From 3b51fc7c66574ca4b4e4b1f6aa738290ae5e1040 Mon Sep 17 00:00:00 2001 From: DotNaos Date: Sun, 29 Jun 2025 22:00:47 +0200 Subject: [PATCH] Saving --- .DS_Store | Bin 0 -> 6148 bytes code/anim.py | 128 +++++++++ code/shadertoy.glsl | 18 ++ formulas/Formelsammlung.md | 228 +++++++++++++++ formulas/Formulas.pur | Bin 0 -> 389341 bytes formulas/Schemas.md | 565 +++++++++++++++++++++++++++++++++++++ notes/Komplexe Zahlen.md | 54 ++++ notes/Matrizen.md | 52 ++++ notes/Prompts.md | 6 + notes/Vektorfelder.md | 107 +++++++ notes/Zusammenfassung.md | 27 ++ pyproject.toml | 11 + uv.lock | 392 +++++++++++++++++++++++++ 13 files changed, 1588 insertions(+) create mode 100644 .DS_Store create mode 100644 code/anim.py create mode 100644 code/shadertoy.glsl create mode 100644 formulas/Formelsammlung.md create mode 100644 formulas/Formulas.pur create mode 100644 formulas/Schemas.md create mode 100644 notes/Komplexe Zahlen.md create mode 100644 notes/Matrizen.md create mode 100644 notes/Prompts.md create mode 100644 notes/Vektorfelder.md create mode 100644 notes/Zusammenfassung.md create mode 100644 pyproject.toml create mode 100644 uv.lock diff --git a/.DS_Store b/.DS_Store new file mode 100644 index 0000000000000000000000000000000000000000..9b540ab5bc49c72a593132df35ddc2395cc06bd9 GIT binary patch literal 6148 zcmeHK%}N6?5T3MEQ;N`oLXQEj1#7Ef@v_wV0 zk}O#4!IOxcf#jRaPqOpD<_7>oXWBmir~-h*Mkq*GA!J_ZD!5=mVdj`Z3<0F!EFGj^ZetG#VeGP%Le1Zb>OC^3H$KGk+3HrfDY_-_Yn% z>p0ByAiRnOvtD)QOed2dN(N(-5cLNba(f*meLd^wY0^(ku5TT(TrT&jwfVf+KB}u$ z`(#m9^LDdQSBI_R#iA^C_x4XOde8AN(XWnmC@v!f1@)y%01jv$sTrsB cE7CE}bC_AAS;%hE5&0ru2;q(y_yq<&0a5o&L;wH) literal 0 HcmV?d00001 diff --git a/code/anim.py b/code/anim.py new file mode 100644 index 0000000..b1a9ec4 --- /dev/null +++ b/code/anim.py @@ -0,0 +1,128 @@ +import numpy as np +import plotly.graph_objects as go + +""" +Interactive 3D Plotly animation of the scalar line integral + ∮_C f(x, y) ds with f(x, y) = x · y +along the unit circle C, plotted on the surface z = f(x,y). +No files are written; the animation is displayed with interactive controls. +""" + +# Parameters +t_frame = np.linspace(0.0, 2 * np.pi, 120) + + +# Scalar field definition +def scalar_field(x, y): + return x * y + + +# Surface mesh for scalar field +lin = np.linspace(-1.2, 1.2, 50) +X, Y = np.meshgrid(lin, lin) +Z = scalar_field(X, Y) + +# Compute running integral along circle +# Midpoint rule +mid_ts = (t_frame[1:] + t_frame[:-1]) / 2 +f_mid = scalar_field(np.cos(mid_ts), np.sin(mid_ts)) +dt = t_frame[1:] - t_frame[:-1] +running = np.concatenate([[0], np.cumsum(f_mid * dt)]) + +# Prepare static surface trace +surface = go.Surface( + x=X, y=Y, z=Z, colorscale="RdBu", cmin=-1, cmax=1, opacity=0.6, showscale=False +) + +# Full path on surface +curve_x = np.cos(t_frame) +curve_y = np.sin(t_frame) +curve_z = scalar_field(curve_x, curve_y) +path = go.Scatter3d( + x=curve_x, y=curve_y, z=curve_z, mode="lines", line=dict(color="black", width=2) +) + +# Initial marker at t=0 +t0 = t_frame[0] +marker = go.Scatter3d( + x=[np.cos(t0)], + y=[np.sin(t0)], + z=[scalar_field(np.cos(t0), np.sin(t0))], + mode="markers", + marker=dict(size=5, color="black"), +) + +# Build frames for animation +frames = [] +for i, t in enumerate(t_frame): + x0, y0 = np.cos(t), np.sin(t) + z0 = scalar_field(x0, y0) + frame = go.Frame( + data=[marker.update(x=[x0], y=[y0], z=[z0])], + layout=go.Layout(title_text=f"Running integral: {running[i]:.3f}"), + name=f"frame{i}", + ) + frames.append(frame) + +# Layout with sliders and buttons +graph = go.Figure( + data=[surface, path, marker], + layout=go.Layout( + title="Running integral: 0.000", + scene=dict( + xaxis=dict(range=[-1.3, 1.3]), + yaxis=dict(range=[-1.3, 1.3]), + zaxis=dict(range=[Z.min(), Z.max()]), + ), + updatemenus=[ + dict( + type="buttons", + showactive=False, + buttons=[ + dict( + label="Play", + method="animate", + args=[ + None, + dict( + frame=dict(duration=50, redraw=True), + fromcurrent=True, + transition=dict(duration=0), + ), + ], + ) + ], + ) + ], + ), + frames=frames, +) + +# Add slider +sliders = [ + dict( + steps=[ + dict( + method="animate", + args=[ + [f.name], + dict( + mode="immediate", + frame=dict(duration=50, redraw=True), + transition=dict(duration=0), + ), + ], + label=str(i), + ) + for i, f in enumerate(frames) + ], + transition=dict(duration=0), + x=0, + y=0, + currentvalue=dict(prefix="Frame: "), + len=1.0, + ) +] +graph.update_layout(sliders=sliders) + +graph.show() diff --git a/code/shadertoy.glsl b/code/shadertoy.glsl new file mode 100644 index 0000000..8db891f --- /dev/null +++ b/code/shadertoy.glsl @@ -0,0 +1,18 @@ +/* Pixel unit conversion function */ +vec2 pos(in float x, in float y) { return st + vec2(x * rx, y * rx); } +vec2 pos(in float x) { return pos(x, x); } +vec2 pos(in vec2 p) { return pos(p.x, p.y); } +float size(in float x) { return x * rx; } +vec2 size(in float x, in float y) { return vec2(x * rx, y * rx); } + +void mainImage( out vec4 fragColor, in vec2 fragCoord ) +{ + // Normalized pixel coordinates (from 0 to 1) + vec2 uv = fragCoord/iResolution.xy; + + // Time varying pixel color + vec3 col = 0.5 + 0.5*cos(iTime+uv.xyx+vec3(0,2,4)); + + // Output to screen + fragColor = vec4(col,1.0); +} diff --git a/formulas/Formelsammlung.md b/formulas/Formelsammlung.md new file mode 100644 index 0000000..dde8aa8 --- /dev/null +++ b/formulas/Formelsammlung.md @@ -0,0 +1,228 @@ + +## Woche 1 +- Ich kenne die Begriffe Integral, Stammfunktion, lineare Substitution und deren wichtigste Eigenschaften. +- Ich kenne die Begriffe komplexe Zahl, Realteil, Imaginärteil, Betrag und komplex konjugiert und deren wichtigste Eigenschaften. +- Ich kenne die elementaren Rechenregeln für die komplexe Konjugation. +- Ich kann die Methode der linearen Substitution anwenden, um bestimmte und unbestimmte Integrale von linear substituierten Elementarfunktionen zu berechnen. +- Ich kann den Flächeninhalt zwischen einer Funktion und den Koordinatenachsen bestimmen. +- Ich kann den Flächeninhalt zwischen zwei sich schneidenden Funktionen