Merge branch 'main' of gitea.mintcalc.com:alexander/TUM-Formelsammlungen
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This commit is contained in:
alexander
9043
src/Analysis_rewrite.pdf
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9043
src/Analysis_rewrite.pdf
Normal file
File diff suppressed because one or more lines are too long
@@ -1,3 +1,5 @@
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#import "lib/common_rewrite.typ" : *
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#set page(
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paper: "a4",
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margin: (
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@@ -19,26 +21,6 @@
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[Analysis 1 (IE)]
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))
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|
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|
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|
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#let subHeading(body, fill: color) = {
|
||||
box(
|
||||
align(
|
||||
top+center,
|
||||
text(
|
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body,
|
||||
size: 10pt,
|
||||
weight: "regular",
|
||||
style: "italic",
|
||||
)
|
||||
),
|
||||
fill: fill,
|
||||
width: 100%,
|
||||
inset: 1mm,
|
||||
height: auto
|
||||
)
|
||||
}
|
||||
|
||||
#let SeperatorLine = line(length: 100%, stroke: (paint: black, thickness: 0.3mm))
|
||||
#let MathAlignLeft(e) = {
|
||||
align(left, block(e))
|
||||
@@ -50,7 +32,6 @@
|
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#let colorAbleitung = color.hsl(356.92deg, 92.13%, 75.1%)
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#let colorIntegral = color.hsl(34.87deg, 92.13%, 75.1%)
|
||||
|
||||
#let bgBlock(body, fill: color) = block(body, fill:fill.lighten(80%), width: 100%, inset: (bottom: 2mm))
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||||
|
||||
#columns(4, gutter: 2mm)[
|
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#bgBlock(fill: colorAllgemein)[
|
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@@ -67,7 +48,7 @@
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$abs(x dot y) <= abs(abs(x) dot abs(y))$
|
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],
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[Geometrische Summenformel], [
|
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#MathAlignLeft($ sum_(k=1)^(n) k = (n(n+1))/2 $)
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#MathAlignLeft($ limits(sum)_(k=1)^(n) k = (n(n+1))/2 $)
|
||||
],
|
||||
[Bernoulli-Ungleichung ], [
|
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$(1 + a)^n x in RR >= 1 + n a$
|
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@@ -83,6 +64,10 @@
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[Gausklammer], [
|
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$floor(x) = text("floor")(x)$ \
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$ceil(x) = text("ceil")(x)$
|
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],
|
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[Bekannte Werte], [
|
