From 1573913f3fc186170e97220f1c9544a2e62969ef Mon Sep 17 00:00:00 2001
From: alexander <alexander@xstation>
Date: Sat, 31 Jan 2026 19:17:45 +0100
Subject: [PATCH 1/6] Added verschaltung

---
 src/cheatsheets/Digitaltechnik.typ    |   7 +-
 src/cheatsheets/Schaltungstheorie.typ |  69 +++++++-
 src/lib/circuit.typ                   |   2 +-
 src/lib/common_rewrite.typ            |  13 +-
 src/lib/fetModel.typ                  | 242 ++++++++++++++++++++++++++
 5 files changed, 318 insertions(+), 15 deletions(-)
 create mode 100644 src/lib/fetModel.typ

diff --git a/src/cheatsheets/Digitaltechnik.typ b/src/cheatsheets/Digitaltechnik.typ
index 688e720..8958d3e 100644
--- a/src/cheatsheets/Digitaltechnik.typ
+++ b/src/cheatsheets/Digitaltechnik.typ
@@ -1,8 +1,10 @@
-#import "../lib/common_rewrite.typ" : *
 #import "@preview/mannot:0.3.1"
 #import "@preview/cetz:0.4.2"
 #import "@preview/zap:0.5.0"
+
+#import "../lib/common_rewrite.typ" : *
 #import "../lib/truthtable.typ" : *
+#import "../lib/fetModel.typ" : *
 
 #show math.integral: it => math.limits(math.integral)
 #show math.sum: it => math.limits(math.sum)
@@ -565,10 +567,7 @@
         (cycle_start,signal_hight), (cycle_time*(t_setup + 2), signal_hight), 
         (cycle_time*(t_setup + 2) + switch_offset, 0), (cycle_end + switch_offset, 0), stroke: signal_storke
       )
-    
-
     })
-
   ]
 
 
diff --git a/src/cheatsheets/Schaltungstheorie.typ b/src/cheatsheets/Schaltungstheorie.typ
index 186cc16..cd5a139 100644
--- a/src/cheatsheets/Schaltungstheorie.typ
+++ b/src/cheatsheets/Schaltungstheorie.typ
@@ -108,7 +108,63 @@
   #bgBlock(fill: colorAllgemein)[
     #subHeading(fill: colorAllgemein)[Verschaltung]
 
+    $1/x_"ges" = 1/x_1 + 1/x_2 + 1/x_3 + ...$
 
+    $x_1 parallel x_2 = 1/(1/x_1 + 1/x_2) = (x_1 x_2)/(x_1 + x_2)$
+
+    $x_1 parallel x_2 parallel x_3 = (x_1 x_2 x_3)/(x_1 x_2 + x_2 x_3 + x_1 x_3)$
+
+    #table(
+      columns: (1fr, 1fr),
+      fill: (x, y) => if calc.rem(x, 2) == 1 { tableFillLow } else { tableFillHigh },
+
+      [*Serie*],
+      [*Reihe/Parrallel*], 
+      
+      
+      align(
+        horizon+center,
+        scale(100%,
+          zap.circuit({
+            import zap : *
+
+            resistor("R1", (0,0.375), (1.5,0.375), fill: none, width: 1, height: 0.4)
+            resistor("R2", (1.5,0.375), (3,0.375), fill: none, width: 1, height: 0.4)
+          })
+        )
+      ),
+      align(
+        horizon+center,
+        scale(
+          100%,
+          zap.circuit({
+            import zap : *
+
+            resistor("R1", (0,0), (2,0), fill: none, width: 1, height: 0.4)
+            resistor("R2", (0,0.75), (2,0.75), fill: none, width: 1, height: 0.4)
+            wire("R1.in", "R2.in")
+            wire("R1.out", "R2.out")
+
+            node("N2", (0,0.375))
+            node("N3", (2,0.375))
+
+            wire("N2", (rel: (-0.3, 0)))
+            wire("N3", (rel: (0.3, 0)))
+          }),
+        )
+      ),
+
+
+
+      $R_"ges" = R_1+R_2$,
+      $R_"ges" = R_1 parallel R_2$,
+      $G_"ges" = G_1 parallel G_2$,
+      $G_"ges" = G_1 + G_2$,
+      $L_"ges" = L_1 + L_2$,
+      $L_"ges" = L_1 parallel L_2$,
+      $C_"ges" = C_1 parallel C_2$,
+      $C_"ges" = C_1 + C_2$,
+    )
   ]
 
   // Quell Wandlung
@@ -133,8 +189,6 @@
     )
   ]
 
-  #colbreak()
-
   // Quell Wandlung
   #bgBlock(fill: colorEineTore)[
     #subHeading(fill: colorEineTore)[Quelle Wandlung]
@@ -785,13 +839,14 @@
 
     *Levi's Lustig Leistung*
 
-    $underline(S) = underline(U) dot underline(I)^*$\
+    $(P =) space underline(S) = underline(U) dot underline(I)^*$\
 
     #table(
       columns: (auto, 1fr, auto),
-      [Scheinleitsung], [$abs(underline(S))$], [$["VA"]$],
-      [Wirkleistung], [$P = "Real"(underline(S)) $], [$["W"]$],
-      [Blindleistung], [$Q = "Imag"(underline(S))$], [$["var"]$]
+      fill: (x, y) => if calc.rem(y, 2) == 0 { tableFillLow } else { tableFillHigh },
+      [Scheinleitsung], [$(P_s =) space abs(underline(S))$], [$["VA"]$],
+      [Wirkleistung], [$(P_w =) space P = "Real"(underline(S)) $], [$["W"]$],
+      [Blindleistung], [$(P_b =) space Q = "Imag"(underline(S))$], [$["var"]$]
     )
   ]
 
@@ -829,6 +884,8 @@
       columns: (auto, auto, auto, auto),
       inset: 2mm,
       align: horizon,
+      fill: (x, y) => if calc.rem(y, 2) == 1 { rgb("#c5c5c5") } else { white },
+
       table.header([], [*Ein-Tor*], [*Zwei-Tor*], [*Reaktive Elemente*]),
       [*passiv*\ (nimmt Energie auf)\ $not$aktiv],
       [$forall (u,i) in cal(F): u dot i >= 0$],
diff --git a/src/lib/circuit.typ b/src/lib/circuit.typ
index f09b286..c5efbb3 100644
--- a/src/lib/circuit.typ
+++ b/src/lib/circuit.typ
@@ -13,7 +13,7 @@
 
     ctx.zap.style.insert("einTor", (
       scale: auto,
-      fill: auto,
+      fill: none,
       height: 3mm,
       width: 6mm,
     ))
diff --git a/src/lib/common_rewrite.typ b/src/lib/common_rewrite.typ
index 8272635..e19e38f 100644
--- a/src/lib/common_rewrite.typ
+++ b/src/lib/common_rewrite.typ
@@ -27,6 +27,9 @@
   align(left, block(e))
 }
 
+#let tableFillHigh = white
+#let tableFillLow = color.lighten(gray, 30%)
+
 #let sinTable = [
   #let data = json("../sintable.json")
   #table(
@@ -34,10 +37,11 @@
     rows: data.keys().len(),
     stroke: none,
     table.hline(stroke: (thickness: 0.3mm)),
-    fill: (x, y) => if (calc.rem(y, 2) == 0) { color.lighten(gray, 50%) } else { white },
+    fill: (x, y) => if (calc.rem(y, 2) == 0) { tableFillLow } else { tableFillHigh },
 
