without$), + ($+-$, $plus.minus$), + ($=>$, $arrow.r.double.long$), + ($=<$, $arrow.l.double.long$), + ($<=>$, $arrow.l.r.double.long$), + ($..$, $quad$), + ) + + // shiroa/zeta setup + body +} + +// symbol shortcuts +#let cmt = align(right, $wide |$) +#let Abb = math.op("Abb") +#let rel = math.op("R") +#let lin = math.op("lin") +#let arccot = math.op("arccot") + +// flashcards +#let flashcard(id, front, back) = { + back +} diff --git a/template.typ b/template.typ new file mode 100644 index 0000000..aabec9e --- /dev/null +++ b/template.typ @@ -0,0 +1,8 @@ +#import "./preamble.typ": * + +#show: conf.with(num: 1) + += Uebersicht + + + diff --git a/theorems.typ b/theorems.typ new file mode 100644 index 0000000..cb98965 --- /dev/null +++ b/theorems.typ @@ -0,0 +1,24 @@ +#import "@preview/ctheorems:1.1.3": * + +// What is this for? +// #show: thmrules.with(qed-symbol: $square$) + + +// TODO: Set this up so that it looks good +#let theorem = thmbox("theorem", "Theorem", fill: rgb("#eeffee")).with(numbering: "1.1") //tem +#let corollary = thmplain( //cor + "corollary", + "Corollary", + base: "theorem", + titlefmt: strong +) +#let definition = thmbox("definition", "Definition", inset: (x: 1.2em, top: 1em)) //def +#let example = thmplain("example", "Example").with(numbering: none) //exa +#let proof = thmproof("proof", "Proof") //pro +#let remark = thmbox("remark", "Remark") //rem +#let axiom = thmbox("axiom", "Axiom") //axi +#let lemma = thmbox("lemma", "Lemma") //lem + +// Add a note +#let note = thmbox("note", "Note") //nte +
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vorrechnen + += Integration + +#flashcard(0)[onw][ + sf +] + +Idee: + +- Differenzieren: +$ + F: I -> RR \ + F'(x) = lim_(x -> oo) +$ + +Die Idee ist nach einer Funktion zu fragen, welche die Vorschrift $F' = f$ erfuellt. + +Dazu kann in 2 Dimensionen der Flaecheninhalt unter dem Graphen der Funktion ermittelt werden. + +== Unbestimmte Integration + +Frage: + +Gegeben sei ein Intervall $I$ in RR und eine Funktion $f: I -> RR$ + +Finden wir $F: I -> RR$ mit $F' = f$? + +#definition[ + Gegeben: $I <= RR$ Intervall + + $F, f: I -> RR$ + + - $F$ eine Stammfunktion zu f auf I oder unbestimmtes +] + +Beachte: + +Gegeben seien Stammfunktionen $F_1, F_2$ zu $f$ + +$ + ==> (F_1 - F_2)'(x) = f(x) - f(x) = 0\ + ==> F_1 - F_2 "konstant auf" I +$ + +Man ueberprueft ob eine Funktion eine Stammfunktion ist anhand der Definition. + +Nicht jede Funktion hat eine Stammfunktion. + + +#theorem[ + Zwischenwertsatz fuer Ableitungen + + Sei $F: [a,b] -> RR$ diffb. + + $ + ==> F' "nimmt auf" (a,b) "jeden Wert zwischen" F'(a) "und" F'(b) "an". + $ +] + +#proof[ + // TODO: write this proof + +] + +Die Funktion, welche eine Stufenfunktion ist hat keine Stammfunktion, da sie im Widerspruch zu Satz @zws steht. + +#theorem[ + Summenregen fuer Integration + + Seien $I, I_0 "Intervalle in" RR$ uns $alpha_1, ... alpha_n in RR$ + + - Gegeben: + + $f_1, ...,f_n I -> RR $ + // TODO: zuende schreiben + +] + + diff --git a/S2/AnaMech/nolting.typ b/S2/AnaMech/nolting.typ new file mode 100644 index 0000000..ccbe0df --- /dev/null +++ b/S2/AnaMech/nolting.typ @@ -0,0 +1,32 @@ += Einfuehrung + +In der Newton-Mechanik werden physikalische Systeme mithilfe aller Massepunkte beschrieben. + +#flashcard(0)[Zwangsbedingung][ + Zwangsbedingungen sind Bedingungen, welche die freie Bewegung der Systemteilchen einschraenken (geometrische Bindungen). +] + +#flashcard(0)[Zwangskraefte][ + Zwangskraefte sind Kraefte, die die Zwangsbedingungen bewirken, also die freie Teilchenbewegung behindern (z.B. Afulagekraefte, Fadenspannungen). + +] + +#flashcard(0)[Holonome Zwangsbedingungen][ + Verknuepfungen der Teilchenkoordinaten und eventuell der Zeit in der Form + $ f_nu(r_1, r_2, ..., r_n, t) = 0, nu = 1, 2, ..., p. $ +] + +#flashcard(0)[Holonom, skleronome Zwangsbedingungen][ + Holonome Zwangsbedinungen mit zusaetzlich + $ + + $ + +] + + += Lagrange Formalismus + +Was muss ich auswendig lernen fuer den Lagrangeforalismus? + + diff --git a/S2/AnaMech/preamble.typ b/S2/AnaMech/preamble.typ new file mode 100644 index 0000000..ba829b3 --- /dev/null +++ b/S2/AnaMech/preamble.typ @@ -0,0 +1,16 @@ +#import "../../default.typ": * + +#let conf(num: none, ueb: false, body) = { + // Global settings + show: default + + // Set the header + if (ueb == false) [AnaMech \ Vorlesung #(num)] + if (ueb == true) [AnaMech \ Uebung #(num)] + + // Make the outline + outline() + + // load the document + body +} diff --git a/S2/DiffII/.anki b/S2/DiffII/.anki new file mode 100644 index 0000000..04caf8c --- 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--- /dev/null +++ b/S2/DiffII/VL/DiffVL1.typ @@ -0,0 +1,131 @@ +#import "../preamble.typ": * +#show: conf.with(num: 1) +#set heading(numbering: "1.1.") + +Tutorium ist immer Mi 2-6pm + +Tutoren sind Oscar Cossarat und Simon Fischer + +Das Hauptziel ist das Wissen aus dem ersten Semester ueber $f: RR -> RR$ auf meherer Dimensionen zu verallgemeinern. + += Topologische Grundbegriffe + +== Euklidischer Abstand im $RR^n$ + +In der Diff 1 haben wir Stetigkeit Diffbarkeit und Integrierbarkeit von Funktionen $f: [a, b] -> RR, a < b$ besprochen. + +Diese sind hier eindimensional und auf einem Kompaktum definiert. + +$==>$ Verallgemeinrerung zu Funktionen $f: DD = RR^m -> RR^n$ ? + +$==>$ Konvergenz im $RR^n$ ? + +#example[ + Auf $CC$ haben wir die Abstandsfunktion + $ + d(a, b) = abs(a-b), quad a,b in CC, + $ + wobei $abs(z) = abs(z_1+i z_2) := sqrt(z_1^2+z_2^2)$, $z_1, z_2 in RR$. +] + +#definition[ + Sei $n in NN$. Wir definieren die Euklidische Norm als die Funktion $||dot||: RR^n -> RR^+, quad (x_1, ..., x_n) |-> sqrt(x_1^2 + ... + x_n^2 )$. + + Wir definieren die euklidische Metrik $d: RR^n times RR^n -> RR, quad x, y |-> d(x, y) = norm(x+y)$. +] + +Wir schreiben $underline(x)$ fuer einen Vektor $x$. Ich werde einfache Symbole verwenden und nur im Notfall des Kontexts eine Unterscheidung machen.