This commit is contained in:
Christoph J. Scherr 2024-09-12 16:28:12 +02:00
parent 985b0a9e74
commit 13d4b597ff
Signed by: cscherrNT
GPG Key ID: 8E2B45BC51A27EA7
13 changed files with 837 additions and 5 deletions

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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)

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== 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 =
$

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== 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. $qed$

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#include "1.typ" #include "1.typ"
#pagebreak()
#include "2.typ"
#pagebreak()
#include "3.typ"

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// headcolor // headcolor
#let headcolor = rgb("80b3ff") #let headcolor = rgb("80b3ff")
#set heading(numbering: "1.")
// reset counter at each chapter // reset counter at each chapter
#show heading.where(level:1): it => { #show heading.where(level:1): it => {
counter(math.equation).update(0) counter(math.equation).update(0)
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#set footnote(numbering: "*") #set footnote(numbering: "*")
#set math.equation(numbering: n => { #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 => { #set figure(numbering: n => {

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url = "https://matheguru.com/integralrechnung/herleitung-der-stammfunktion-des-naturlichen-logarithmus.html", url = "https://matheguru.com/integralrechnung/herleitung-der-stammfunktion-des-naturlichen-logarithmus.html",
urldate = "2024-09-09", 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",
}

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== Basics & Outlook @Vorlesung[Foliensatz 2]
- Größtenteils Wiederholung von Analysis

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#include "1.typ" #include "1.typ"
#include "2.typ"