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$e approx 2.71828$ ($2 < e < 3$) \
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$pi approx 3.14159$ ($3 < pi < 4$)
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]
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)
|
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]
|
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@@ -92,8 +77,6 @@
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]
|
||||
|
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#bgBlock(fill: colorAllgemein)[
|
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#subHeading(fill: colorAllgemein)[Sinus-Tabel]
|
||||
|
||||
#table(
|
||||
inset: 1.5mm,
|
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stroke: (thickness: 0.2mm),
|
||||
@@ -126,6 +109,47 @@
|
||||
[$ sin(x) = (e^(i x) - e^(-i x))/(2i) $],
|
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[$ cos(x) = (e^(i x) + e^(-i x))/(2) $]
|
||||
)
|
||||
#subHeading(fill: colorAllgemein)[Trigonmetrie]
|
||||
$sin(x+y) = cos(x)sin(y) + sin(x)cos(y)$ \
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$cos(x+y) = cos(x)cos(y) - sin(x)sin(y)$ \
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|
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$cos(2x) = cos^2(x) - sin^2(x)$ \
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$sin(2x) = 2sin(x)cos(x)$
|
||||
|
||||
#grid(
|
||||
gutter: 5mm,
|
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columns: (auto, auto),
|
||||
[$cos^2(x) = (1 + cos(2x))/2$],
|
||||
[$sin^2(x) = (1 - cos(2x))/2$]
|
||||
)
|
||||
|
||||
$cos^2(x) + sin^2(x) = 1$
|
||||
git config pull.rebase falsegit config pull.rebase false
|
||||
#grid(
|
||||
gutter: 5mm,
|
||||
columns: (auto, auto),
|
||||
[$cos(-x) = cos(x)$],
|
||||
[$sin(-x) = -sin(x)$],
|
||||
)
|
||||
|
||||
Subsitution mit Hilfsvariable
|
||||
|
||||
#grid(
|
||||
gutter: 5mm,
|
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row-gutter: 3mm,
|
||||
columns: (auto, auto),
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||||
[$tan(x)=sin(x)/cos(x)$],
|
||||
[$cot(x)=cos(x)/sin(x)$],
|
||||
[$tan(x)=-cot(x + pi/2)$],
|
||||
[$cot(x)=-tan(x + pi/2)$],
|
||||
[$cos(x - pi/2) = sin(x)$],
|
||||
[$sin(x + pi/2) = cos(x)$],
|
||||
)
|
||||
$sin(x)cos(y) = 1/2sin(x - y) + 1/2sin(x + y)$
|
||||
|
||||
Für $x in [-1, 1]$ \
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$arcsin(x) = -arccos(x) - pi/2 in [-pi/2, pi/2]$ \
|
||||
$arccos(x) = -arcsin(x) + pi/2 in [0, pi]$
|
||||
]
|
||||
|
||||
#bgBlock(fill: colorFolgen)[
|
||||
@@ -235,6 +259,56 @@
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|
||||
#bgBlock(fill: colorReihen)[
|
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#subHeading(fill: colorReihen)[Reihen]
|
||||
$limits(lim)_(n->infinity) a_n != 0 => limits(sum)_(n=1)^infinity a_n$ konverigiert NICHT \
|
||||
|
||||
- *Absolute Konvergenz* \
|
||||
$limits(sum)_(n=1)^infinity abs(a_n) = a => limits(sum)_(n=1)^infinity a_n$ konvergent
|
||||
|
||||
|
||||
|
||||
- *Partialsummen* \
|
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ALLE Partialsummen von $limits(sum)_(k=1)^infinity abs(a)$ beschränkt\
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$=>$ _Absolute Konvergent_
