3

scripts/.ipynb_checkpoints/norms-checkpoint.py

@@ -0,0 +1,17 @@
+from math import sqrt
+
+def l1(x: list[float]) -> float:
+    a=0
+    for i in x:
+        a+= abs(i)
+    return a
+
+def l2(x: list[float]) -> float:
+    a=0
+    for i in x:
+        a+= i**2
+    return sqrt(a)
+
+
+def li(x: list[float]) -> float:
+    return max(x)

src/exercise/2.typ

@@ -0,0 +1,58 @@
+== Exercise Sheet 2 - Introduction to Multivariable Functions
+
+=== Exercise 1 @Exercise[2, 1]
+#block(
+  fill: luma(230),
+  inset: 8pt,
+  radius: 4pt,
+[
+  Consider the function
+
+$
+  f: RR^2 -> RR \
+  (x,y) |-> x^2 - y^2
+$
+
+Determine the levels sets at $c = 1$ and $c = −1$. What does it look like (sketch?)
+])
+
+For $c = 1$:
+$
+  f(x,y) =^! 1 \
+  => x^2 - y^2 = 1 <=>  y = sqrt(x^2 - 1) \
+$
+
+#figure(
+  image("../media/img/ex2-1a.png", height: 20%), caption: [Zeichnung von $y = sqrt(x^2-1)$], 
+)
+
+For $c = -1$:
+$
+  f(x,y) =^! -1 \
+  => x^2 - y^2 = -1 <=>  y = sqrt(x^2 + 1) \
+$
+
+#figure(
+  image("../media/img/ex2-1b.png", height: 20%), caption: [Zeichnung von $y = sqrt(x^2+1)$]
+)
+
+#pagebreak()
+=== Exercise 2 @Exercise[2, 2]
+#block(
+  fill: luma(230),
+  inset: 8pt,
+  radius: 4pt,
+[
+  Draw the following sets (what will be domains). Recall the lecture about curves.
+
+  #set enum(numbering: "(a)")
+  + $ D = {(x,y) thin|thin (x-2)^2 + (y+1)^2 <= 9} $
+  + $ D = {(x,y) thin|thin 0 <= x <= 1 and x^2 < y < sqrt(x)} $
+])
+
+#set enum(numbering: "(a)")
++ $ 
+  (x-2)^2 + (y+1)^2 &=^! 9 \
+  => (x-2)^2 + (y+1)^2 &= x^2-4x+4+y^2+2y+1 \
+  => y = 
+$