\ Erfuellt $d(x, y) = norm(x - y)$ die Eigenschaften einer Metrik? + +#definition[ + Ein metrischer Raum ist ein Tupel $(X, d_x)$ aus einer Menge $X$ und einer Funktion $d_x: X times X -> R^+$ mit drei Eigenschaften. + + + $d(x,x) = 0 and d(x,y) != 0, x != y $ + + Symetrie: $d(x,y) = d(y,x)$ + + Dreiecksungleichung +] + +Wir definieren das *Standard-Skalarprodukt* als $angle.l dot, dot angle.r: CC^n times CC^n -> CC$. + + +// TODO: add angles to typstar +#lemma[ + Cauchy-Schwarz Ungleichung + + Fuer $x, y in CC^n$ gilt $abs(angle.l x\, y angle.r) <= abs(x) dot abs(y)$ + + Die beiden Vektoren sind genau dann linear abhaengig, wenn Gleichheit gilt +] + +#proof[ + #highlight[TODO] +] + +#lemma[ + Die euklidische metrik $d: RR^n times RR^n -> RR, (x, y) -> abs(x - y)$ ist eine Metrik auf $RR^n$. +] + +#proof[ + #highlight[TODO] + // Hier wird der Teil 3 ueber das Ausschreiben von abs(x+y)^2 gemacht + // Dannach kann die Cauchy Schwarz ungleichung verwendet werden um eine Abschaetzung nach oben zu gewinnen + // Mit der binomischen +] + +== Konvergenz im $RR^n$ + +#remark[ + Sei $(X, d)$ ein metrischer Raum, $(a_k )_(k in NN)$ eine Folge in $X$ und $a in X$. Wir sagen, dass die Folge gegen $a$ konvergiert falls gilt: + $ + forall epsilon > 0 exists k_0: d(a_k, a) < epsilon, forall k >= k_0 + $ + alternativ: $lim_(k -> oo) d (a_k, a) = 0$. + + In dem Fall schreiben wir $lim_(n -> oo) a_k = a$. +] + +#lemma[ +Sei $x_k, k in NN$ eine Folge im $RR^n$ + mit $x_k = (x_(k,1), x_(k,2), ...)$ und $a = (a_1, ..., a_n)$. + + Dann konvergiert die Folge genau dann gegen $a$, wenn $lim_(n -> oo) x_(n,j) = a_j, forall 1 <= j <= n$. +] + +#proof[ + #highlight[TODO] + + Idee: verwende die Ungleichung $abs(x_k-a_l) <= abs(x_k-a) <= sum_(i=0)^(n) abs(x_k-a_l )$ +] + +#definition[ + Wir werden eine Folge im $RR^n$ beschraenkt, falls es eine Konstante $R>0$ gibt, sodass $d(0, a_k) < R, forall k in NN$ + + d.h $a_k in K_R(0), forall k in NN$ +] + +#remark[ + Ist eine Folge im $RR^n$ konvergent, so ist diese beschraenkt. +] + +#theorem[ + Bolzano-Weierstrass + + Eine beschraenkte Folge im $RR^n$ besitzt eine konvergente Teilfolge. + +] + +#proof[ + Induktion nach $n$. Fuer $n=1$ siehe Diff 1. + Angenommen @bolz gilt fuer ein $n in NN$ und es ist eine beschraenkte Folge im $RR^(n+1)$ gegeben. + + Dann ist diese Folge beschraenkt auf $RR^n$ eine beschraenkte folge mit konvergenter Teilfolge. + + #highlight[TODO] +] + +#theorem[ + $RR^n$ mit der euklidischen Metrik ist vollstaendig. +] + +#proof[ + Fuer eine Cauchyfolge F von Vektoren im $RR^n$ sind die Folgen der Komponenten wieder Cauchy-Folgen im $RR$. Diese haben wegen der Vollstaendigkeit von $RR$ einen Grenzwert. 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