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|
||||
- *(Cauchy-Kriterium)*\
|
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konvergent wenn $forall epsilon > 0 space exists n_epsilon in NN$ \
|
||||
sodass $abs(s_n - s_m) = abs(limits(sum)_(k=m+1)^(n)) < epsilon space$ \
|
||||
$forall n_epsilon < m < n $
|
||||
|
||||
- *Leibnitzkriterium* \
|
||||
Alternierend + Nullfolge \
|
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$=> limits(sum)_(n=1)^infinity (-1)^n dot a_n$ konvergent
|
||||
|
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- *Vergleichskriterium* \
|
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$a_n, b_n : abs(a_n) <= b_n space forall n in NN > N_0, N_0 in NN$
|
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1. $limits(sum)_(n=0)^infinity b_n$ konvergent $=> limits(sum)_(n=0)^infinity abs(a_n)$ konvergent \
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Suche $b_n$ für Konvergenz
|
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2. $limits(sum)_(n=0)^infinity abs(a_n)$ divergent $=> limits(sum)_(n=0)^infinity b_n$ divergent \
|
||||
Suche $abs(a_n)$ für Divergenz
|
||||
|
||||
Nützlich:
|
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- Dreiecksungleichung
|
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- $forall space n > N_0 in NN space exists k,q in RR$ \
|
||||
sodass $q > 1$: $n^k <= q^n$ (Potenz stärker Polynom)
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|
||||
- *Quotientenkriterium und Wurzelkriterium*
|
||||
1. $rho = lim_(n -> infinity) abs((a_(n+1))/(a_n)) $
|
||||
2. $rho = lim_(n -> infinity) root(n, abs(a_(n+1))) $ \
|
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|
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divergent: $rho > 1$, keine Aussage $rho = 1$, konvergent $rho < 1$
|
||||
|
||||
- *Geometrische Reihe*
|
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$limits(sum)_(n=0)^infinity q^n$
|
||||
- konvergent $abs(q) < 1$, divergent $abs(q) >= 1$
|
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- Grenzwert: (Muss $n=0$) $=1/(1-q)$
|
||||
- *Harmonische Reihe* $limits(sum)_(n=0)^infinity 1/n = +infinity$
|
||||
|
||||
- *Reihendarstellungen*
|
||||
1. $e^x = limits(sum)_(n=0)^infinity (x^n)/(n!)$
|
||||
2. $ln(x) = limits(sum)_(n=0)^infinity (-1)^n x^(n+1)$
|
||||
3. $sin(x) = limits(sum)_(n=0)^infinity $
|
||||
4. $cos(x) = limits(sum)_(n=0)^infinity $
|
||||
|
||||
]
|
||||
|
||||
#bgBlock(fill: colorReihen)[
|
||||
@@ -258,6 +332,23 @@
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|
||||
#bgBlock(fill: colorAbleitung)[
|
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#subHeading(fill: colorAbleitung)[Funktionen]
|
||||
Sei $f : [a,b] -> RR$, stetig auf $x in [a,b]$
|
||||
- *Zwischenwertsatz* \
|
||||
$=> forall y in [f(a), f(b)] exists text("min. ein") x in [a,b] : f(x) = y$ \
|
||||
_Beweiß für mindest. n Nst_
|
||||
- *Satze von Rolle* \
|
||||
diffbar $x in (a,b)$\
|
||||
$f(a) = f(b) => exists text("min. ein") x_0 in (a,b) : f'(x_0) = 0$
|
||||
_Beweiß für max. n Nst, durchWiederspruchsbweiß mit $f(a)=f(b)=0$ und Wiederholte Ableitung_
|
||||
|
||||
- *Mittelwertsatz*
|
||||
diffbar $x in (a,b)$ \
|
