added eintor liste
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This commit is contained in:
levi
@@ -1,9 +1,7 @@
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#import "../lib/common_rewrite.typ" : *
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#import "@preview/mannot:0.3.1"
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#show math.integral: it => math.limits(math.integral)
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#show math.sum: it => math.limits(math.sum)
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#let lim = $limits("lim")$
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#import "../lib/common_rewrite.typ" : *
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#import "../lib/mathExpressions.typ" : *
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#set text(7.5pt)
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@@ -106,9 +104,8 @@
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// Complex Zahlen
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#bgBlock(fill: colorAllgemein)[
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#subHeading(fill: colorAllgemein)[Complexe Zahlen]
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$z = r dot e^(phi i) = r (cos(phi) + i sin(phi))$
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$z^n = r^n dot e^(phi i dot n) = r^n (cos(n phi) + i sin(n phi))$
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#ComplexNumbersSection()
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#grid(
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columns: (1fr, 1fr),
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@@ -1,8 +1,10 @@
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#import "../lib/common_rewrite.typ" : *
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#import "@preview/mannot:0.3.1"
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#import "@preview/cetz:0.4.2"
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#import "@preview/zap:0.5.0"
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#import "../lib/common_rewrite.typ" : *
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#import "../lib/truthtable.typ" : *
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#import "../lib/fetModel.typ" : *
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#show math.integral: it => math.limits(math.integral)
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#show math.sum: it => math.limits(math.sum)
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@@ -565,10 +567,7 @@
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(cycle_start,signal_hight), (cycle_time*(t_setup + 2), signal_hight),
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(cycle_time*(t_setup + 2) + switch_offset, 0), (cycle_end + switch_offset, 0), stroke: signal_storke
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)
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})
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]
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@@ -1,7 +1,11 @@
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#import "@preview/biceps:0.0.1" : *
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#import "@preview/mannot:0.3.1"
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#import "@preview/fletcher:0.5.8"
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#import "@preview/cetz:0.4.2"
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#import "../lib/styles.typ" : *
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#import "../lib/common_rewrite.typ" : *
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#import "../lib/mathExpressions.typ" : *
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#set page(
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paper: "a4",
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@@ -23,8 +27,9 @@
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))
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#let colorAllgemein = color.hsl(105.13deg, 92.13%, 75.1%)
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#let colorMatrixVerfahren = color.hsl(330.19deg, 100%, 68.43%)
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#let colorMatrix = color.hsl(202.05deg, 92.13%, 75.1%)
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#let colorReihen = color.hsl(280deg, 92.13%, 75.1%)
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#let colorVR = color.hsl(280deg, 92.13%, 75.1%)
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#let colorAbbildungen = color.hsl(356.92deg, 92.13%, 75.1%)
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#let colorGruppen = color.hsl(34.87deg, 92.13%, 75.1%)
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@@ -35,9 +40,11 @@
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}
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#columns(4, gutter: 2mm)[
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#bgBlock(fill: colorAllgemein)[
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#subHeading(fill: colorAllgemein)[Notation]
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#subHeading(fill: colorAllgemein)[Komplexe Zahlen]
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#ComplexNumbersSection()
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#sinTable
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]
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#bgBlock(fill: colorGruppen)[
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@@ -75,8 +82,6 @@
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- $(R, dot)$ Halbgruppe
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- $(a + b) dot c = (a dot c) + (a dot b) space$ (Distributiv Gesetz)
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#colbreak()
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*Körper:* Menge $K$ mit:
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- $(K, +), (K without {0} , dot)$ kommutativ Gruppe \
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($0$ ist Neutrales Element von $+$)
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@@ -84,8 +89,8 @@
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_Beweiß durch Überprüfung der Eigneschaften_
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]
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#bgBlock(fill: colorReihen)[
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#subHeading(fill: colorReihen)[Vektorräume (VR)]
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#bgBlock(fill: colorVR)[
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#subHeading(fill: colorVR)[Vektorräume (VR)]
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$(V, plus.o, dot.o)$ ist ein über Körper $K$
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- $+: V times V -> V, (v,w) -> v + w$
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- $dot: K times V -> V, (lambda,v) -> lambda v$
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@@ -102,8 +107,8 @@
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- $(U inter W) subset V$
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]
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#bgBlock(fill: colorReihen)[
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#subHeading(fill: colorReihen)[Basis und Dim]
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#bgBlock(fill: colorVR)[
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#subHeading(fill: colorVR)[Basis und Dim]
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*Linear Abbildung:* $Phi: V -> V$
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- $Phi(0) = 0$
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- $Phi(lambda v + w) = lambda Phi(v) + Phi(w)$
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@@ -154,6 +159,7 @@