-    ..for (label) in data.keys() {
-      ([*#eval(label, mode: "math")*], )
+    table.vline(),
+    ..for (i, label) in data.keys().enumerate() {
+      ([*#eval(label, mode: "math")*], if i > 0 { table.vline() } else { table.vline(stroke: none) })
     },
 
     table.hline(stroke: (thickness: 0.3mm)),
@@ -46,6 +50,7 @@
     for (col) in data.keys() {
         (eval(data.at(col).at(i), mode: "math"),)
       }
-    }
+    },
+    table.hline(stroke: (thickness: 0.3mm)),
   )
 ]
\ No newline at end of file
diff --git a/src/lib/fetModel.typ b/src/lib/fetModel.typ
new file mode 100644
index 0000000..6661e0d
--- /dev/null
+++ b/src/lib/fetModel.typ
@@ -0,0 +1,242 @@
+#import "@preview/zap:0.5.0"
+
+#set page(width: auto, height: auto)
+
+#let FetModelSubstrate = zap.circuit({
+  import zap: *
+  import cetz.draw: *
+
+  rect(
+    (0, 0),
+    (12, 4),
+    fill: rgb("#ffb1b1"),
+    name: "p",
+  )
+  rect((0, -0.05), (12, -0.05), stroke: 3pt, name: "substrate")
+  earth("g1", (11.5, 0))
+
+  content("substrate", [Bulk], anchor: "north", padding: 0.2)
+})
+
+#let FetModel1 = zap.circuit({
+  import zap: *
+  import cetz.draw: *
+
+  rect(
+    (0, 0),
+    (12, 4),
+    fill: rgb("#ffb1b1"),
+    name: "p",
+  )
+  rect((0, -0.05), (12, -0.05), stroke: 3pt, name: "substrate")
+  earth("g1", (11.5, 0))
+
+  rect((2.75, 3), (5.25, 4), fill: rgb("#61ff9f"), name: "kanal1", radius: (
+    south: 0.2,
+  ))
+  rect((6.75, 3), (9.25, 4), fill: rgb("#61ff9f"), name: "kanal2", radius: (
+    south: 0.2,
+  ))
+
+  content("kanal1", [n+], anchor: "center", padding: 0.2)
+  content("kanal2", [n+], anchor: "center", padding: 0.2)
+  content("substrate", [Bulk], anchor: "north", padding: 0.2)
+})
+
+#let FetModel2 = zap.circuit({
+  import zap: *
+  import cetz.draw: *
+
+  rect(
+    (0, 0),
+    (12, 4),
+    fill: rgb("#ffb1b1"),
+    name: "p",
+  )
+  rect((0, -0.05), (12, -0.05), stroke: 3pt, name: "substrate")
+  earth("g1", (11.5, 0))
+
+  rect((2.75, 3), (5.25, 4), fill: rgb("#61ff9f"), name: "kanal1", radius: (
+    south: 0.2,
+  ))
+  rect((6.75, 3), (9.25, 4), fill: rgb("#61ff9f"), name: "kanal2", radius: (
+    south: 0.2,
+  ))
+
+  content("kanal1", [n+], anchor: "center", padding: 0.2)
+  content("kanal2", [n+], anchor: "center", padding: 0.2)
+  content("substrate", [Bulk], anchor: "north", padding: 0.2)
+
+
+  rect((0, 4), (12, 4.5), fill: rgb("#fffc61"), name: "isolator")
+  content("isolator.west", [Isolator ($"SiO"_2$)], anchor: "west", padding: .2)
+})
+
+#let FetModel3 = zap.circuit({
+  import zap: *
+  import cetz.draw: *
+
+  rect(
+    (0, 0),
+    (12, 4),
+    fill: rgb("#ffb1b1"),
+    name: "p",
+  )
+  rect((0, -0.05), (12, -0.05), stroke: 3pt, name: "substrate")
+  earth("g1", (11.5, 0))
+
+  rect((2.75, 3), (5.25, 4), fill: rgb("#61ff9f"), name: "kanal1", radius: (
+    south: 0.2,
+  ))
+  rect((6.75, 3), (9.25, 4), fill: rgb("#61ff9f"), name: "kanal2", radius: (
+    south: 0.2,
+  ))
+
+  content("kanal1", [n+], anchor: "center", padding: 0.2)
+  content("kanal2", [n+], anchor: "center", padding: 0.2)
+  content("substrate", [Bulk], anchor: "north", padding: 0.2)
+
+
+  rect((0, 4), (3, 4.5), fill: rgb("#fffc61"), name: "isolator2")
+  rect((5, 4), (7, 4.5), fill: rgb("#fffc61"))
+  rect((9, 4), (12, 4.5), fill: rgb("#fffc61"), name: "isolator")
+
+  rect((3, 4), (5, 4.25), fill: gray)
+  rect((7, 4), (9, 4.25), fill: gray)
+  rect((5.1, 4.5), (6.9, 4.75), fill: gray)
+
+  content("isolator", [$"SiO"_2$])
+  content("isolator2", [Isolator])
+})
+
+#let FetModel(type: "N", s: 100%) = zap.circuit({
+  import zap: *
+  import cetz.draw: rect, content
+
+  cetz.draw.scale(s)
+
+  let pTypeFill = rgb("#ffb1b1");
+  let pTypeFill = rgb("#ffb1b1");
+  let nTypeFill = rgb("#61ff9f")
+
+  rect(
+    (0, 0),
+    (12, 4),
+    fill: if(type == "N") { pTypeFill } else { nTypeFill },
+    name: "p",
+  )
+  rect((0, -0.05), (12, -0.05), stroke: 3pt, name: "substrate")
+  earth("g1", (11.5, 0))
+
+  wire((13, 5.5), (13, 0), mark: (end: ">"))
+  node("n-r1", (13, 5.75))
+
+  wire((6, 4.5), (6, 5.75))
+  node("n-g1", (6, 5.75))
+  wire("n-g1", "n-r1")
+
+  node("n-s1", (4, 5))
+  wire("n-s1", (4, 4.25))
+
+  node("n-d1", (8, 5))
+  wire("n-d1", (8, 4.25))
+
+  rect((0, 4), (3, 4.5), fill: rgb("#fffc61"), name: "isolator2")
+  rect((5, 4), (7, 4.5), fill: rgb("#fffc61"))
+  rect((9, 4), (12, 4.5), fill: rgb("#fffc61"), name: "isolator")
+
+  rect((3, 4), (5, 4.25), fill: gray)
+  rect((7, 4), (9, 4.25), fill: gray)
+  rect((5.1, 4.5), (6.9, 4.75), fill: gray)
+
+  content("n-g1", [*G*\ate], anchor: "south", padding: 0.2, auto-scale: true)
+  content("n-s1", [*S*\ource], anchor: "south", padding: 0.2)
+  content("n-d1", [*D*\rain], anchor: "south", padding: 0.2, auto-scale: true)
+  content("substrate", [Bulk], anchor: "north", padding: 0.2, auto-scale: true)
+  content("p", if type == "N" [p] else [n], auto-scale: true)
+  content("isolator", [$"SiO"_2$], auto-scale: true)
+
+  content("isolator2", [Isolator], auto-scale: true)
+
+  rect((2.75, 3), (5.25, 4), fill: if(type == "N") { nTypeFill }  else { pTypeFill }, name: "kanal1", radius: (
+    south: 0.2,
+  ))
+  rect((6.75, 3), (9.25, 4), fill: if(type == "N") { nTypeFill } else { pTypeFill }, name: "kanal2", radius: (
+    south: 0.2,
+  ))
+
+  content("kanal1", if type == "N" [n+] else [p+], anchor: "center", padding: 0.2, auto-scale: true)
+  content("kanal2", if type == "N" [n+] else [p+], anchor: "center", padding: 0.2, auto-scale: true)
+})
+
+#let FetModelConducting = zap.circuit({
+  import zap: *
+  import cetz.draw: *
+
+  rect(
+    (0, 0),
+    (12, 4),
+    fill: rgb("#ffb1b1"),
+    name: "p",
+  )
+  rect((0, -0.05), (12, -0.05), stroke: 3pt, name: "substrate")
+  earth("g1", (11.5, 0))
+
+  wire((13, 5.5), (13, 0), mark: (end: ">"))
+  node("n-r1", (13, 5.75))
+
+  wire((6, 4.5), (6, 5.75))
+  node("n-g1", (6, 5.75))
+  wire("n-g1", "n-r1")
+
+  node("n-s1", (4, 5))
+  wire("n-s1", (4, 4.25))
+
+  node("n-d1", (8, 5))
+  wire("n-d1", (8, 4.25))
+
+  rect((0, 4), (3, 4.5), fill: rgb("#fffc61"), name: "isolator2")
+  rect((5, 4), (7, 4.5), fill: rgb("#fffc61"))
+  rect((9, 4), (12, 4.5), fill: rgb("#fffc61"), name: "isolator")
+
+  rect((3, 4), (5, 4.25), fill: gray)
+  rect((7, 4), (9, 4.25), fill: gray)
+  rect((5.1, 4.5), (6.9, 4.75), fill: gray)
+
+  content("n-g1", [*G*\ate], anchor: "south", padding: 0.2)
+  content("n-s1", [*S*\ource], anchor: "south", padding: 0.2)
+  content("n-d1", [*D*\rain], anchor: "south", padding: 0.2)
+  content("substrate", [Bulk], anchor: "north", padding: 0.2)
+  content("p", [p])
+  content("isolator", [$"SiO"_2$])
+  content("isolator2", [Isolator])
+
+  rect((2.75, 3), (5.25, 4), fill: rgb("#61ff9f"), name: "kanal1", radius: (
+    south: 0.2,
+  ))
+  rect((6.75, 3), (9.25, 4), fill: rgb("#61ff9f"), name: "kanal2", radius: (
+    south: 0.2,
+  ))
+  rect((5.20, 3.99), (6.8, 3.9), fill: rgb("#61ff9f"), stroke: none)
+
+  content("kanal1", [n+], anchor: "center", padding: 0.2)
+  content("kanal2", [n+], anchor: "center", padding: 0.2)
+
+  rect((0.5, -1), (1, -0.70), fill: gray, stroke: none, name: "metal")
+  content("metal", [metal], anchor: "west", padding: 0.3)
+
+  rect((0.5, -1.2), (1, -1.5), fill: rgb("#61ff9f"), stroke: none, name: "n")
+  content("n", [n+], anchor: "west", padding: 0.3)
+
+  rect(
+    (2.5, -1),
+    (3, -0.7),
+    fill: rgb("#ffb1b1"),
+    stroke: none,
+    name: "p-substrate",
+  )
+  content("p-substrate", [p], anchor: "west", padding: 0.3)
+
+  rect((2.5, -1.2), (3, -1.5), fill: rgb("#fffc61"), stroke: none, name: "siO2")
+  content("siO2", [oxide], anchor: "west", padding: 0.3)
+})