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$=> exists x_0 : f'(x_0)=(f(b) - f(a))/(a-b)$
|
||||
|
||||
- *Monotonie* \
|
||||
$x in I : f'(x) < 0$: Streng monoton steigended \
|
||||
$x_0,x_1 in I, x_0 < x_1 => f(x_0) < f(x_1)$ \
|
||||
(Analog bei (streng ) steigned/fallended)
|
||||
]
|
||||
|
||||
#bgBlock(fill: colorAbleitung)[
|
||||
@@ -334,7 +425,47 @@
|
||||
[$e^(x) : e^(x)$],
|
||||
)
|
||||
- Kettenregel: $f(g(x)) : f'(g(x)) dot g'(x)$
|
||||
]
|
||||
],
|
||||
|
||||
#block([
|
||||
#set text(size: 10pt)
|
||||
#table(
|
||||
align: horizon,
|
||||
columns: (1fr, 1fr, 1fr),
|
||||
table.header([*$F(x)$*], [*$f(x)$*], [*$f'(x)$*]),
|
||||
row-gutter: 1mm,
|
||||
fill: (x, y) => if x == 0 { color.hsl(180deg, 89.47%, 88.82%) }
|
||||
else if x == 1 { color.hsl(180deg, 100%, 93.14%) } else
|
||||
{ color.hsl(180deg, 81.82%, 95.69%) },
|
||||
[$1/(q + x) x^(q+1)$], [$x^q$], [$q x^(q-1)$],
|
||||
[$ln abs(x)$], [$1/x$], [$-1/x^2$],
|
||||
[$x ln(a x) - x$], [$ln(a x)$], [$1 / x$],
|
||||
[$2/3 sqrt(a x^3)$], [$sqrt(a x)$], [$a/(2 sqrt(a x))$],
|
||||
[$e^x$], [$e^x$], [$e^x$],
|
||||
[$a^x/ln(a)$], [$a^x$], [$a^x ln(a)$],
|
||||
|
||||
[$x arcsin(x) + sqrt(1 - x^2)$],
|
||||
[$arcsin(x)$], [$1/sqrt(1 - x^2)$],
|
||||
|
||||
[$x arccos(x) - sqrt(1 - x^2)$],
|
||||
[$arccos(x)$], [$-1/sqrt(1 - x^2)$],
|
||||
|
||||
[$x arctan(x) - 1/2 ln abs(1 + x^2)$],
|
||||
[$arctan(x)$], [$1/(1 + x^2)$],
|
||||
|
||||
[$x op("arccot")(x) + \ 1/2 ln abs(1 + x^2)$],
|
||||
[$op("arccot")(x)$], [$-1/(1 + x^2)$],
|
||||
|
||||
[$x op("arsinH")(x) + \ sqrt(1 + x^2)$],
|
||||
[$op("arsinH")(x)$], [$1/sqrt(1 + x^2)$],
|
||||
|
||||
[$x op("arcosH")(x) + \ sqrt(1 + x^2)$],
|
||||
[$op("arcosH")(x)$], [$1/sqrt(x^2-1)$],
|
||||
|
||||
[$x op("artanH")(x) + \ 1/2 ln(1 - x^2)$],
|
||||
[$op("artanH")(x)$], [$1/(1 - x^2)$],
|
||||
)
|
||||
])
|
||||
|
||||
#colbreak()
|
||||
]
|
||||
|
||||
@@ -1,42 +1,177 @@
|
||||
#import "@preview/biceps:0.0.1" : *
|
||||
#import "@preview/mannot:0.3.1"
|
||||
#import "lib/styles.typ" : *
|
||||
#import "lib/common_rewrite.typ" : *
|
||||
|
||||
#show: stdTemplate
|
||||
#set page(
|
||||
paper: "a4",
|
||||
margin: (
|
||||
bottom: 10mm,
|
||||
top: 5mm,
|
||||
left: 5mm,
|
||||
right: 5mm
|
||||
),
|
||||
flipped:true,
|
||||
numbering: "— 1 —",
|
||||
number-align: center
|
||||
)
|
||||
|
||||
#place(top+center, scope: "parent", float: true, heading(
|
||||
[Linear Algebra EI]
|
||||
))
|
||||
|
||||
#let colorAllgemein = color.hsl(105.13deg, 92.13%, 75.1%)
|
||||
#let colorFolgen = color.hsl(202.05deg, 92.13%, 75.1%)
|
||||
#let colorReihen = color.hsl(280deg, 92.13%, 75.1%)
|
||||
#let colorAbbildungen = color.hsl(356.92deg, 92.13%, 75.1%)
|
||||
#let colorGruppen = color.hsl(34.87deg, 92.13%, 75.1%)
|
||||
|
||||
|
||||
#let SeperatorLine = line(length: 100%, stroke: (paint: black, thickness: 0.3mm))
|
||||
#let MathAlignLeft(e) = {
|
||||
align(left, block(e))
|
||||
}
|
||||
#columns(4, gutter: 2mm)[
|
||||
#bgBlock(fill: colorAllgemein)[
|
||||
#subHeading(fill: colorAllgemein)[Notation]
|
||||
|
||||
|
||||
]
|
||||
|
||||
#bgBlock(fill: colorGruppen)[
|
||||
#subHeading(fill: colorGruppen)[Gruppen]
|
||||
|
||||
#flexwrap(
|
||||
main-spacing: 1mm,
|
||||
