Second week without fest

S3/MaPhyIII/VL/MaPhIIIVL2.typ

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+// Main VL template
+#import "../preamble.typ": *
+
+// Fix theorems to be shown the right way in this document
+#import "@preview/ctheorems:1.1.3": *
+#show: thmrules
+
+// Main settings call
+#show: conf.with(
+	// May add more flags here in the future
+	num: 5,
+	type: 0, // 0 normal, 1 exercise
+	date: datetime.today().display(),
+	//date: datetime(
+	//	year: 2025,
+	//	month: 5,
+	//	day: 1,
+	//).display(),
+)
+
+= Uebersicht
+
+Diese Vorlesung ist ausgefallen.

S3/MaPhyIII/VL/MaPhIIIVL3.typ

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+// Main VL template
+#import "../preamble.typ": *
+
+// Fix theorems to be shown the right way in this document
+#import "@preview/ctheorems:1.1.3": *
+#show: thmrules
+
+// Main settings call
+#show: conf.with(
+	// May add more flags here in the future
+	num: 3,
+	type: 0, // 0 normal, 1 exercise
+	date: datetime.today().display(),
+	//date: datetime(
+	//	year: 2025,
+	//	month: 5,
+	//	day: 1,
+	//).display(),
+)
+
+= Uebersicht
+
+Wiederholung der *Bessel'schen Ungleichung*. Betrachte $f in RR [- pi,  pi] \, space  c^(k) =  1/(2 pi) integral _(- pi) ^(pi) f (x) e ^(i k x)  dif x$
+$
+  ==> sum  abs(c^(k) )^2 <= norm(f)_(L^2 ) ^2  \ 
+  norm(f)_(L^2 ) ^2  =  lr(angle.l f, f angle.r) \ 
+  lr(angle.l f, g angle.r) =  1/(2 pi) integral _(- pi) ^(pi)  overline(f (x)) g (x) dif x
+$
+
+Im Beweis 
+$
+  norm(f -  sum  _(- n) ^(n) c_(k) e_(k) )^2 = lr(angle.l f -  sum  c_(k) e_(k) , f -  sum  c_(k) e_(k)   angle.r) \ 
+  = norm(f)^2 - sum abs(c_(k))^2  -> 0  .. (n ->  oo) "wenn" sum  c_(k) e_(k) "gegen" f "bezueglich" norm(*)_(L^2 )  \ 
+  e_(k) (x) =  e ^(i k x).
+$
+
+Es folgt die *Parseval'sche Gleichung*
+$
+   norm(f)_(2) ^(2)  =  lim_(n -> oo) sum_(- n)^(n) abs(c_(k) )^2 \ 
+   ==> sum abs(c_(k) )^2 < oo.
+$
+$==>$ Eindeutigkeit der Fourierkoeffizienten.
+
+#proof[
+  Sei $f =  sum  c_(k) e_(k)  \, space g =  sum  d_(k) e_(k) $. Mit $c_(k)  =  lr(angle.l e_(k) , f angle.r) \, space d_(k) =  lr(angle.l e_(k) , g angle.r)$.
+  Ist $f != g$ so gilt 
+  $
+    0 !=   norm(f - g)_(L^2 ) ^2 =  sum  abs(lr(angle.l e_(k) , f - g angle.r))^2  = sum  abs(c_(k)  -  d_(k) )^2.
+  $
+  Angenommen $c_(k) = d_(k) forall k ==> 0 != 0 $. Widerspruch.
+  $f ~ g$ bezueglich der $norm(*)_(L_(1) ) :<==> f "und" g "auf Nullmenge verschieden sein duerfen" $.
+
+  Also $f, g in C [- pi,  pi] ==>( f != g ==> exists x_0 in [- pi,  pi]: f (x_0 ) != g (x_0 ))$.
+  Parseval $==>$ $sum_(k=1)^(oo) 1/k^2 =  pi^2 /6$.
+
+  
+#figure(
+  image("typst-assets/drawing-2025-11-07-10-46-32.rnote.svg"),
+)
+
+  $
+    f (x) =  x "auf" [- pi,  pi] \, space c_(k) =  (- 1)^(k) ) i/k \, space k != 0, c_0 = 0 \ 
+    ==> sum_(- oo)^(oo) abs(c_(k) )^2  =  2 sum_(k=1)^(oo) 1/k^2 =^("Parseval") lr(angle.l f, f angle.r) =  1/(2 pi) integral _(- pi) ^(pi)  x^2  dif x = pi^2 /3 
+  $
+]
+
+Die Schwingende Rechtecksmembran durch den Tayloransatz loesen
+
+$
+  u (t, x, y) =  sum _(n = 1)  ^(oo) sum_(m = 1)^(oo) (a_(m, n) cos (sqrt(lambda_(n, m) ) c t) +  b_(n, m) sin (sqrt(lambda_(n, m) ) c t)) nu_(n, m).
+$
+
+Cladni-Figuren in Wolframalpha.
+
+= Kreisfoernige Membran
+
+Wellengleichung 
+$
+  arrow(nabla) ^2 u =  1/v^2 partial _(t) ^2 u.
+$
+Produktansatz $==>$ $u =  v (t) w (x, y)$.
+
+Wenn man auf Polarkoordinaten transformiert, dann muessen die Randbedingungen alle erfuellt sein, was man durch den Limes prueft.
+
+$
+  (partial _(x) ^2 +  partial _(y) ^2 ) v =  - lambda v
+$
+
+Es wird dann wieder ein Ansatz gemacht
+$
+  V (r, theta) =  R (r) A (theta) \ 
+  ==>  (r^2 R'' +  v R')1/R +  lambda r^2 =  - (A'') / (A).
+$
+
+$
+  1/r partial _(r)  (r partial _(r)  (R (r) A (theta))) +  1 / r^2 partial _(theta) ^2  (R (r) A (theta)) +  lambda R (r) A (theta) = ^(!) 0
+$
+
+Potenzreihenansatz als 
+$
+   f (rho ) =  sum a_(k)  rho^(k).
+
+$
+Es wird gleichmaessige Konvergenz angennommen. Es folgt 
+$
+  rho^2 f'' (rho) =  rho^2 (sum a_(k) k rho ^(k - 1) )' =  rho^2  sum  a_(k) k (k - 1) rho ^(k - 2).
+$
+Nachdem die anderen Teile ausgerechnet wurden kann ein Koeffizientenvergleich gemacht werden.

S3/MaPhyIII/VL/typst-assets/drawing-2025-11-07-10-46-32.rnote.svg

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