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*Vektorraum-Homomorphismus:* linear Abbildung zwischen VR
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]
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// Spann und Bild, Kern
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#bgBlock(fill: colorAbbildungen)[
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#subHeading(fill: colorAbbildungen)[Spann und Bild]
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*Spann:*
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@@ -173,131 +179,258 @@
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*Rang*
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$op("Rang") f := dim op("Bild") f$
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*Dimensionssatz:* Sei $A$ lineare Abbildung \
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$dim(V) = dim(kern(A)) + dim(Bild(A))$ \
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$dim(V) = dim(kern(A)) + Rang(A)$ \
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$dim(V) = dim(Bild(A)) "oder" dim(kern(A)) = 0 \ <=> A "bijektiv" <=> "invertierbar"$
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]
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#bgBlock(fill: colorAbbildungen)[
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#subHeading(fill: colorAbbildungen)[Determinate und Bilinearform]
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]
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#bgBlock(fill: colorVR)[
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#subHeading(fill: colorVR)[Eukldische Vektorräume]
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]
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#bgBlock(fill: colorVR)[
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#subHeading(fill: colorVR)[Unitair Vektorräume ]
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]
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// Matrix Typem
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#bgBlock(fill: colorMatrix)[
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#let colred(x) = text(fill: red, $#x$)
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#let colblue(x) = text(fill: blue, $#x$)
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#subHeading(fill: colorMatrix)[Matrix Typen]
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#align(center, scale($colred(m "Zeilen") colblue(n "Splate")\ A in KK^(colred(m) times colblue(n))$, 120%)) #grid(columns: (1fr, 1fr),
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$quad mat(
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a_11, a_12, ..., a_(1n);
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a_21, a_22, ..., a_(2n);
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dots.v, dots.v, dots.down, dots.v;
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a_(m 1), a_(m 2), ..., a_(m n)
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)
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$,
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*Einheits Matrix* $I,E$
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cetz.canvas({
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import cetz.draw : *
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*Diagonalmatrix*
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rect((0, 0), (1, 1), fill: rgb("#9292926b"))
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*Symetrisch* $S$: \
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$A A^T$ ist symetrisch
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set-style(mark: (end: (symbol: "straight")))
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line((0, -0.2), (1, -0.2), stroke: (paint: blue, thickness: 0.3mm))
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line((-0.2, 1), (-0.2, 0), stroke: (paint: red, thickness: 0.3mm))
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*Orthogonal* $O$:
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content((-0.45, 0.5), $colred(bold(m))$)
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content((0.5, -0.35), $colblue(bold(n))$)
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content((0.5, 0.5), $A$)
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})
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)
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*Unitair:*
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#table(
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columns: (auto, 1fr),
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inset: 2mm,
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fill: (x, y) => if (calc.rem(y, 2) == 0) { tableFillLow } else { tableFillHigh },
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[*Einheits Matrix*\ $I,E$], [],
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[*Diagonalmatrix* \ $Sigma,D$], [
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Nur Einträger auf Hauptdiagonalen \
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$det(D) = d_00 dot d_11 dot d_22 dot ...$
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],
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[*Symetrisch*\ $S$], [
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$S = S^T$, $S in KK^(n times n)$\
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$A A^T$, $A^T A$ ist symetrisch \
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$S$ immer diagonaliserbar \
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EW immer $in RR$, EV orthogonal
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],
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[*Invertierbar*], [
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$exists A^(-1) : A A^(-1) = A^(-1) A = E$ \
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*postiv-semi-definit* \
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$forall$ Eigenwerte $>= 0$
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*Invertierbar wenn:* \
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$A$ bijektiv, $det(A) = 0$ \
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$"Spalten Vekoren lin. unabhänig"$ \
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$det(A) = 0$ \
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*Nicht Invertierbar wenn:*\
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$exists$ EW $!= 0 => not "invertierbar"$
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Keine Qudratische Matrix
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],
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[*Orthogonal*\ $O$], [
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$O^T = O^(-1)$ \
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$ip(O v, O w) = ip(v, w)$
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],
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[*Unitair*], [
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$V^* )$
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],
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[*Diagonaliserbar*], [
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$exists A = B D B^(-1)$, $D$ diagonal,
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$B$: Splaten sind EV von $A$
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- Selbst-Adujunkte diagonalisierbar
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- Symetrisch Matrix
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- $A in KK^(n times n) "AND" alg(lambda) = geo(lambda)$
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],
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[*postiv-semi-definit*], [
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$forall$ EW $>= 0$
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],