From d7703597bbdf3d9e1df2b6c65fac2afebe4ffecf Mon Sep 17 00:00:00 2001
From: alexander <alexander@xstation>
Date: Sun, 1 Feb 2026 11:49:45 +0100
Subject: [PATCH 2/6] Added Qullen Plot

---
 src/cheatsheets/Schaltungstheorie.typ | 141 ++++++++++++++++++++++++--
 src/lib/common_rewrite.typ            |   2 +-
 src/lib/fetModel.typ                  |  48 +++++++++
 3 files changed, 181 insertions(+), 10 deletions(-)

diff --git a/src/cheatsheets/Schaltungstheorie.typ b/src/cheatsheets/Schaltungstheorie.typ
index cd5a139..a4460f5 100644
--- a/src/cheatsheets/Schaltungstheorie.typ
+++ b/src/cheatsheets/Schaltungstheorie.typ
@@ -1,9 +1,11 @@
-#import "../lib/common_rewrite.typ" : *
 #import "@preview/mannot:0.3.1"
 #import "@preview/zap:0.5.0"
 #import "@preview/cetz:0.4.2"
-#import "../lib/circuit.typ" : *
 #import "@preview/unify:0.7.1": num,qty,unit
+#import "@preview/cetz-plot:0.1.3"
+
+#import "../lib/common_rewrite.typ" : *
+#import "../lib/circuit.typ" : *
 
 #set math.mat(delim: "[") 
 #show math.equation.where(block: true): it => math.inline(it)
@@ -164,6 +166,13 @@
       $L_"ges" = L_1 parallel L_2$,
       $C_"ges" = C_1 parallel C_2$,
       $C_"ges" = C_1 + C_2$,
+
+      $U_"ges" = U_1 + U_2$,
+      $U_"ges" = U_1 = U_2$,
+      $I_"ges" = I_1 = I_2$,
+      $I_"ges" = I_1 + I_2$,
+      [In $U$-Richtung Addieren],
+      [In $I$-Richtung Addieren],
     )
   ]
 
@@ -189,6 +198,108 @@
     )
   ]
 