src/exercise/3.typ

@@ -0,0 +1,117 @@
+== Exercise Sheet 3 - Topology of Metric Spaces
+
+=== Exercise 1 @Exercise[3, 1]
+#block(
+  fill: luma(230),
+  inset: 8pt,
+  radius: 4pt,
+[
+  Let $(X, d)$ be a metric space, and let $x_1, . . . , x_4 in X$. Show that
+
+$
+  |d(x_1,x_2) - d(x_2,x_3)| <= d(x_1,x_3)
+$
+])
+
+$
+  #text("z.z.") &|d(x_1,x_2) - d(x_2,x_3)| <= d(x_1,x_3) \
+  d(x_1,x_3) <= &|d(x_1,x_2) + d(x_2,x_3)| \
+  <=> 0 <= &|d(x_1,x_2) + d(x_2,x_3)| - d(x_1,x_3) \
+  ?????????????????&???????????????
+$
+
+#pagebreak()
+=== Exercise 2 @Exercise[3, 2]
+#block(
+  fill: luma(230),
+  inset: 8pt,
+  radius: 4pt,
+[
+  We define the metric
+$
+  d(x, y) = arctan(|x − y|)
+$
+
+  on $RR$. Show that $d$ indeed defines $a$ metric, i. e. it satisfies the axioms of a metric.
+
+  Hint: You can and should use that
+
+  $ arctan(x + y) ≤ arctan(x) + arctan(y); x, y in RR $
+
+])
+
+#set enum(numbering: "(I)")
++ $
+  #text("z.z.") wide d(x,y) = 0 &<=> x=y \
+  d(x,y) = 0 &<=> arctan(|x-y|) = 0 \
+  &<=> |x-y| 0 \
+  &<=> x-y = 0 \
+  &<=> x = y space qed
+$
++ $
+  #text("z.z.") wide d(x,y) &= d(y,x) \
+  d(x,y) &= arctan(|x-y|) \
+  &= arctan(| -1 dot (x-y) |) \
+  &= arctan(|y-x|) \
+  &= d(y,x) space qed
+$
++ $
+  #text("z.z.") wide d(x , y ) &<= d(x , z) + d(z, y ) \ \
+  #text("es gilt:") |x-y| &= |x-z+z-y| <= |x-z| + |z-y| \
+  arctan &#text("ist streng monoton steigend.") \ \
+  d(x,y) &= arctan(|x-y|) \
+  &= arctan(|x-z+z-y|) \
+  &<= arctan(|x-z|+|z-y|) \
+  &= arctan(|x-z|) + arctan(|z-y|) #text("Zauber") \
+  &= d(x,z) + d(z,y) space qed
+$
+
+Somit gelten alle Eigenschaften einer Metrik für $d(x, y) = arctan(|x − y|)$ $qed$.
+
+#pagebreak()
+=== Exercise 3 @Exercise[3, 3]
+#block(
+  fill: luma(230),
+  inset: 8pt,
+  radius: 4pt,
+[
+  Show that the set $U=[0, 1) subset RR$ is neither open nor closed.
+])
+
+$U$ ist nicht offen, weil es keinen offenen Ball um $x=0$ mit radius $epsilon 
+> 0$ geben kann, der vollständig in $U$ liegt.
+
+Damit $U$ geschlossen ist, müsste $RR\\U$ offen sein. Man kann jedoch keinen 
+offenen Ball um $x=1; x in RR\\U$ mit radius $epsilon >0$ legen, sodass dieser 
+vollständig in $RR\\U$ liegen würde. $qed$
+
+=== Exercise 4 @Exercise[3, 4]
+#block(
+  fill: luma(230),
+  inset: 8pt,
+  radius: 4pt,
+[
+  Show that the union
+  $ み = union.big space U_i $
+  of the sequence of open sets $U$ is open.
+])
+
+@Vorlesung[3, Theorem 2.2] sagt das ist bereits wahr.
+
+Trotzdem: Jeses $x$ ist auf jeden fall teil mindestens eines $U_i$. Dann existiert ein offener Ball 
+$B_epsilon(x) subset U_i subset み$ für alle $x subset U_i subset み$. Somit ist $み$ offen. $qed$
+
+=== Exercise 4 @Exercise[3, 5]
+#block(
+  fill: luma(230),
+  inset: 8pt,
+  radius: 4pt,
+[
+  Show that the intersection
+  $ ま = sect.big space U_i $
+  of the sequence of closed sets $U$ is closed.
+])
+
+@Vorlesung[3, Theorem 2.2] sagt das ist bereits wahr.
+
+Geschlossen bedeutet das Komplement ist offen. $overline(ま) = overline(sect.big space U_i) = union.big overline(U_i)$ Wie in Ex 4 gezeigt, ist $overline(ま)$ offen, da $overline(U_i)$ offen ist, da $U_i$ geschlossen ist. $qed$

src/exercise/index.typ

@@ -1,2 +1,5 @@
-
 #include "1.typ"
+#pagebreak()
+#include "2.typ"
+#pagebreak()
+#include "3.typ"

src/main.typ

@@ -32,8 +32,6 @@
 // headcolor
 #let headcolor = rgb("80b3ff")
 
-#set heading(numbering: "1.")
-
 // reset counter at each chapter
 #show heading.where(level:1): it => {
   counter(math.equation).update(0)
@@ -51,8 +49,9 @@
 #set footnote(numbering: "*")
 
 #set math.equation(numbering: n => {
-  let h1 = counter(heading).get().first()
+  let h1 = counter(heading.where(level: 1)).get().first()
-  numbering("(1.1)", h1, n)
+  // let h2 = counter(heading.where(level: 2)).get().first()
+  numbering("(1.1.1)", h1, n)
 })
 
 #set figure(numbering: n => {

src/refs.bib

@@ -23,3 +23,9 @@
     url = "https://matheguru.com/integralrechnung/herleitung-der-stammfunktion-des-naturlichen-logarithmus.html",
     urldate = "2024-09-09",
 }
+
+@online{trig-formeln,
+    title = "Formelsammlung Trigonometrie",
+    url = "https://de.wikipedia.org/wiki/Formelsammlung_Trigonometrie#Produkte_der_Winkelfunktionen",
+    urldate = "2024-09-10",
+}

src/vorlesungen/2.typ

@@ -0,0 +1,3 @@
+== Basics & Outlook @Vorlesung[Foliensatz 2]
+
+- Größtenteils Wiederholung von Analysis

src/vorlesungen/index.typ

@@ -1 +1,2 @@
 #include "1.typ"
+#include "2.typ"