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)
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]
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#colbreak()
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#bgBlock(fill: colorMatrixVerfahren)[
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#bgBlock(fill: colorMatrix)[
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#subHeading(fill: colorMatrix)[Eigenwert und Eigenvektoren ]
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#subHeading(fill: colorMatrixVerfahren)[Eigenwert und Eigenvektoren ]
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$A in CC^(n times n):$ $n$ Complexe Eigenwerte \
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$A in RR^(n times n)$
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*Eigentwete bestimmen*
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*1. Eigentwete bestimmen*
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$A v = lambda v$
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$A v = lambda v => det(A-E lambda) = 0$
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Lösen: $0 = det mat(#mannot.markhl($x_11 - lambda_1$, color: red), x_12, ..., x_(1n);
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$0 = det mat(#mannot.markhl($x_11 - lambda_1$, color: red), x_12, ..., x_(1n);
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x_21, #mannot.markhl($x_22 - lambda_2$, color: red), ..., x_(2n);
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dots.v, dots.v, dots.down, dots.v;
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x_(n 1), x_(n 2), ..., #mannot.markhl($x_(n n) -lambda_n$, color: red)
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)$
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Charakteristisches Polynom: $chi_(A)$
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$--> chi_A = (lambda_0 - lambda)^(n_0) dot (lambda_1 - lambda)^(n_1) ... $
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*Eigenvektor bestimmen*
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Eigentwerte einsetzen: $lambda in {lambda_1, lambda_2, ... lambda}$
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$lambda_0, lambda_1, ... = $ Nst von $chi_A$
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*Algebrasche Vielfacheit:* \
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$sum$ Häufikeit der Nsts von $chi_A$
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*2. Eigenvektor bestimmen*
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*Geometrische Vielfacheit:*\
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$dim(op("spann")(v_lambda_1, v_lambda_2 ..., v_lambda_n))$ \
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$Eig(lambda_k) = kern(A - lambda_k E)$
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Anzahl der linearunabhänige $v_lambda_i$
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$mat(#mannot.markhl($x_11 - lambda_k$, color: red), x_12, ..., x_(1n);
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x_21, #mannot.markhl($x_22 - lambda_k$, color: red), ..., x_(2n);
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dots.v, dots.v, dots.down, dots.v;
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x_(n 1), x_(n 2), ..., #mannot.markhl($x_(n n) -lambda_k$, color: red)
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) vec(v_1, v_2, dots.v, v_n) = vec(0, 0, dots.v, 0)$
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$"Geometrische" <= "Algebrasche"$
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*Algebrasche Vielfacheit:* $alg(lambda) = n_0 + n_1 + ...$ \
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*Geometrische Vielfacheit:* $geo(lambda) = dim("Eig"_A (lambda))$ \
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$1 <= geo(lambda) <= alg(lambda)$
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]
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#bgBlock(fill: colorMatrix)[
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#subHeading(fill: colorMatrix)[Diagonalisierung]
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#bgBlock(fill: colorMatrixVerfahren)[
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#subHeading(fill: colorMatrixVerfahren)[Gram-Schmit ONB]
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$A = R D R^T$
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$D$: Diagonalmatrix
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]
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#bgBlock(fill: colorMatrix)[
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#subHeading(fill: colorMatrix)[Schur-Zerlegung]
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#bgBlock(fill: colorMatrixVerfahren)[
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#subHeading(fill: colorMatrixVerfahren)[Diagonalisierung]
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$A = R D R^(-1)$
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#grid(
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columns: (1fr, 1fr),
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$D = mat(
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lambda_1, 0, 0,...;
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0, lambda_1, 0, ...;
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0, 0, lambda_2, ...;
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dots.v, dots.v, dots.v, dots.down
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)$,
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$D^n = mat(
|
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lambda_1^n, 0, 0,...;
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0, lambda_1^n, 0, ...;
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0, 0, lambda_2^n, ...;
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dots.v, dots.v, dots.v, dots.down
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)$
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) \
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*Rezept Diagonalisierung*
|
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|
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1. *EW* bestimmen
|
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2. $chi_A$ bestimmen und in $(lambda_0 - lambda)^(n_0) dot (lambda_1 - lambda)^(n_1) ...$ Form bringen \
|
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$chi_A$ nicht vollstandig zerfällt (in $RR$), $=>$ NICHT diagonalisierbar
|
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3. *EV* bestimmen
|
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4. $R = mat( bar, bar, ..; v_(lambda 0), v_(lambda 1), ...; bar, bar, ...)$
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5. $R^(-1)$ bestimmen
|
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]
|
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|
||||
|
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#bgBlock(fill: colorMatrixVerfahren)[
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#subHeading(fill: colorMatrixVerfahren)[Schur-Zerlegung]
|
||||
immer anwendbar;
|
||||
|
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$A in RR^(n times n)\/CC^(n times n)$ zerlegbar in $O^T R O$
|
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|
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Orthogonal $O,O^T$, Dreiecksmatrix $R$
|
||||
|
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$R = mat(lambda_1, *, *,..., *;
|
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0, lambda_2, *, ..., *;
|
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0, 0, lambda_3, ..., *;
|
||||
dots.v, dots.v, dots.v, dots.down, dots.v;
|
||||
0, 0, 0, ..., lambda_n;
|
||||
)$
|
||||
|
||||
1. Eigenwerte bestimmen $lambda_1, lambda_2, ... lambda_n$
|
||||
2. Eigenvektor $v_lambda_1, v_lambda_2 ..., v_lambda_n$
|
||||
3.