+  // Lineare Quelle
+  #bgBlock(fill: colorEineTore)[
+    #subHeading(fill: colorEineTore)[Lineare Quelle]
+
+    #align(
+      center+horizon,
+      cetz.canvas({
+        import cetz.draw: *
+        import cetz-plot: *
+        plot.plot(size: (3, 3), name: "plot",
+            axis-style: "school-book",
+            x-label: "u", y-label: "i",
+            x-tick-step: none, y-tick-step: none,
+            axes: ("u", "i"),
+            x-min: -1, x-max: 2, x-grid: "both",
+            y-min: -2, y-max: 1, y-grid: "both", {
+          plot.add(((-2, -3), (3,2)))
+          plot.add-anchor("u0", (1,0))
+          plot.add-anchor("i0", (0,-1))
+        })
+
+        content("plot.u0", $U_0$, anchor: "south", padding: .2)
+        content("plot.i0", $-I_0$, anchor: "east", padding: .2)
+
+        mark("plot.u0", 0deg, symbol: "+", fill: black)
+        mark("plot.i0", 0deg, symbol: "+", fill: black)
+
+        line("plot.i0", (horizontal: "plot.u0", vertical: "plot.i0"), "plot.u0", stroke: (dash: "dashed", paint: rgb("#005c00")))
+
+        content((horizontal: "plot.u0", vertical: "plot.i0"), anchor: "south-west", text(rgb("#005c00"))[$R$], padding: 0.1)
+      })
+    )
+
+    $U_0$: LL-Spannung ($i = 0 => u = U_0$) \
+    $I_0$: KS-Strom ($u = 0 => i = -I_0$)
+
+    $R_i$: Innenwiderstand $R_i = U_0/I_0$ 
+
+    #table(
+      columns: (1fr, 1fr),
+      fill: (x, y) => if calc.rem(x, 2) == 1 { tableFillLow } else { tableFillHigh },
+      inset: 3mm,
+      align(center, [*$u$-gesteuert*]),
+      align(center, [*$i$-gesteuert*]),
+      
+      align(
+        horizon+center,
+        zap.circuit({
+          import zap : *
+          import cetz.draw
+
+          zap.resistor("R1", (1, 0), (1, -1.5), fill: none, width: 0.8, height: 0.3)
+          zap.isource("I0", (0, 0), (0, -1.5), fill: none, scale: 0.6, i: (content: $-I_0$, distance: 6pt, label-distance: -11pt, anchor: "west", invert: true))
+          node("N0", "R1.in")
+          node("N0", "R1.out")
+
+          wire("I0.out", "R1.out", (rel: (0.5, 0)))
+          wire("I0.in", "R1.in")
+          wire("R1.in", (rel: (0.5, 0)), i: (content: $i$, invert: true))
+
+          cetz.draw.content((0.62, -0.75), [$R$])
+          cetz.draw.set-style(mark: (end: ">", fill: black, scale: 0.6))
+          cetz.draw.content((1.7, -0.75), [$u$])
+          cetz.draw.line((1.5, -0.1), (1.5, -1.4), stroke: 0.5pt)
+        })
+      ),
+      align(
+        horizon+center,
+        zap.circuit({
+          import zap : *
+          import cetz.draw
+
+          zap.vsource("U0", (0, 0), (0, -1.5), fill: none, scale: 0.6, u: (content: $U_0$, distance: -4pt, label-distance: -8pt, anchor: "south-west", invert: true))
+          zap.resistor("R1", (0, 0), (1.75, 0), fill: none, width: 0.8, height: 0.3,
+            i: (content: $i$, invert: true, distance: 0.3)
+          )
+          wire((0, -1.5), (1.75, -1.5))
+      
+          cetz.draw.content((0.62, -0.75), [$R$])
+          cetz.draw.set-style(mark: (end: ">", fill: black, scale: 0.6))
+          cetz.draw.content((1.95, -0.75), [$u$])
+          cetz.draw.line((1.75, -0.1), (1.75, -1.4), stroke: 0.5pt)
+        })
+      ),
+
+      [
+        $u = R_i i + u_0$ \
+
+      ],
+      [
+        $i = 1/R_i u - I_0$
+      ],
+
+      table.cell(colspan: 2)[
+        #align($-->$)
+      ],
+      table.cell(colspan: 2)[
+        $<--$
+      ],
+    )
+  ]
+
   // Quell Wandlung
   #bgBlock(fill: colorEineTore)[
     #subHeading(fill: colorEineTore)[Quelle Wandlung]
@@ -245,14 +356,14 @@
         
         $R_i=1/G_i$
 
-        $r(i) = R_i i + U_0$
+        $u = r(i) = R_i i + U_0$
       ],
       [
         $I_0 = U_0 G_i$
 
         $G_i=1/R_i$
 
-        $g(u) = G_i u + I_0$
+        $i = g(u) = G_i u + I_0$
       ]
     );
   ]
@@ -786,12 +897,16 @@
       $E_L = Phi^2/2L = (L i^2)/2$
     ],
     [
-      $C$: Admetanz $hat(=) G$
+      $C$: Admittanz $hat(=) G$
     ], [],
     [
       $L$: Impedanz $hat(=) R$
     ]
     )
+
+    Admittanz: $Y = I/U$\
+    Impedanz: $Z = U/I$
+
   ]
 
   // Reaktive Dual Wandlung
@@ -836,18 +951,26 @@
       U_"ges"^2 = U_1^2 + 2 U_1 U_2 + U_2^2 \
       tan(phi) = (U_2 sin(phi))/(U_1 + U_2 cos(phi))
     $
+  ]
 
-    *Levi's Lustig Leistung*
+  #bgBlock(fill: colorComplexAC)[
+    #subHeading(fill: colorComplexAC)[*Levi's Lustig Leistung*]
 
-    $(P =) space underline(S) = underline(U) dot underline(I)^*$\
+    $P = 1/2 U dot I^*$\
 
     #table(
       columns: (auto, 1fr, auto),
       fill: (x, y) => if calc.rem(y, 2) == 0 { tableFillLow } else { tableFillHigh },
       [Scheinleitsung], [$(P_s =) space abs(underline(S))$], [$["VA"]$],
-      [Wirkleistung], [$(P_w =) space P = "Real"(underline(S)) $], [$["W"]$],
-      [Blindleistung], [$(P_b =) space Q = "Imag"(underline(S))$], [$["var"]$]
+      [Wirkleistung], [$(P_w =) space P = "Re"{} $], [$["W"]$],
+      [Blindleistung], [$(P_b =) space Q = "Im"{}$], [$["var"]$]
     )
+
+    Bei Wiederstand: $R$
+
+    $P_w = U_m^2 / 2R = (I_m^2 R)/2$
+
+    $U_"eff" = U_m/sqrt(2), I_"eff" = I_m / sqrt(2)$
   ]
 
   // Komplexe Zahlen
diff --git a/src/lib/common_rewrite.typ b/src/lib/common_rewrite.typ
index e19e38f..71d53be 100644
--- a/src/lib/common_rewrite.typ
+++ b/src/lib/common_rewrite.typ
@@ -28,7 +28,7 @@
 }
 
 #let tableFillHigh = white
-#let tableFillLow = color.lighten(gray, 30%)
+#let tableFillLow = color.lighten(gray, 50%)
 
 #let sinTable = [
   #let data = json("../sintable.json")
diff --git a/src/lib/fetModel.typ b/src/lib/fetModel.typ
index 6661e0d..8639b6a 100644
--- a/src/lib/fetModel.typ
+++ b/src/lib/fetModel.typ
@@ -1,4 +1,6 @@
 #import "@preview/zap:0.5.0"
+#import "@preview/cetz-plot:0.1.3"
+
 
 #set page(width: auto, height: auto)
 
@@ -240,3 +242,49 @@
   rect((2.5, -1.2), (3, -1.5), fill: rgb("#fffc61"), stroke: none, name: "siO2")
   content("siO2", [oxide], anchor: "west", padding: 0.3)
 })
+
+#let FetPlot() = {
+  let u_gs = 1
+  let beta = 1
+
+  cetz.canvas({
+    import cetz-plot: plot
+    import cetz: draw.content
+
+    cetz.draw.set-style(axes: (
+      shared-zero: false,
+      overshoot: 0.2,
+      x: (mark: (end: ">", fill: black, scale: 0.6)),
+      y: (mark: (end: ">", fill: black, scale: 0.6)),
+    ))
+
+    plot.plot(
+      size: (2, 2),
+      name: "plot",
+      axis-style: "school-book",
+      x-min: 0,
+      x-tick-step: none,
+      y-tick-step: none,
+      x-label: $U_"GS"$,
+      y-label: $U_"DS"$,
+      {
+        plot.add-fill-between(domain: (1, 6), ((1, 0), (1, 5)), u_gs => u_gs - u_t)
+
+        plot.add(domain: (0, 5), fill: true, axes: ("y", "x"), _ => 1)
+
+        plot.add(domain: (1, 6), fill: true, u_gs => u_gs - u_t)
+
+        plot.add-anchor("I", (0.5, 2.5))
+        plot.add-anchor("II", (4.5, 1.5))
+        plot.add-anchor("III", (2.5, 3.5))
+
+        plot.add-anchor("ut", (u_t, 0))
+      }
+    )
+    content("plot.ut", $U_t$, anchor: "north", padding: 0.1)
+
+    content("plot.I", [I])
+    content("plot.II", [II])
+    content("plot.III", [III])
+  })
+}
\ No newline at end of file