cross-spacing: 1mm,
|
||||
stdBlock([
|
||||
*Halbgruppe:* $(M, compose): M times M arrow M$
|
||||
- Assoziativgesetz: $a dot (b dot c) = (a dot b) dot c$
|
||||
*Monoid* Halbgruppe $M$ mit:
|
||||
- Identitätselment: $e in M : a e = e a = a$
|
||||
*Kommutativ/abelsch:* Halbgruppe/Monoid mit
|
||||
- Kommutativgesetz; $a dot b = b dot a$
|
||||
|
||||
#SeperatorLine
|
||||
|
||||
*Gruppe:* Monoid mit
|
||||
- Inverse: $forall a in M : exists space a a^(-1) = a^(-1)a = e$
|
||||
- Eindeutig Lösung für Gleichungen
|
||||
- Auch kommutativ wenn: $a dot a = e$
|
||||
*Ring:* Menge $M$ mit:
|
||||
- Kommutativ Gruppe unter $(M, +)$,
|
||||
- Halbgruppe unter $(M, dot)$
|
||||
- Distributiv Gesetz: $(a + b) dot c = (a dot c) + (a dot b)$
|
||||
*Körper:* Menge $M$ mit:
|
||||
- Kommutativ Gruppe unter $(M, +)$
|
||||
- Kommutativ Gruppe unter $(M, times)$
|
||||
- Distributiv Gesetz: $(a + b) dot c = (a dot c) + (a dot b)$
|
||||
]),
|
||||
stdBlock(
|
||||
[
|
||||
*Injectiv:* one to one \
|
||||
$f(x) = f(y) <=> x = y$
|
||||
- Inverse: $forall a in G : exists space a a^(-1) = a^(-1)a = e$
|
||||
- Eindeutig Lösung für Gleichungen
|
||||
Zusatz:
|
||||
- Inverseregel: $(a dot b)^(-1) = b^(-1) dot a^(-1)$
|
||||
*Untergruppe:*
|
||||
- Gruppe: $(G, dot)$, $U subset G$
|
||||
- $a,b in U <=> a dot b in U$
|
||||
- $a in U <=> a^(-1) in U$
|
||||
- $e in U$ (Neutrales Element)
|
||||
|
||||
*Surjectiv:* Output space coverered \
|
||||
- Zeigen das $f(f^(-1)(x)) = x$ für $x in DD$
|
||||
*Direktes Produkt:*\
|
||||
$(G_1,dot_1) times (G_2,dot_2) times ... $ \
|
||||
$(a_1,b_1,...)(a_2,b_2,...)= (a_1 dot_1 b_1, a_2 dot_2 b_2, ...)$
|
||||
|
||||
Beweiß durch Wiederspruch \
|
||||
für Gegenbeweiß
|
||||
]
|
||||
),
|
||||
|
||||
)
|
||||
#SeperatorLine
|
||||
|
||||
*Ring:* (auch Schiefkörper) Menge $R$ mit:
|
||||
- $(R, +)$ kommutativ Gruppe
|
||||
- $(R, dot)$ Halbgruppe
|
||||
- $(a + b) dot c = (a dot c) + (a dot b) space$ (Distributiv Gesetz)
|
||||
|
||||
#colbreak()
|
||||
|
||||
*Körper:* Menge $K$ mit:
|
||||
- $(K, +), (K without {0} , dot)$ kommutativ Gruppe \
|
||||
($0$ ist Neutrales Element von $+$)
|
||||
- $(a + b) dot c = (a dot c) + (a dot b) space$ (Distributiv Gesetz)
|
||||
_Beweiß durch Überprüfung der Eigneschaften_
|
||||
]
|
||||
|
||||
#bgBlock(fill: colorReihen)[
|
||||
#subHeading(fill: colorReihen)[Vektorräume (VR)]
|
||||
$(V, plus.o, dot.o)$ ist ein über Körper $K$
|
||||
- $+: V times V -> V, (v,w) -> v + w$
|
||||
- $dot: K times V -> V, (lambda,v) -> lambda v$
|
||||
Es gilt: $lambda,mu in K, space v,w in V$
|
||||
- $(lambda mu)v = lambda (mu v)$
|
||||
- $lambda(v + w) = lambda v + lambda w$\
|
||||
$(lambda + mu)v = lambda v + lambda mu$
|
||||
- $1v = v$, $arrow(0) in V$
|
||||
Bsp: $KK^n$ ($RR^n, CC^n$)
|
||||
|
||||
*Untervektorraum:* $U subset V$ \
|
||||
$v,w in U, lambda in K$ \
|
||||
$ <=> v + w in U$, $arrow(0) in U$ UND $lambda v in U$
|
||||
- $(U inter W) subset V$
|
||||
]
|
||||
|
||||
#bgBlock(fill: colorReihen)[
|
||||
#subHeading(fill: colorReihen)[Basis und Dim]
|
||||
*Linear Abbildung:* $Phi: V -> V$
|
||||
- $Phi(0) = 0$
|
||||
- $Phi(lambda v + w) = lambda Phi(v) + Phi(w)$
|
||||
- Menge aller linearen Abbildung: $L(V,W)$
|
||||
|
||||
