From eff0c7da6b06bb591f0c7243583d2bcbf1c2b98c Mon Sep 17 00:00:00 2001 From: alexander Date: Sun, 21 Dec 2025 19:51:38 +0100 Subject: [PATCH 1/3] asd --- src/Analysis_rewrite.typ | 3 ++- src/lib/common_rewrite.typ | 24 ++++++++++++++++++++++++ 2 files changed, 26 insertions(+), 1 deletion(-) create mode 100644 src/lib/common_rewrite.typ diff --git a/src/Analysis_rewrite.typ b/src/Analysis_rewrite.typ index b1bee0b..3704924 100644 --- a/src/Analysis_rewrite.typ +++ b/src/Analysis_rewrite.typ @@ -1,3 +1,5 @@ +#import "lib/common_rewrite.typ" : * + #set page( paper: "a4", margin: ( @@ -50,7 +52,6 @@ #let colorAbleitung = color.hsl(356.92deg, 92.13%, 75.1%) #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)) #columns(4, gutter: 2mm)[ #bgBlock(fill: colorAllgemein)[ diff --git a/src/lib/common_rewrite.typ b/src/lib/common_rewrite.typ new file mode 100644 index 0000000..2521372 --- /dev/null +++ b/src/lib/common_rewrite.typ @@ -0,0 +1,24 @@ +#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(it: content, fill: color) = { + box( + align( + top+center, + text( + it, + size: 10pt, + weight: "regular", + style: "italic", + ) + ), + fill: fill, + width: 100%, + inset: 1mm, + height: auto + ) +} From 76ab900cac597574c835530bec8154017f67acf7 Mon Sep 17 00:00:00 2001 From: alexander Date: Tue, 6 Jan 2026 20:23:25 +0100 Subject: [PATCH 2/3] Further expended linal Sammlung --- src/Analysis_rewrite.typ | 20 ---- src/LinearAlgebra.typ | 191 +++++++++++++++++++++++++++++++------ src/lib/common_rewrite.typ | 8 +- 3 files changed, 169 insertions(+), 50 deletions(-) diff --git a/src/Analysis_rewrite.typ b/src/Analysis_rewrite.typ index 3704924..8af6623 100644 --- a/src/Analysis_rewrite.typ +++ b/src/Analysis_rewrite.typ @@ -21,26 +21,6 @@ [Analysis 1 (IE)] )) - - -#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 SeperatorLine = line(length: 100%, stroke: (paint: black, thickness: 0.3mm)) #let MathAlignLeft(e) = { align(left, block(e)) diff --git a/src/LinearAlgebra.typ b/src/LinearAlgebra.typ index 92a67f6..acdb14e 100644 --- a/src/LinearAlgebra.typ +++ b/src/LinearAlgebra.typ @@ -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* + $op("Rang") f := dim op("Bild") f$ + ] + +] + diff --git a/src/lib/common_rewrite.typ b/src/lib/common_rewrite.typ index 2521372..7c29f45 100644 --- a/src/lib/common_rewrite.typ +++ b/src/lib/common_rewrite.typ @@ -5,12 +5,12 @@ align(left, block(e)) } -#let subHeading(it: content, fill: color) = { +#let subHeading(body, fill: color) = { box( align( top+center, text( - it, + body, size: 10pt, weight: "regular", style: "italic", @@ -22,3 +22,7 @@ height: auto ) } + +#let MathAlignLeft(e) = { + align(left, block(e)) +} \ No newline at end of file From c4696dde4b9cd00491b0ca8d1b317b1ea2ba8ae3 Mon Sep 17 00:00:00 2001 From: alexander Date: Tue, 13 Jan 2026 00:20:43 +0100 Subject: [PATCH 3/3] aaa --- src/Analysis_rewrite.pdf | 