From 72e31ef355c56c4c426410fe767f8b0af305f76a Mon Sep 17 00:00:00 2001
From: alexander <alexander@xstation>
Date: Sun, 1 Feb 2026 23:56:11 +0100
Subject: [PATCH 3/6] Added alot of linAlg

---
 src/cheatsheets/Analysis1.typ         |   3 +-
 src/cheatsheets/LinearAlgebra.typ     | 328 ++++++++++++++++++--------
 src/cheatsheets/Schaltungstheorie.typ |   8 +-
 src/lib/common_rewrite.typ            |  15 ++
 4 files changed, 247 insertions(+), 107 deletions(-)

diff --git a/src/cheatsheets/Analysis1.typ b/src/cheatsheets/Analysis1.typ
index 8a899f2..f742bb5 100644
--- a/src/cheatsheets/Analysis1.typ
+++ b/src/cheatsheets/Analysis1.typ
@@ -106,9 +106,8 @@
   // Complex Zahlen
   #bgBlock(fill: colorAllgemein)[
     #subHeading(fill: colorAllgemein)[Complexe Zahlen]
-    $z = r dot e^(phi i) = r (cos(phi) + i sin(phi))$
 
-    $z^n = r^n dot e^(phi i dot n) = r^n (cos(n phi) + i sin(n phi))$
+    #ComplexNumbersSection()
 
     #grid(
       columns: (1fr, 1fr),
diff --git a/src/cheatsheets/LinearAlgebra.typ b/src/cheatsheets/LinearAlgebra.typ
index 4b047dc..ee46336 100644
--- a/src/cheatsheets/LinearAlgebra.typ
+++ b/src/cheatsheets/LinearAlgebra.typ
@@ -1,5 +1,8 @@
 #import "@preview/biceps:0.0.1" : *
 #import "@preview/mannot:0.3.1"
+#import "@preview/fletcher:0.5.8"
+#import "@preview/cetz:0.4.2"
+
 #import "../lib/styles.typ" : *
 #import "../lib/common_rewrite.typ" : *
 
@@ -23,11 +26,22 @@
 ))
 
 #let colorAllgemein = color.hsl(105.13deg, 92.13%, 75.1%)
+#let colorMatrixVerfahren = color.hsl(330.19deg, 100%, 68.43%)
 #let colorMatrix = color.hsl(202.05deg, 92.13%, 75.1%)
-#let colorReihen = color.hsl(280deg, 92.13%, 75.1%)
+#let colorVR = 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%)
 
+// Math macors
+#let kern(x) = $op("kern")(#x)$
+#let alg(x) = $op("alg")(#x)$
+#let geo(x) = $op("geo")(#x)$
+#let spann(x) = $op("spann")(#x)$
+#let Bild(x) = $op("Bild")(#x)$
+#let Rang(x) = $op("Rang")(#x)$
+#let Eig(x) = $op("Eig")(#x)$
+
+#let ip(x, y) = $lr(angle.l #x, #y angle.r)$
 
 #let SeperatorLine = line(length: 100%, stroke: (paint: black, thickness: 0.3mm))
 #let MathAlignLeft(e) = {
@@ -35,11 +49,19 @@
 }
 #columns(4, gutter: 2mm)[
     #bgBlock(fill: colorAllgemein)[
-        #subHeading(fill: colorAllgemein)[Notation]
+        #subHeading(fill: colorAllgemein)[Algemein]
         
 
     ]
 
+    #bgBlock(fill: colorAllgemein)[
+        #subHeading(fill: colorAllgemein)[Komplexe Zahlen]
+        
+        #ComplexNumbersSection()
+        
+        #sinTable
+    ]
+
     #bgBlock(fill: colorGruppen)[
          #subHeading(fill: colorGruppen)[Gruppen]
 
@@ -75,8 +97,6 @@
         - $(R, dot)$ 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 $+$)
@@ -84,8 +104,8 @@
         _Beweiß durch Überprüfung der Eigneschaften_
     ]
 
-    #bgBlock(fill: colorReihen)[
-        #subHeading(fill: colorReihen)[Vektorräume (VR)]
+    #bgBlock(fill: colorVR)[
+        #subHeading(fill: colorVR)[Vektorräume (VR)]
         $(V, plus.o, dot.o)$ ist ein über Körper $K$
         - $+: V times V -> V, (v,w) -> v + w$
         - $dot: K times V -> V, (lambda,v) -> lambda v$
@@ -102,8 +122,8 @@
         - $(U inter W) subset V$
     ]
 
-    #bgBlock(fill: colorReihen)[
-        #subHeading(fill: colorReihen)[Basis und Dim]
+    #bgBlock(fill: colorVR)[
+        #subHeading(fill: colorVR)[Basis und Dim]
         *Linear Abbildung:* $Phi: V -> V$ 
         - $Phi(0) = 0$
         - $Phi(lambda v + w) = lambda Phi(v) + Phi(w)$
@@ -154,6 +174,7 @@
         *Vektorraum-Homomorphismus:* linear Abbildung zwischen VR
     ]
 
+    // Spann und Bild, Kern
     #bgBlock(fill: colorAbbildungen)[
         #subHeading(fill: colorAbbildungen)[Spann und Bild]
         *Spann:* 
@@ -173,131 +194,242 @@
 
         *Rang*
         $op("Rang") f := dim op("Bild") f$
+
+        *Dimensionssatz:* Sei $A$ lineare Abbildung \
+        $dim(V) = dim(kern(A)) + dim(Bild(A))$ \
+        $dim(V) = dim(kern(A)) + Rang(A)$ \
+
+        $dim(V) = dim(Bild(A)) "oder" dim(kern(A)) = 0 \ <=> A "bijektiv" <=> "invertierbar"$
     ]
 
+    #bgBlock(fill: colorAbbildungen)[
+        #subHeading(fill: colorAbbildungen)[Determinate und Bilinearform]
+    ]
+
+    #bgBlock(fill: colorVR)[
+        #subHeading(fill: colorVR)[Eukldische Vektorräume]
+    ]
+
+    #bgBlock(fill: colorVR)[
+        #subHeading(fill: colorVR)[Unitair Vektorräume ]
+    ]
+
+
+    // Matrix Typem
     #bgBlock(fill: colorMatrix)[
+        #let colred(x) = text(fill: red, $#x$)
+        #let colblue(x) = text(fill: blue, $#x$)
+
         #subHeading(fill: colorMatrix)[Matrix Typen]
+        #align(center, scale($colred(m "Zeilen") colblue(n "Splate")\ A in KK^(colred(m) times colblue(n))$, 120%)) #grid(columns: (1fr, 1fr),
+        $quad mat(
+                a_11, a_12, ..., a_(1n);
+                a_21, a_22, ..., a_(2n);
+                dots.v, dots.v, dots.down, dots.v;
+                a_(m 1), a_(m 2), ..., a_(m n)
+            )
+        $,
+            
+            cetz.canvas({
+                    import cetz.draw : *
 
-        *Einheits Matrix* $I,E$
+                    rect((0, 0), (1, 1), fill: rgb("#9292926b"))
 
-        *Diagonalmatrix*
+                    set-style(mark: (end: (symbol: "straight")))
+                    line((0, -0.2), (1, -0.2), stroke: (paint: blue, thickness: 0.3mm))
+                    line((-0.2, 1), (-0.2, 0), stroke: (paint: red, thickness: 0.3mm))
 