*Basis:*\
|
||||
linear unabhänige Menge $B$ an $v in V$, sodass $op("spann")(v_1, ..., v_n) = op("spann")(V)$
|
||||
- $B$ ist Erzeugerssystem von $V$
|
||||
- Endliche Erzeugerssystem: $abs(B_1)=abs(B_2)...$
|
||||
|
||||
*Linear unabhänige:*
|
||||
Linearkombintation in welcher $lambda_0 = 0, ..., lambda_n = 0$ die EINZIEGE Lösung für $lambda_0 v_0 + ... + lambda_1 v_1 = 0$
|
||||
|
||||
*Basisergänzungssatz:* \
|
||||
Sei ${v_1, ... v_n}$ lin. unabhänig und $M$ kein Basis. Dann $exists v_(n+1)$ sodass ${v_1, ... v_n, v_(n+1)}$ lin unabhänig (aber evt. eine Basis ist)
|
||||
|
||||
*Dimension:* $dim V = \#$Vektoren der Basis
|
||||
- $dim V = infinity$, wenn $V$ nicht endlich erzeugt ist
|
||||
]
|
||||
|
||||
#bgBlock(fill: colorAbbildungen)[
|
||||
#subHeading([Abbildungen], fill: colorAbbildungen)
|
||||
|
||||
$f(x)=y, f: A -> B$
|
||||
|
||||
*Injectiv (Monomorphismus):*\
|
||||
_one to one_ \
|
||||
$f(x) = f(y) <=> x = y$
|
||||
|
||||
*Surjectiv (Epimorhismis):* \
|
||||
_Output space coverered_ \
|
||||
- Zeigen das $f(f^(-1)(x)) = x$ für $x in DD$
|
||||
- $forall x in B: exists x in A : f(x) = y$
|
||||
NICHT surjektiv wenn $abs(a) < abs(b)$
|
||||
|
||||
*Bijektiv (Isomorphismus):* \
|
||||
_Injectiv und Surjectiv_ \
|
||||
- In einer Gruppe ist $f(x) = x c$ für $c,x in G$ bijektiv
|
||||
- isomorph: $V,W$ VRs, $f$ bijektiv $f(V) = W => V tilde.equiv W$
|
||||
|
||||
Beweiß durch Wiederspruch \
|
||||
für Gegenbeweiß
|
||||
|
||||
*Endomorphismus:* $A -> B$ mit $A, B subset.eq C$
|
||||
|
||||
*Automorphismus:* Endomorphismus und Bijektiv (Isomorphismus)
|
||||
|
||||
*Vektorraum-Homomorphismus:* linear Abbildung zwischen VR
|
||||
]
|
||||
|
||||
#bgBlock(fill: colorAbbildungen)[
|
||||
#subHeading(fill: colorAbbildungen)[Spann und Bild]
|
||||
*Spann:*
|
||||
- Vektorraum $V : op("spann")(V) = limits(inter)_(M subset V) U$
|
||||
- $B : op("spann")(U) = {lambda_0 v_0 + ... + lambda_n v_n, lambda_0, ... lambda_n in K}$
|
||||
- $op("spann")(Phi(M)) = Phi(op("spann")(M))$
|
||||
|
||||
*Urbild:* $f^(-1)(I subset B) subset.eq A$
|
||||
|
||||
*Bild:* Wertemenge $WW$
|
||||
- $f(I subset A) = B$ (Oft $I = A$)
|
||||
- Basis $B : op("spann")(B)$
|
||||
- $op("Bild") Phi := {Phi in W | v in V}$
|
||||
|
||||
*Nullraum/Kern:* \
|
||||
$op("Kern") Phi := {v in V | Phi(v) = 0}$
|
||||
|
||||
*Rang*
|
||||
$op("Rang") f := dim op("Bild") f$
|
||||
]
|
||||
|
||||
]
|
||||
|
||||
|
||||
28
src/lib/common_rewrite.typ
Normal file
28
src/lib/common_rewrite.typ
Normal file
@@ -0,0 +1,28 @@
|
||||
#let bgBlock(body, fill: color) = block(body, fill:fill.lighten(80%), width: 100%, inset: (bottom: 2mm))
|
||||
|
||||
#let SeperatorLine = line(length: 100%, stroke: (paint: black, thickness: 0.3mm))
|
||||
#let MathAlignLeft(e) = {
|
||||
align(left, block(e))
|
||||
}
|
||||
|
||||
#let subHeading(body, fill: color) = {
|
||||
box(
|
||||
align(
|
||||
top+center,
|
||||
text(
|
||||
body,
|
||||
size: 10pt,
|
||||
weight: "regular",
|
||||
style: "italic",
|
||||
)
|
||||
),
|
||||
fill: fill,
|
||||
width: 100%,
|
||||
inset: 1mm,
|
||||
height: auto
|
||||
)
|
||||
}
|
||||
|
||||
#let MathAlignLeft(e) = {
|
||||
align(left, block(e))
|
||||
}
|
||||
Reference in New Issue
Block a user