bestimmen. +- Ich kann das Volumen von Rotationskörpern bestimmen. +- Ich kann für jede komplexe Zahl ihren Realteil, Imaginärteil, Betrag und ihre komplex Konjugierte bestimmen. +- Ich kann Summe, Differenz, Produkt und Quotient von komplexen Zahlen berechnen. +- Ich kann einfache Brüche und Potenzen von komplexen Zahlen durch Anwenden der Rechenregeln vereinfachen. + + +## Woche 2 +- Ich kenne die Begriffe Integral, Stammfunktion, Substitution, Trapezformel und deren wichtigste Eigenschaften. +- Ich kenne die Begriffe Gaußsche Zahlenebene, arithmetische und trigonometrische Form einer komplexen Zahl und Arg-Funktion und deren Eigenschaften. +- Ich kann die Methode der Substitution anwenden, um bestimmte und unbestimmte Integrale zu berechnen. +- Ich kann bestimmte Integrale näherungsweise auf eine vorgegebene Anzahl Dezimalstellen mit Python/Numpy berechnen. + +- Ich kann komplexe Zahlen in der Gaußschen Zahlenebene darstellen. +- Ich kann komplexe Zahlen von der arithmetischen in die trigonometrische Form und umgekehrt umwandeln. +- Ich kann einfache Brüche und Potenzen von komplexen Zahlen durch Anwenden der Rechenregeln vereinfachen. +- Ich kann quadratische Gleichungen mit reellen Koeffizienten lösen. + +## Woche 3 + +- Ich kenne die Begriffe Integral, Stammfunktion, partielle Integration und deren wichtigste Eigenschaften. +- Ich kenne die Begriffe Exponentialform einer komplexen Zahl, Eulersche Formel, Potenzgleichungen und deren Eigenschaften. +- Ich kann die partielle Integration anwenden, um bestimmte und unbestimmte Integrale zu berechnen. +- Ich kann unbestimmte Integrale mit Python/Sympy berechnen. +- Ich kann komplexe Zahlen in der arithmetischen, trigonometrischen und Exponentialform darstellen und von einer in die andere Form umwandeln. +- Ich kann die Grundrechenarten für die komplexen Zahlen anwenden. +- Ich kann komplexe Zahlen sowohl potenzieren als auch aus ihnen die Wurzel ziehen. + +## Woche 4 +- Ich kenne den Begriff uneigentliches Integral und dessen wichtigste Eigenschaften. +- Ich kenne die Begriffe Matrix, symmetrische/schiefsymmetrische Matrix, Einheitsmatrix, inverse Matrix, Transposition und deren wichtigste Eigenschaften. + +- Ich kann die Existenz eines uneigentlichen Integrals beurteilen und gegebenenfalls seinen Wert berechnen. +- Ich kann Matrizen addieren, subtrahieren und mit einem Skalar bzw. mit einer anderen Matrix multiplizieren und bestimmen, ob diese Operationen für gegebene Matrizen durchführbar sind oder nicht. + +## Woche 5 +- Ich kenne die Begriffe Bogenlänge, Mantelfläche von Rotationskörpern, Massenmittelpunkt, Schwerpunkt, Flächenschwerpunkt und deren wichtigste Eigenschaften. +- Ich kenne die Begriffe Matrix, inverse Matrix, Drehmatrix, lineare Abbildung und deren wichtigste Eigenschaften. +- Ich kenne die 2x2 Standardmatrizen und die dadurch beschriebenen linearen Abbildungen in 2D. +- Ich kann mit Hilfe der Integration die Bogenlänge einer ebenen Kurve, die Mantelfläche, den Flächenschwerpunkt und das Volumen von