-        *Symetrisch* $S$: \
-        $A A^T$ ist symetrisch
-        
-        *Orthogonal* $O$:
+                    content((-0.45, 0.5), $colred(bold(m))$)
+                    content((0.5, -0.35), $colblue(bold(n))$)
+                    content((0.5, 0.5), $A$)
+                })
+        )
 
-        *Unitair:*
+        #table(
+            columns: (auto, 1fr),
+            inset: 2mm,
+            fill: (x, y) => if (calc.rem(y, 2) == 0) { tableFillLow } else { tableFillHigh },
+            [*Einheits Matrix*\ $I,E$], [],
+            [*Diagonalmatrix* \ $Sigma,D$], [
+                Nur Einträger auf Hauptdiagonalen \
+                $det(D) = d_00 dot d_11 dot d_22 dot ...$
+            ],
+            [*Symetrisch*\ $S$], [
+                $S = S^T$, $S in KK^(n times n)$\
+                $A A^T$, $A^T A$ ist symetrisch \
+                $S$ immer diagonaliserbar \
+                EW immer $in RR$, EV orthogonal
+            ],
+            [*Invertierbar*], [
+                $exists A^(-1) : A A^(-1) = A^(-1) A = E$ \
 
-        *postiv-semi-definit* \
-        $forall$ Eigenwerte $>= 0$
+                *Invertierbar wenn:* \
+                $A$ bijektiv, $det(A) = 0$ \
+                $"Spalten Vekoren lin. unabhänig"$ \
+                $det(A) = 0$ \                
+
+                *Nicht Invertierbar wenn:*\
+                $exists$ EW $!= 0 => not "invertierbar"$
+                Keine Qudratische Matrix
+            ],
+            [*Orthogonal*\ $O$], [
+                $O^T = O^(-1)$ \
+                $ip(O v, O w) = ip(v, w)$
+            ],
+            [*Unitair*], [
+
+            ],
+            [*Diagonaliserbar*], [
+                $exists A = B D B^(-1)$, $D$ diagonal,
+
+                $B$: Splaten sind EV von $A$
+
+                - Selbst-Adujunkte diagonalisierbar
+                - Symetrisch Matrix
+                - $A in KK^(n times n) "AND" alg(lambda) = geo(lambda)$
+            ],
+            [*postiv-semi-definit*], [
+                $forall$ EW $>= 0$
+            ],
+        )
     ]
 
-    #colbreak()
+    #bgBlock(fill: colorMatrixVerfahren)[
 
-    #bgBlock(fill: colorMatrix)[
-
-        #subHeading(fill: colorMatrix)[Eigenwert und Eigenvektoren ]
+        #subHeading(fill: colorMatrixVerfahren)[Eigenwert und Eigenvektoren ]
 
         $A in CC^(n times n):$ $n$ Complexe Eigenwerte \
         $A in RR^(n times n)$
 
-        *Eigentwete bestimmen*
+        *1. Eigentwete bestimmen*
         
-        $A v = lambda v$
+        $A v = lambda v => det(A-E lambda) = 0$
 
-        Lösen: $0 = det mat(#mannot.markhl($x_11 - lambda_1$, color: red), x_12, ..., x_(1n);
+        $0 = det mat(#mannot.markhl($x_11 - lambda_1$, color: red), x_12, ..., x_(1n);
             x_21, #mannot.markhl($x_22 - lambda_2$, color: red), ..., x_(2n);
             dots.v, dots.v, dots.down,  dots.v;
             x_(n 1), x_(n 2), ..., #mannot.markhl($x_(n n) -lambda_n$, color: red)
         )$
 
-        Charakteristisches Polynom: $chi_(A)$
+        $--> chi_A = (lambda_0 - lambda)^(n_0) dot (lambda_1 - lambda)^(n_1) ... $
 
 
-        *Eigenvektor bestimmen*
+        $lambda_0, lambda_1, ... = $ Nst von $chi_A$
+
+
+        *2. Eigenvektor bestimmen*
         
-        Eigentwerte einsetzen: $lambda in {lambda_1, lambda_2, ... lambda}$
+        $Eig(lambda_k) = kern(A - lambda_k E)$
+        
+        $mat(#mannot.markhl($x_11 - lambda_k$, color: red), x_12, ..., x_(1n);
+            x_21, #mannot.markhl($x_22 - lambda_k$, color: red), ..., x_(2n);
+            dots.v, dots.v, dots.down,  dots.v;
+            x_(n 1), x_(n 2), ..., #mannot.markhl($x_(n n) -lambda_k$, color: red)
+        ) vec(v_1, v_2, dots.v, v_n) = vec(0, 0, dots.v, 0)$
 
 
-        *Algebrasche Vielfacheit:* \
-        $sum$ Häufikeit der Nsts von $chi_A$
 
-        *Geometrische Vielfacheit:*\
-        $dim(op("spann")(v_lambda_1, v_lambda_2 ..., v_lambda_n))$ \
+        *Algebrasche Vielfacheit:* $alg(lambda) = n_0 + n_1 + ...$ \
+        *Geometrische Vielfacheit:* $geo(lambda) = dim("Eig"_A (lambda))$ \
 
-        Anzahl der linearunabhänige $v_lambda_i$ 
+        $1 <= geo(lambda) <= alg(lambda)$
 