9043 ++++++++++++++++++++++++++++++++++++++ src/Analysis_rewrite.typ | 160 +- 2 files changed, 9200 insertions(+), 3 deletions(-) create mode 100644 src/Analysis_rewrite.pdf diff --git a/src/Analysis_rewrite.pdf b/src/Analysis_rewrite.pdf new file mode 100644 index 0000000..e170b1f --- /dev/null +++ b/src/Analysis_rewrite.pdf @@ -0,0 +1,9043 @@ +%PDF-1.7 +% + +1 0 obj +<< + /Type /Pages + /Count 2 + /Kids [692 0 R 694 0 R] +>> +endobj + +2 0 obj +<< + /Type /Outlines + /First 3 0 R + /Last 3 0 R + /Count 1 +>> +endobj + +3 0 obj +<< + /Parent 2 0 R + /Title (Analysis 1 (IE)) + /Dest 691 0 R +>> +endobj + +4 0 obj +<< + /Nums [0 655 0 R 1 656 0 R] +>> +endobj + +5 0 obj +<< + /Type /StructTreeRoot + /RoleMap << + /Datetime /Span + /Terms /Part + /Title /P + /Strong /Span + /Em /Span + >> + /K [8 0 R] + /ParentTree << + /Nums [0 6 0 R 1 7 0 R] + >> + /IDTree << + /Names [(U1x0y0) 208 0 R (U1x1y0) 206 0 R (U2x0y0) 110 0 R (U2x1y0) 107 0 R (U2x2y0) 104 0 R] + >> + /ParentTreeNextKey 2 +>> +endobj + +6 0 obj +[654 0 R 653 0 R 652 0 R 650 0 R 650 0 R 650 0 R 650 0 R 650 0 R 650 0 R 650 0 R 650 0 R 650 0 R 650 0 R 650 0 R 650 0 R 650 0 R 649 0 R 649 0 R 649 0 R 649 0 R 649 0 R 649 0 R 649 0 R 649 0 R 649 0 R 649 0 R 649 0 R 649 0 R 649 0 R 649 0 R 649 0 R 647 0 R 645 0 R 645 0 R 645 0 R 645 0 R 645 0 R 645 0 R 645 0 R 645 0 R 645 0 R 645 0 R 645 0 R 645 0 R 645 0 R 645 0 R 645 0 R 643 0 R 641 0 R 641 0 R 641 0 R 641 0 R 641 0 R 641 0 R 641 0 R 641 0 R 641 0 R 641 0 R 641 0 R 641 0 R 641 0 R 641 0 R 639 0 R 637 0 R 637 0 R 637 0 R 637 0 R 637 0 R 637 0 R 637 0 R 637 0 R 637 0 R 637 0 R 637 0 R 637 0 R 637 0 R 637 0 R 635 0 R 633 0 R 633 0 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106 0 R 106 0 R 106 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 99 0 R 99 0 R 97 0 R 97 0 R 97 0 R 97 0 R 97 0 R 94 0 R 94 0 R 94 0 R 94 0 R 92 0 R 92 0 R 90 0 R 90 0 R 90 0 R 90 0 R 87 0 R 87 0 R 87 0 R 87 0 R 87 0 R 87 0 R 87 0 R 87 0 R 85 0 R 85 0 R 85 0 R 85 0 R 85 0 R 83 0 R 83 0 R 80 0 R 80 0 R 80 0 R 80 0 R 80 0 R 80 0 R 78 0 R 78 0 R 78 0 R 76 0 R 76 0 R 76 0 R 76 0 R 76 0 R 73 0 R 73 0 R 71 0 R 71 0 R 69 0 R 69 0 R 66 0 R 66 0 R 66 0 R 66 0 R 66 0 R 66 0 R 64 0 R 64 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 57 0 R 57 0 R 57 0 R 57 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 52 0 R 52 0 R 52 0 R 52 0 R 52 0 R 52 0 R 52 0 R 52 0 R 52 0 R 52 0 R 52 0 R 50 0 R 50 0 R 50 0 R 50 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 43 0 R 43 0 R 43 0 R 43 0 R 41 0 R 41 0 R 41 0 R 41 0 R 41 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 36 0 R 36 0 R 36 0 R 36 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 29 0 R 29 0 R 29 0 R 29 0 R 27 0 R 27 0 R 27 0 R 27 0 R 27 0 R 27 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 22 0 R 22 0 R 22 0 R 22 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 20 