Rotationskörpern berechnen. +- Ich kann lineare Gleichungssysteme mit Hilfe von Matrizen darstellen. +- Ich kann bestimmen, ob eine Matrix invertierbar ist oder nicht und gegebenenfalls die inverse Matrix bestimmen. +- Ich kann Verknüpfungen von linearen Abbildungen durch Matrix-Produkte ausdrücken. + +## Woche 6 +- Ich kenne die Begriffe Polarkoordinaten, Zylinderkoordinaten, Schwerpunkt, Flächenschwerpunkt und deren wichtigste Eigenschaften. +- Ich kenne die Begriffe orthogonale Matrix, Drehmatrix, Spiegelmatrix und deren wichtigste Eigenschaften. +- Ich kenne diejenigen der 2x2 Standardmatrizen, die orthogonal sind. +- Ich kenne das Spaltenvektor Konstruktionsverfahren zur Bestimmung von Matrizen und kann dieses anwenden. +- Ich kann mit Hilfe der Integration den Schwerpunkt homogener Flächen und von Rotationskörpern berechnen. +- Ich kann kartesische Koordinaten in Polar- bzw. Zylinderkoordinaten umwandeln und umgekehrt. +- Ich kann Dreh- und Spiegelmatrizen zur Lösung konkreter Fragestellungen anwenden. + +## Woche 7 +- Ich kenne die Begriffe Mehrfachintegral, Integrationsgebiet und deren wichtigste Eigenschaften. +- Ich kenne die Begriffe Spur, Determinante, Regel von Sarrus und deren wichtigste Eigenschaften. +- Ich kann - z. B. für die Vereinfachung von Zweifach- und Dreifachintegralen - kartesische Koordinaten in Polar- bzw. Zylinderkoordinaten umwandeln. +- Ich kann Mehrfachintegrale auf einfachen Gebieten in 2D und 3D berechnen und die Integrationsreihenfolge vertauschen. +- Ich kann Masse, Volumen und Schwerpunkt mittels Mehrfachintegralen bestimmen. +- Ich kann die Eigenschaften einer Matrix anhand ihrer Spur und Determinante beurteilen. +- Ich kann die Determinante quadratischer Matrizen in 2D und 3D berechnen. +- Ich kann die Determinante einer quadratischen Matrix mit Hilfe des Gaußschen Verfahrens berechnen + +## Woche 8 +- Ich kenne die Begriffe Skalarfeld, Vektorfeld, Kurve, eindimensionale Schnittkurve, Höhenlinie, Niveaufläche, Niveaumenge und deren wichtigste Eigenschaften. +- Ich kenne die Begriffe charakteristisches Polynom, Eigenwert, Eigenvektor, Eigenraum, Spektrum sowie deren wichtigste Eigenschaften. +- Ich kann die natürliche Definitionsmenge und Wertemenge einer Funktion mehrerer Variabler bestimmen. +- Ich kann Höhenlinien und Niveauflächen von Funktionen von zwei bzw. drei Variablen bestimmen und skizzieren. +- Ich kann das charakteristische Polynom, die Eigenwerte und Eigenvektoren einer quadratischen Matrix berechnen. +- Ich kann die Eigenschaften einer Matrix bzw. linearen Abbildung anhand ihrer Eigenwerte und Eigenvektoren beurteilen und umgekehrt. + +## Woche 9 +- Ich kenne die Begriffe partielle Ableitung, Tangentialebene, Gradient, totales Differential und deren wichtigste Eigenschaften. +- Ich kenne die Begriffe Axiom, Skalarkörper, Vektorraum, Linearkombination, lineare Hülle, linear abhängig, linear unabhängig, erzeugend, Basis, Dimension und deren wichtigste Eigenschaften. +- Ich kann die partiellen Ableitungen von Funktionen in