-        $"Geometrische" <= "Algebrasche"$
+    ]
+
+    #bgBlock(fill: colorMatrixVerfahren)[
+        #subHeading(fill: colorMatrixVerfahren)[Gram-Schmit ONB]
+
+
+
+    ]
+
+    #bgBlock(fill: colorMatrixVerfahren)[
+        #subHeading(fill: colorMatrixVerfahren)[Diagonalisierung]
+        $A = R D R^(-1)$
+
+        #grid(
+            columns: (1fr, 1fr),
+            $D = mat(
+                lambda_1, 0, 0,...;
+                0, lambda_1, 0, ...;
+                0, 0, lambda_2, ...;
+                dots.v, dots.v, dots.v, dots.down
+            )$,
+
+            $D^n = mat(
+                lambda_1^n, 0, 0,...;
+                0, lambda_1^n, 0, ...;
+                0, 0, lambda_2^n, ...;
+                dots.v, dots.v, dots.v, dots.down
+            )$
+        ) \
+
+        *Rezept Diagonalisierung*
+
+        1. *EW* bestimmen
+        2. $chi_A$ bestimmen und in $(lambda_0 - lambda)^(n_0) dot (lambda_1 - lambda)^(n_1) ...$ Form bringen \
+            $chi_A$ nicht vollstandig zerfällt (in $RR$), $=>$ NICHT diagonalisierbar
+        3. *EV* bestimmen
+        4. $R = mat( bar, bar, ..; v_(lambda 0), v_(lambda 1), ...; bar, bar, ...)$
+        5. $R^(-1)$ bestimmen 
+    ]
+
+
+    #bgBlock(fill: colorMatrixVerfahren)[
+        #subHeading(fill: colorMatrixVerfahren)[Schur-Zerlegung]
+        immer anwendbar; 
+    ]
+
+    #bgBlock(fill: colorMatrixVerfahren)[
+        #subHeading(fill: colorMatrixVerfahren)[SVD]
+
+        $A in RR^(m times n)$ zerlegbar in $A = U Sigma V^T$ \
+
+
+        $U in RR^(m times m)$ Orthogonal \
+        $Sigma in RR^(m times n)$ Diagonal \
+        $V in RR^(n times n)$ Orthogonal
+
+
+        1. $A^T A$ berechnen $A^T A in RR^(k times k), k = min(n, m)$
+
+        2. Eigenwerte bestimmen $det(A^T A - E lambda) = 0$ \
+            $lambda_0, lambda_1 ... lambda_k$ nach Größe sortieren \
+            Singulärewerte: $sigma_i = sqrt(lambda_i)$
+
+        3. Eigenvekoren von $A^T A$ bestimmen und *Normalisieren*
+            $v_(lambda 0), v_(lambda 1), ... v_(lambda k)$ 
+
+        4. $V = mat( |, |, ..., |; v_0, v_1, ..., v_n; |, |, ..., |) --> V^T$ \
+            (Evt. zu ONB ergenze mit Gram-Schmit/Kreuzprodukt)
+
+        5. $u_i = 1/sigma_i A v_(lambda i) quad quad L = mat( |, |, ..., |; u_0, u_1, ..., u_m; |, |, ..., |)$ \
+            (Evt. zu ONB ergenze mit Gram-Schmit/Kreuzprodukt)
+
+        6. $S in RR^(n times m)$ (wie $A$):
+
+            $S = mat(sigma_0, 0; 0, sigma_1; dots.v, dots.v; 0, 0) quad quad quad S = mat(sigma_0, 0, dots, 0; 0, sigma_1, ..., 0)$
+    ]
+
+    #bgBlock(fill: colorMatrix)[
+        #subHeading(fill: colorMatrix)[Matrix Normen]
+    ]
+
+    #bgBlock(fill: colorMatrix)[
+        #subHeading(fill: colorMatrix)[Rekursive Folgen]
+
+        E.g: $a_1 x_(n-1) + a_2 x_(n) = x_(n+1)$
+
+        1. Als Matrix Schreiben $F: vec(x_(n-1), x_(n)) = vec(x_n, x_(n+1))$ \
+            $F s_(n-1) = s_(n)$
+
+        2. Diagonaliseren: $F = R D R^(-1) $ \
+        3. Wiederholte Anwendung: $F^n = R D^n R^(-1)$
 
     ]
 
     #bgBlock(fill: colorMatrix)[
-        #subHeading(fill: colorMatrix)[Diagonalisierung]
-
-        $A = R D R^T$
-
-        $D$: Diagonalmatrix
+        #subHeading(fill: colorMatrix)[Differenzialgleichungen]
     ]
-
-    #bgBlock(fill: colorMatrix)[
-        #subHeading(fill: colorMatrix)[Schur-Zerlegung]
-        immer anwendbar;
-
-        $A in RR^(n times n)\/CC^(n times n)$ zerlegbar in $O^T R O$
-    
-        Orthogonal $O,O^T$, Dreiecksmatrix $R$
-
-        $R = mat(lambda_1, *, *,..., *;
-            0, lambda_2, *, ..., *;
-            0, 0, lambda_3, ..., *;
-            dots.v, dots.v, dots.v, dots.down, dots.v;
-            0, 0, 0, ..., lambda_n;
-        )$
-
-        1. Eigenwerte bestimmen $lambda_1, lambda_2, ... lambda_n$
-        2. Eigenvektor $v_lambda_1, v_lambda_2 ..., v_lambda_n$
-        3. 
-    ]
-
-    #colbreak()
-    #bgBlock(fill: colorMatrix)[
-        #subHeading(fill: colorMatrix)[SVD]
-
-        $A in RR^(n times m)$ zerlegbar in $A = L S R^T$ \
-        $L$ Orthogonal, $S$ Diagonalmatrix, $R$ Orthogonal \
-        $A^T = R S^T L^T$
-
- 
-        *1. $A A^T$ berechnen* $A A^T in RR^(n times n) $
-
-        *2. Eigenwerte von $A A^T$ bestimmen* $lambda_1, lambda_2, ... lambda_n$
-
-        *3. $S$ aufstellen* ($S$ hat gleiche Form wie $A$)
-        
-        $sigma_i = sqrt(lambda_i) =  S in RR^(n x m) =\ mat(
-            sigma_1, 0, 0, ..., 0, 0, ..., 0;
-            0, sigma_2, 0, ..., 0, 0, ..., 0;
-            0, 0, sigma_3, ..., 0, 0, ..., 0;
-            dots.v, dots.v, dots.v, dots.down, dots.v, dots.v, dots.down, dots.v;
-            0, 0, 0, ..., sigma_m, 0, ... , 0
-        )$
-
-        *4. $R$ bestimmen*
-
-        $op("Eig")(lambda_i) = op("kern")(A A^T - lambda_i) ->$
-
-        $A A^T - lambda_i = 0$ (Gaußverfahren)
-
-        $R = 1/sqrt(lambda_i)$
-
-        *5. $L$ bestimmen*
-        
-        $L = 1/sqrt(lambda_i) $
-    ]
-
-
-
-
 ]
 
diff --git a/src/cheatsheets/Schaltungstheorie.typ b/src/cheatsheets/Schaltungstheorie.typ
index a4460f5..720899e 100644
--- a/src/cheatsheets/Schaltungstheorie.typ
+++ b/src/cheatsheets/Schaltungstheorie.typ
@@ -977,13 +977,7 @@
   #bgBlock(fill: colorAllgemein)[
     #subHeading(fill: colorAllgemein)[Komplexe Zahlen]
 
-    #grid(
-      columns: (auto, auto),
-      row-gutter: 2mm,
-      column-gutter: 3mm,
-      [Euler-Darstellung], $A e^(j phi)$,
-      [Catesiche-Darstellung], $a + b j$
-    )
+    #ComplexNumbersSection(i: $j$)
 
 
   ]
diff --git a/src/lib/common_rewrite.typ b/src/lib/common_rewrite.typ
index 71d53be..a501c8a 100644
--- a/src/lib/common_rewrite.typ
+++ b/src/lib/common_rewrite.typ
@@ -53,4 +53,19 @@
     },
     table.hline(stroke: (thickness: 0.3mm)),
   )
+]
+
+#let ComplexNumbersSection(i: $i$) = [
+  $1/#i = #i^(-1) = -#i quad quad #i^2=-1 quad quad sqrt(#i) = 1/sqrt(2) + 1/sqrt(2)#i$
+
+  $z in CC = a + b #i quad quad quad z = r dot e^(phi #i)$ \
+  $z_0 + z_1 = (a_0 + a_1) + (b_0 + b_1) #i$\
+  $z_0 dot z_1 = (a_1 a_2 - b_1 b_2) + #i (a_1b_2 + a_2 b_1) = r_0 r_1 e^(#i (phi_0 + phi_1))$\
+  $z^x = r^x dot e^(phi #i dot x) quad x in RR$ \
+  $z_0/z_1 = r_0/r_1 e^(#i (phi_0 - phi_1)) quad quad quad$
+
+  $r = abs(z) quad phi = cases(
+    + arccos(a/r) space : space b >= 0,
+    - arccos(a/r) space : space b < 0,
+  )$
 ]
\ No newline at end of file

From 446be9a38f1c5e4ff9c720ffcdc5f4f0e26dc00d Mon Sep 17 00:00:00 2001
From: alexander <alexander@xstation>
Date: Sun, 1 Feb 2026 23:56:42 +0100
Subject: [PATCH 4/6] removed allgemein from LinearAlgebra