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 15 0 R 15 0 R 15 0 R 15 0 R 13 0 R 13 0 R 13 0 R 13 0 R 13 0 R] +endobj + +8 0 obj +<< + /Type /StructElem + /S /Document + /P 5 0 R + /K [654 0 R 653 0 R 607 0 R 604 0 R 601 0 R 596 0 R 594 0 R 589 0 R 588 0 R 575 0 R 573 0 R 569 0 R 568 0 R 567 0 R 563 0 R 552 0 R 550 0 R 546 0 R 533 0 R 531 0 R 530 0 R 528 0 R 512 0 R 510 0 R 508 0 R 502 0 R 500 0 R 484 0 R 482 0 R 451 0 R 450 0 R 434 0 R 433 0 R 417 0 R 416 0 R 415 0 R 397 0 R 395 0 R 273 0 R 272 0 R 269 0 R 236 0 R 234 0 R 232 0 R 231 0 R 227 0 R 225 0 R 223 0 R 212 0 R 210 0 R 171 0 R 169 0 R 136 0 R 134 0 R 119 0 R 114 0 R 113 0 R 9 0 R] +>> +endobj + +9 0 obj +<< + /Type /StructElem + /S /Table + /P 8 0 R + /A [<< + /O /Layout + /BorderColor [0 0 0] + /BorderThickness 1 + >>] + /K [102 0 R 10 0 R] +>> +endobj + +10 0 obj +<< + /Type /StructElem + /S /TBody + /P 9 0 R + /K [95 0 R 88 0 R 81 0 R 74 0 R 67 0 R 60 0 R 53 0 R 46 0 R 39 0 R 32 0 R 25 0 R 18 0 R 11 0 R] +>> +endobj + +11 0 obj +<< + /Type /StructElem + /S /TR + /P 10 0 R + /K [16 0 R 14 0 R 12 0 R] +>> +endobj + +12 0 obj +<< + /Type /StructElem + /S /TD + /P 11 0 R + /A [<< + /O /Table + /Headers [(U2x2y0)] + >> << + /O /Layout + /BackgroundColor [0.92156863 0.99215686 0.99215686] + /BorderStyle /Solid + >>] + /K [13 0 R] +>> +endobj + +13 0 obj +<< + /Type /StructElem + /S /Formula + /P 12 0 R + /K [621 622 623 624 625] + /Pg 694 0 R +>> +endobj + +14 0 obj +<< + /Type /StructElem + /S /TD + /P 11 0 R + /A [<< + /O /Table + /Headers [(U2x1y0)] + >> << + /O /Layout + /BackgroundColor [0.8627451 1 1] + /BorderStyle /Solid + >>] + /K [15 0 R] +>> +endobj + +15 0 obj +<< + /Type /StructElem + /S /Formula + /P 14 0 R + /K [617 618 619 620] + /Pg 694 0 R +>> +endobj + +16 0 obj +<< + /Type /StructElem + /S /TD + /P 11 0 R + /A [<< + /O /Table + /Headers [(U2x0y0)] + >> << + /O /Layout + /BackgroundColor [0.7882353 0.9882353 0.9882353] + /BorderStyle /Solid + >>] + /K [17 0 R] +>> +endobj + +17 0 obj +<< + /Type /StructElem + /S /Formula + /P 16 0 R + /K [602 603 604 605 606 607 608 609 610 611 612 613 614 615 616] + /Pg 694 0 R +>> +endobj + +18 0 obj +<< + /Type /StructElem + /S /TR + /P 10 0 R + /K [23 0 R 21 0 R 19 0 R] +>> +endobj + +19 0 obj +<< + /Type /StructElem + /S /TD + /P 18 0 R + /A [<< + /O /Table + /Headers [(U2x2y0)] + >> << + /O /Layout + /BackgroundColor [0.92156863 0.99215686 0.99215686] + /BorderStyle /Solid + >>] + /K [20 0 R] +>> +endobj + +20 0 obj +<< + /Type /StructElem + /S /Formula + /P 19 0 R + /K [596 597 598 599 600 601] + /Pg 694 0 R +>> +endobj + +21 0 obj +<< + /Type /StructElem + /S /TD + /P 18 0 R + /A [<< + /O /Table + /Headers [(U2x1y0)] + >> << + /O /Layout + /BackgroundColor [0.8627451 1 1] + /BorderStyle /Solid + >>] + /K [22 0 R] +>> +endobj + +22 0 obj +<< + /Type /StructElem + /S /Formula + /P 21 0 R + /K [592 593 594 595] + /Pg 694 0 R +>> +endobj + +23 0 obj +<< + /Type /StructElem + /S /TD + /P 18 0 R + /A [<< + /O /Table + /Headers [(U2x0y0)] + >> << + /O /Layout + /BackgroundColor [0.7882353 0.9882353 0.9882353] + /BorderStyle /Solid + >>] + /K [24 0 R] +>> +endobj + +24 0 obj +<< + /Type /StructElem + /S /Formula + /P 23 0 R + /K [581 582 583 584 585 586 587 588 589 590 591] + /Pg 694 0 R +>> +endobj + +25 0 obj +<< + /Type /StructElem + /S /TR + /P 10 0 R + /K [30 0 R 28 0 R 26 0 R] +>> +endobj + +26 0 obj +<< + /Type /StructElem + /S /TD + /P 25 0 R + /A [<< + /O /Table + /Headers [(U2x2y0)] + >> << + /O /Layout + /BackgroundColor [0.92156863 0.99215686 0.99215686] + /BorderStyle /Solid + >>] + /K [27 0 R] +>> +endobj + +27 0 obj +<< + /Type /StructElem + /S /Formula + /P 26 0 R + /K [575 576 577 578 579 580] + /Pg 694 0 R +>> +endobj + +28 0 obj +<< + /Type /StructElem + /S /TD + /P 25 0 R + /A [<< + /O /Table + /Headers [(U2x1y0)] + >> << + /O /Layout + /BackgroundColor [0.8627451 1 1] + /BorderStyle /Solid + >>] + /K [29 0 R] +>> +endobj + +29 0 obj +<< + /Type /StructElem + /S /Formula + /P 28 0 R + /K [571 572 573 574] + /Pg 694 0 R +>> +endobj + +30 0 obj +<< + /Type /StructElem + /S /TD + /P 25 0 R + /A [<< + /O /Table + /Headers [(U2x0y0)] + >> << + /O /Layout + /BackgroundColor [0.7882353 0.9882353 0.9882353] + /BorderStyle /Solid + >>] + /K [31 0 R] +>> +endobj + +31 0 obj +<< + /Type /StructElem + /S /Formula + /P 30 0 R + /K [560 561 562 563 564 565 566 567 568 569 570] + /Pg 694 0 R +>> +endobj + +32 0 obj +<< + /Type /StructElem + /S /TR + /P 10 0 R + /K [37 0 R 35 0 R 33 0 R] +>> +endobj + +33 0 obj +<< + /Type /StructElem + /S /TD + /P 32 0 R + /A [<< + /O /Table + /Headers [(U2x2y0)] + >> << + /O /Layout + /BackgroundColor [0.92156863 0.99215686 0.99215686] + /BorderStyle /Solid + >>] + /K [34 0 R] +>> +endobj + +34 0 obj +<< + /Type /StructElem + /S /Formula + /P 33 0 R + /K [554 555 556 557 558 559] + /Pg 694 0 R +>> +endobj + +35 0 obj +<< + /Type /StructElem + /S /TD + /P 32 0 R + /A [<< + /O /Table + /Headers [(U2x1y0)] + >> << + /O /Layout + /BackgroundColor [0.8627451 1 1] + /BorderStyle /Solid + >>] + /K [36 0 R] +>> +endobj + +36 0 obj +<< + /Type /StructElem + /S /Formula + /P 35 0 R + /K [550 551 552 553] + /Pg 694 0 R +>> +endobj + +37 0 obj +<< + /Type /StructElem + /S /TD + /P 32 0 R + /A [<< + /O /Table + /Headers [(U2x0y0)] + >> << + /O /Layout + /BackgroundColor [0.7882353 0.9882353 0.9882353] + /BorderStyle /Solid + >>] + /K [38 0 R] +>> +endobj + +38 0 obj +<< + /Type /StructElem + /S /Formula + /P 37 0 R + /K [535 536 537 538 539 540 541 542 543 544 545 546 547 548 549] + /Pg 694 0 R +>> +endobj + +39 0 obj +<< + /Type /StructElem + /S /TR + /P 10 0 R + /K [44 0 R 42 0 R 40 0 R] +>> +endobj + +40 0 obj +<< + /Type /StructElem + /S /TD + /P 39 0 R + /A [<< + /O /Table + /Headers [(U2x2y0)] + >> << + /O /Layout + /BackgroundColor [0.92156863 0.99215686 