|
||||
]
|
||||
|
||||
#colbreak()
|
||||
#bgBlock(fill: 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)[SVD]
|
||||
#subHeading(fill: colorMatrix)[Matrix Normen]
|
||||
|
||||
$A in RR^(n times m)$ zerlegbar in $A = L S R^T$ \
|
||||
$L$ Orthogonal, $S$ Diagonalmatrix, $R$ Orthogonal \
|
||||
$A^T = R S^T L^T$
|
||||
$|| dot ||_M$ Matrix Norm, $|| dot ||_V$ Vektornorm
|
||||
|
||||
Generisch Vektor Norm: $|| v ||_p = root(p, sum_(k=1)^n (x_k)^p)$
|
||||
|
||||
*1. $A A^T$ berechnen* $A A^T in RR^(n times n) $
|
||||
- submultiplikativ: $||A B||_"M" <= ||A||||B||$
|
||||
- verträglich mit einer Vektornorm: $||A v||_"V" <= ||A||_"M" ||v||_"V"$
|
||||
|
||||
*2. Eigenwerte von $A A^T$ bestimmen* $lambda_1, lambda_2, ... lambda_n$
|
||||
*Frobenius-Norm* $||A||_"M" = sqrt(sum_(i=1)^m sum_(j=1)^n a_(m n)^2)$
|
||||
|
||||
*3. $S$ aufstellen* ($S$ hat gleiche Form wie $A$)
|
||||
*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 $||dot||_V$
|
||||
|
||||
$sigma_i = sqrt(lambda_i) = S in RR^(n x m) =\ mat(
|
||||
sigma_1, 0, 0, ..., 0, 0, ..., 0;
|
||||
0, sigma_2, 0, ..., 0, 0, ..., 0;
|
||||
0, 0, sigma_3, ..., 0, 0, ..., 0;
|
||||
dots.v, dots.v, dots.v, dots.down, dots.v, dots.v, dots.down, dots.v;
|
||||
0, 0, 0, ..., sigma_m, 0, ... , 0
|
||||
)$
|
||||
|
||||
*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) $
|
||||
*maximale Spaltensumme* $||A||_r = max_(1<= i <= n) sum_(j=1)^n |a_(j)|$
|
||||
]
|
||||
|
||||
#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)[Differenzialgleichungen]
|
||||
]
|
||||
]
|
||||
|
||||
|
||||
@@ -1,10 +1,13 @@
|
||||
#import "../lib/common_rewrite.typ" : *
|
||||
#import "@preview/mannot:0.3.1"
|
||||
#import "@preview/zap:0.5.0"
|
||||
#import "@preview/cetz:0.4.2" :*
|
||||
#import "@preview/cetz-plot:0.1.3"
|
||||
#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)
|
||||
@@ -109,7 +112,70 @@
|
||||
#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)))
|
||||
}),
|
||||
)
|
||||
),
|
||||
|
||||
|
||||
|
||||
$R_"ges" = R_1+R_2$,
|
||||
$R_"ges" = R_1 parallel R_2$,
|
||||
$G_"ges" = G_1 parallel G_2$,
|
||||
$G_"ges" = G_1 + G_2$,
|
||||
$L_"ges" = L_1 + L_2$,
|
||||
$L_"ges" = L_1 parallel L_2$,
|
||||
$C_"ges" = C_1 parallel C_2$,
|
||||
$C_"ges" = C_1 + C_2$,
|
||||
|
||||
$U_"ges" = U_1 + U_2$,
|
||||
$U_"ges" = U_1 = U_2$,
|
||||
$I_"ges" = I_1 = I_2$,
|
||||
$I_"ges" = I_1 + I_2$,
|
||||
[In $U$-Richtung Addieren],
|
||||
[In $I$-Richtung Addieren],
|
||||
)
|
||||
]
|
||||
|
||||
// Quell Wandlung
|
||||
@@ -134,7 +200,107 @@
|
||||
)
|
||||
]
|
||||
|
||||
#colbreak()
|
||||
// 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)[
|
||||
@@ -192,14 +358,14 @@
|
||||
|
||||
$R_i=1/G_i$
|
||||
|
||||
$r(i) = R_i i + U_0$
|
||||
$u = r(i) = R_i i + U_0$
|
||||
],
|
||||
[
|
||||
$I_0 = U_0 G_i$
|
||||
|
||||
$G_i=1/R_i$
|