---
 src/cheatsheets/LinearAlgebra.typ | 6 ------
 1 file changed, 6 deletions(-)

diff --git a/src/cheatsheets/LinearAlgebra.typ b/src/cheatsheets/LinearAlgebra.typ
index ee46336..dc9551f 100644
--- a/src/cheatsheets/LinearAlgebra.typ
+++ b/src/cheatsheets/LinearAlgebra.typ
@@ -48,12 +48,6 @@
   align(left, block(e))
 }
 #columns(4, gutter: 2mm)[
-    #bgBlock(fill: colorAllgemein)[
-        #subHeading(fill: colorAllgemein)[Algemein]
-        
-
-    ]
-
     #bgBlock(fill: colorAllgemein)[
         #subHeading(fill: colorAllgemein)[Komplexe Zahlen]
         

From d3e4df0a3ff0b6570134787f59b39374eb07bc3a Mon Sep 17 00:00:00 2001
From: alexander <alexander@ideapad>
Date: Mon, 2 Feb 2026 07:34:12 +0100
Subject: [PATCH 5/6] Moved Math macros to seperte file

---
 src/cheatsheets/Analysis1.typ     |  6 ++----
 src/cheatsheets/LinearAlgebra.typ | 11 +----------
 src/lib/mathExpressions.typ       | 14 ++++++++++++++
 3 files changed, 17 insertions(+), 14 deletions(-)
 create mode 100644 src/lib/mathExpressions.typ

diff --git a/src/cheatsheets/Analysis1.typ b/src/cheatsheets/Analysis1.typ
index f742bb5..fc35741 100644
--- a/src/cheatsheets/Analysis1.typ
+++ b/src/cheatsheets/Analysis1.typ
@@ -1,9 +1,7 @@
-#import "../lib/common_rewrite.typ" : *
 #import "@preview/mannot:0.3.1"
 
-#show math.integral: it => math.limits(math.integral)
-#show math.sum: it => math.limits(math.sum)
-#let lim = $limits("lim")$
+#import "../lib/common_rewrite.typ" : *
+#import "../lib/mathExpressions.typ" : *
 
 #set text(7.5pt)
 
diff --git a/src/cheatsheets/LinearAlgebra.typ b/src/cheatsheets/LinearAlgebra.typ
index dc9551f..edf95a1 100644
--- a/src/cheatsheets/LinearAlgebra.typ
+++ b/src/cheatsheets/LinearAlgebra.typ
@@ -5,6 +5,7 @@
 
 #import "../lib/styles.typ" : *
 #import "../lib/common_rewrite.typ" : *
+#import "../lib/mathExpressions.typ" : *
 
 #set page(
   paper: "a4", 
@@ -32,16 +33,6 @@
 #let colorAbbildungen = color.hsl(356.92deg, 92.13%, 75.1%)
 #let colorGruppen = color.hsl(34.87deg, 92.13%, 75.1%)
 
-// Math macors
-#let kern(x) = $op("kern")(#x)$
-#let alg(x) = $op("alg")(#x)$
-#let geo(x) = $op("geo")(#x)$
-#let spann(x) = $op("spann")(#x)$
-#let Bild(x) = $op("Bild")(#x)$
-#let Rang(x) = $op("Rang")(#x)$
-#let Eig(x) = $op("Eig")(#x)$
-
-#let ip(x, y) = $lr(angle.l #x, #y angle.r)$
 
 #let SeperatorLine = line(length: 100%, stroke: (paint: black, thickness: 0.3mm))
 #let MathAlignLeft(e) = {
diff --git a/src/lib/mathExpressions.typ b/src/lib/mathExpressions.typ
new file mode 100644
index 0000000..848f623
--- /dev/null
+++ b/src/lib/mathExpressions.typ
@@ -0,0 +1,14 @@
+// Math macors
+#let kern(x) = $op("kern")(#x)$
+#let alg(x) = $op("alg")(#x)$
+#let geo(x) = $op("geo")(#x)$
+#let spann(x) = $op("spann")(#x)$
+#let Bild(x) = $op("Bild")(#x)$
+#let Rang(x) = $op("Rang")(#x)$
+#let Eig(x) = $op("Eig")(#x)$
+
+#let ip(x, y) = $lr(angle.l #x, #y angle.r)$
+
+#show math.integral: it => math.limits(math.integral)
+#show math.sum: it => math.limits(math.sum)
+#let lim = $limits("lim")$
\ No newline at end of file

From 68b599eea48b9010c6f67524f34b96c08e02939c Mon Sep 17 00:00:00 2001
From: alexander <alexander@ideapad>
Date: Mon, 2 Feb 2026 12:40:23 +0100
Subject: [PATCH 6/6] Fixed complex

---
 src/cheatsheets/LinearAlgebra.typ | 18 +++++++++++++++++-
 src/lib/mathExpressions.typ       |  5 ++---
 2 files changed, 19 insertions(+), 4 deletions(-)

diff --git a/src/cheatsheets/LinearAlgebra.typ b/src/cheatsheets/LinearAlgebra.typ
index edf95a1..c98942c 100644
--- a/src/cheatsheets/LinearAlgebra.typ
+++ b/src/cheatsheets/LinearAlgebra.typ
@@ -262,7 +262,7 @@
                 $ip(O v, O w) = ip(v, w)$
             ],
             [*Unitair*], [
-
+                $V^* )$
             ],
             [*Diagonaliserbar*], [
                 $exists A = B D B^(-1)$, $D$ diagonal,
@@ -398,6 +398,22 @@
 
     #bgBlock(fill: colorMatrix)[
         #subHeading(fill: colorMatrix)[Matrix Normen]
+
+        $|| dot ||_M$ Matrix Norm, $|| dot ||_V$ Vektornorm
+
+        Generisch Vektor Norm: $|| v ||_p = root(p, sum_(k=1)^n (x_k)^p)$
+
+        - submultiplikativ: $||A B||_"M" <= ||A||||B||$
+        - verträglich mit einer Vektornorm: $||A v||_"V" <= ||A||_"M" ||v||_"V"$
+
+        *Frobenius-Norm* $||A||_"M" = sqrt(sum_(i=1)^m sum_(j=1)^n a_(m n)^2)$
+
+        *Induzierte Norm* $||A||_"M" = sup_(v in V without {0}) (||A v||_V)/(||v||_V)$\
+        $ = sup_(||v|| = 1) (||A v||_V)/(||v||_V)$
+            - submultiplikativ
+            - verträglich mit einer Vektornorm $||dot||_V$
+
+        *maximale Spaltensumme* $||A||_r = max_(1<= i <= n) sum_(j=1)^n |a_(j)|$
     ]
 
     #bgBlock(fill: colorMatrix)[
diff --git a/src/lib/mathExpressions.typ b/src/lib/mathExpressions.typ
index 848f623..5930fd7 100644
--- a/src/lib/mathExpressions.typ
+++ b/src/lib/mathExpressions.typ
@@ -6,9 +6,8 @@
 #let Bild(x) = $op("Bild")(#x)$
 #let Rang(x) = $op("Rang")(#x)$
 #let Eig(x) = $op("Eig")(#x)$
-
+#let lim = $limits("lim")$
 #let ip(x, y) = $lr(angle.l #x, #y angle.r)$
 
 #show math.integral: it => math.limits(math.integral)
-#show math.sum: it => math.limits(math.sum)
-#let lim = $limits("lim")$
\ No newline at end of file
+#show math.sum: it => math.limits(math.sum)
\ No newline at end of file