0.99215686] + /BorderStyle /Solid + >>] + /K [41 0 R] +>> +endobj + +41 0 obj +<< + /Type /StructElem + /S /Formula + /P 40 0 R + /K [530 531 532 533 534] + /Pg 694 0 R +>> +endobj + +42 0 obj +<< + /Type /StructElem + /S /TD + /P 39 0 R + /A [<< + /O /Table + /Headers [(U2x1y0)] + >> << + /O /Layout + /BackgroundColor [0.8627451 1 1] + /BorderStyle /Solid + >>] + /K [43 0 R] +>> +endobj + +43 0 obj +<< + /Type /StructElem + /S /Formula + /P 42 0 R + /K [526 527 528 529] + /Pg 694 0 R +>> +endobj + +44 0 obj +<< + /Type /StructElem + /S /TD + /P 39 0 R + /A [<< + /O /Table + /Headers [(U2x0y0)] + >> << + /O /Layout + /BackgroundColor [0.7882353 0.9882353 0.9882353] + /BorderStyle /Solid + >>] + /K [45 0 R] +>> +endobj + +45 0 obj +<< + /Type /StructElem + /S /Formula + /P 44 0 R + /K [511 512 513 514 515 516 517 518 519 520 521 522 523 524 525] + /Pg 694 0 R +>> +endobj + +46 0 obj +<< + /Type /StructElem + /S /TR + /P 10 0 R + /K [51 0 R 49 0 R 47 0 R] +>> +endobj + +47 0 obj +<< + /Type /StructElem + /S /TD + /P 46 0 R + /A [<< + /O /Table + /Headers [(U2x2y0)] + >> << + /O /Layout + /BackgroundColor [0.92156863 0.99215686 0.99215686] + /BorderStyle /Solid + >>] + /K [48 0 R] +>> +endobj + +48 0 obj +<< + /Type /StructElem + /S /Formula + /P 47 0 R + /K [504 505 506 507 508 509 510] + /Pg 694 0 R +>> +endobj + +49 0 obj +<< + /Type /StructElem + /S /TD + /P 46 0 R + /A [<< + /O /Table + /Headers [(U2x1y0)] + >> << + /O /Layout + /BackgroundColor [0.8627451 1 1] + /BorderStyle /Solid + >>] + /K [50 0 R] +>> +endobj + +50 0 obj +<< + /Type /StructElem + /S /Formula + /P 49 0 R + /K [500 501 502 503] + /Pg 694 0 R +>> +endobj + +51 0 obj +<< + /Type /StructElem + /S /TD + /P 46 0 R + /A [<< + /O /Table + /Headers [(U2x0y0)] + >> << + /O /Layout + /BackgroundColor [0.7882353 0.9882353 0.9882353] + /BorderStyle /Solid + >>] + /K [52 0 R] +>> +endobj + +52 0 obj +<< + /Type /StructElem + /S /Formula + /P 51 0 R + /K [489 490 491 492 493 494 495 496 497 498 499] + /Pg 694 0 R +>> +endobj + +53 0 obj +<< + /Type /StructElem + /S /TR + /P 10 0 R + /K [58 0 R 56 0 R 54 0 R] +>> +endobj + +54 0 obj +<< + /Type /StructElem + /S /TD + /P 53 0 R + /A [<< + /O /Table + /Headers [(U2x2y0)] + >> << + /O /Layout + /BackgroundColor [0.92156863 0.99215686 0.99215686] + /BorderStyle /Solid + >>] + /K [55 0 R] +>> +endobj + +55 0 obj +<< + /Type /StructElem + /S /Formula + /P 54 0 R + /K [483 484 485 486 487 488] + /Pg 694 0 R +>> +endobj + +56 0 obj +<< + /Type /StructElem + /S /TD + /P 53 0 R + /A [<< + /O /Table + /Headers [(U2x1y0)] + >> << + /O /Layout + /BackgroundColor [0.8627451 1 1] + 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b/src/Analysis_rewrite.typ index 8af6623..918ef9a 100644 --- a/src/Analysis_rewrite.typ +++ b/src/Analysis_rewrite.typ @@ -48,7 +48,7 @@ $abs(x dot y) <= abs(abs(x) dot abs(y))$ ], [Geometrische Summenformel], [ - #MathAlignLeft($ sum_(k=1)^(n) k = (n(n+1))/2 $) + #MathAlignLeft($ limits(sum)_(k=1)^(n) k = (n(n+1))/2 $) ], [Bernoulli-Ungleichung ], [ $(1 + a)^n x in RR >= 1 + n a$ @@ -64,10 +64,58 @@ [Gausklammer], [ $floor(x) = text("floor")(x)$ \ $ceil(x) = text("ceil")(x)$ + ], + [Bekannte Werte], [ + $e approx 2.71828$ ($2 < e < 3$) \ + $pi approx 3.14159$ ($3 < pi < 4$) ] ) ] + #bgBlock(fill: colorAllgemein)[ + #subHeading(fill: colorAllgemein)[Trigonmetrie] + $sin(x+y) = cos(x)sin(y) + sin(x)cos(y)$ \ + $cos(x+y) = cos(x)cos(y) - sin(x)sin(y)$ \ + + $cos(2x) = cos^2(x) - sin^2(x)$ \ + $sin(2x) = 2sin(x)cos(x)$ + + #grid( + gutter: 5mm, + columns: (auto, auto), + [$cos^2(x) = (1 + cos(2x))/2$], + [$sin^2(x) = (1 - cos(2x))/2$] + ) + + $cos^2(x) + sin^2(x) = 1$ + + #grid( + gutter: 5mm, + columns: (auto, auto), + [$cos(-x) = cos(x)$], + [$sin(-x) = -sin(x)$], + ) + + Subsitution mit Hilfsvariable + + #grid( + gutter: 5mm, + row-gutter: 3mm, + columns: (auto, auto), + [$tan(x)=sin(x)/cos(x)$], + [$cot(x)=cos(x)/sin(x)$], + [$tan(x)=-cot(x + pi/2)$], + [$cot(x)=-tan(x + pi/2)$], + [$cos(x - pi/2) = sin(x)$], + [$sin(x + pi/2) = cos(x)$], + ) + $sin(x)cos(y) = 1/2sin(x - y) + 1/2sin(x + y)$ + + Für $x in [-1, 1]$ \ + $arcsin(x) = -arccos(x) - pi/2 in [-pi/2, pi/2]$ \ + $arccos(x) = -arcsin(x) + pi/2 in [0, pi]$ + ] + #bgBlock(fill: colorFolgen)[ #subHeading(fill: colorFolgen)[Folgen] $ lim_(x -> infinity) a_n $ @@ -170,16 +218,82 @@ #bgBlock(fill: colorReihen)[ #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* \ + ALLE Partialsummen von $limits(sum)_(k=1)^infinity abs(a)$ beschränkt\ + $=>$ _Absolute Konvergent_ + + - *(Cauchy-Kriterium)*\ + 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 \ + $=> limits(sum)_(n=1)^infinity (-1)^n dot a_n$ konvergent + + - *Vergleichskriterium* \ + $a_n, b_n : abs(a_n) <= b_n space forall n in NN > N_0, N_0 in NN$ + 1. $limits(sum)_(n=0)^infinity b_n$ konvergent $=> limits(sum)_(n=0)^infinity abs(a_n)$ konvergent \ + Suche $b_n$ für Konvergenz + 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: + - Dreiecksungleichung + - $forall space n > N_0 in NN space exists k,q in RR$ \ + sodass $q > 1$: $n^k <= q^n$ (Potenz stärker Polynom) + + - *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))) $ \ + + divergent: $rho > 1$, keine Aussage $rho = 1$, konvergent $rho < 1$ + + - *Geometrische Reihe* + $limits(sum)_(n=0)^infinity q^n$ + - konvergent $abs(q) < 1$, divergent $abs(q) >= 1$ + - 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)[ #subHeading(fill: colorReihen)[Potenzreihen] ] - #colbreak() #bgBlock(fill: colorAbleitung)[ #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 $x in (a,b)$ \ + $=> exists x_0 : f'(x_0)=(f(b) - f(a))/(a-b)$ + + - *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)[ @@ -256,7 +370,47 @@ [$e^(x) : e^(x)$], ) - 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() ] \ No newline at end of file