||||
|
||||
$g(u) = G_i u + I_0$
|
||||
$i = g(u) = G_i u + I_0$
|
||||
]
|
||||
);
|
||||
]
|
||||
@@ -1071,12 +1237,16 @@
|
||||
$E_L = Phi^2/2L = (L i^2)/2$
|
||||
],
|
||||
[
|
||||
$C$: Admetanz $hat(=) G$
|
||||
$C$: Admittanz $hat(=) G$
|
||||
], [],
|
||||
[
|
||||
$L$: Impedanz $hat(=) R$
|
||||
]
|
||||
)
|
||||
|
||||
Admittanz: $Y = I/U$\
|
||||
Impedanz: $Z = U/I$
|
||||
|
||||
]
|
||||
|
||||
// Reaktive Dual Wandlung
|
||||
@@ -1121,30 +1291,33 @@
|
||||
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*]
|
||||
|
||||
$underline(S) = underline(U) dot underline(I)^*$\
|
||||
$P = 1/2 U dot I^*$\
|
||||
|
||||
#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 = "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
|
||||
#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: $j$)
|
||||
|
||||
|
||||
]
|
||||
@@ -1168,6 +1341,8 @@
|
||||
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],
|
||||
[$forall (u,i) in cal(F): u dot i >= 0$],
|
||||
|
||||
@@ -13,7 +13,7 @@
|
||||
|
||||
ctx.zap.style.insert("einTor", (
|
||||
scale: auto,
|
||||
fill: auto,
|
||||
fill: none,
|
||||
height: 3mm,
|
||||
width: 6mm,
|
||||
))
|
||||
|
||||
@@ -27,6 +27,9 @@
|
||||
align(left, block(e))
|
||||
}
|
||||
|
||||
#let tableFillHigh = white
|
||||
#let tableFillLow = color.lighten(gray, 50%)
|
||||
|
||||
#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 +50,22 @@
|
||||
for (col) in data.keys() {
|
||||
(eval(data.at(col).at(i), mode: "math"),)
|
||||
}
|
||||
}
|
||||
},
|
||||
table.hline(stroke: (thickness: 0.3mm)),
|
||||
)
|
||||
]
|
||||
|
||||
#let ComplexNumbersSection(i: $i$) = [
|
||||
$1/#i = #i^(-1) = -#i quad quad #i^2=-1 quad quad sqrt(#i) = 1/sqrt(2) + 1/sqrt(2)#i$
|
||||
|
||||
$z in CC = a + b #i quad quad quad z = r dot e^(phi #i)$ \
|
||||
$z_0 + z_1 = (a_0 + a_1) + (b_0 + b_1) #i$\
|
||||
$z_0 dot z_1 = (a_1 a_2 - b_1 b_2) + #i (a_1b_2 + a_2 b_1) = r_0 r_1 e^(#i (phi_0 + phi_1))$\
|
||||
$z^x = r^x dot e^(phi #i dot x) quad x in RR$ \
|
||||
$z_0/z_1 = r_0/r_1 e^(#i (phi_0 - phi_1)) quad quad quad$
|
||||
|
||||
$r = abs(z) quad phi = cases(
|
||||
+ arccos(a/r) space : space b >= 0,
|
||||
- arccos(a/r) space : space b < 0,
|
||||
)$
|
||||
]
|
||||
290
src/lib/fetModel.typ
Normal file
290
src/lib/fetModel.typ
Normal file
@@ -0,0 +1,290 @@
|
||||
#import "@preview/zap:0.5.0"
|
||||
#import "@preview/cetz-plot:0.1.3"
|
||||
|
||||
|
||||
#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)
|
||||
})
|
||||
|
||||
#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])
|
||||
})
|
||||
}
|
||||
13
src/lib/mathExpressions.typ
Normal file
13
src/lib/mathExpressions.typ
Normal file
@@ -0,0 +1,13 @@
|
||||
// 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 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)
|
||||
Reference in New Issue
Block a user