mehreren Variablen berechnen. +- Ich kann die Tangentialebene in einem Punkt an ein Skalarfeld bestimmen. +- Ich kann den Gradienten und das totale Differential von Skalarfeldern bestimmen. +- Ich kann  beurteilen, ob gegebene Mengen eine Vektorraumstruktur bilden. +- Ich kann beurteilen, ob die Vektoren einer Teilmenge von ℝ𝑛Rn  linear abhängig, linear unabhängig oder erzeugend sind und ob sie eine Basis bilden. + +## Woche 10 +- Ich kenne die Begriffe Satz von Schwarz, Hesse-Matrix, Richtungsableitung, lokale/globale Extrema, Sattelpunkt, Extrema unter Nebenbedingungen, Lagrange´scher Multiplikator und deren wichtigste Eigenschaften. +- Ich kenne die Begriffe Bild, Kern, algebraische und geometrische Vielfachheit, ähnliche Matrix, Diagonalisierbarkeit einer Matrix und deren wichtigste Eigenschaften. +- Ich kann die Hesse-Matrix von Skalarfeldern bestimmen. +- Ich kann die Richtungsableitung berechnen. +- Ich kann lokale Extrema und Sattelpunkte bestimmen. +- Ich kann Extrema einer Funktion unter Nebenbedingungen bestimmen. +- Ich kann Bild und Kern einer linearen Abbildung berechnen. +- Ich kann bestimmen, ob eine Matrix diagonalisierbar ist oder nicht und die Diagonalmatrix angeben. + +## Woche 11 +- Ich kenne die Begriffe Kurve, Spur, Geschwindigkeits-/Tangentenvektor, Beschleunigungsvektor, Tangenteneinheitsvektor, Hauptnormalenvektor, Binormalenvektor, Parametrisierung, Bogenlänge, Kurven-/Linienintegral, Vektorfeld und deren wichtigste Eigenschaften. +- Ich kann die Spur einer parametrisierten Kurve in 2D skizzieren. +- Ich kann ein Vektorfeld in 2D skizzieren. +- Ich kann die Bogenlänge einer Kurve berechnen. +- Ich kann Linienintegrale berechnen. + +## Woche 12 +- Ich kenne die Begriffe Laplace Operator, Divergenz und Rotation eines Vektorfeldes, quellenfrei, wirbelfrei, konservativ, Potential-/Gradientenfeld und deren wichtigste Eigenschaften. +- Ich kann die Rotation und Divergenz von Vektorfeldern bestimmen. +- Ich kann bestimmen, ob ein Vektorfeld quellen- bzw. wirbelfrei ist. + + +## Probeprüfung Themen + +--- + +#### **Aufgabe 1 – Aussagen zu Integralen** + +- Begriff und Eigenschaften von Integral, Stammfunktion, (linearer) Substitution (1 | 2) + +- Anwendung der linearen Substitution auf bestimmte/unbestimmte Integrale (1 | 2) + +- Beurteilung & Berechnung eines uneigentlichen Integrals (4) + +- Zusammenhang zwischen Flächeninhalt in 2 D und (Doppel-)Integral (7) + + +--- + +#### **Aufgabe 2 – Aussagen über eine Funktion** + +- Begriff Skalarfeld, Niveau-/Höhen­linien, Niveau­mengen (8) + +- Bestimmung des Funktionswertes in einem Punkt (grundlegend, 8) + +- Interpretation des Graphen einer Funktion zweier Variablen im 3 D-Raum (8) + + +--- + +#### **Aufgabe 3 – Aussagen über ein Vektorfeld** + +- Rotation, Divergenz, konservative (= wirbel- & quellenfreie) Vektorfelder, Potentialfeld (12) + +- Prüfen, ob ∇ × **v** = 0 bzw. ∇·**v** = 0 und Existenz eines Potentials (12) + + +--- + +#### **Aufgabe 4 – Aussagen über zwei Matrizen** + +- Begriffe Matrix, symmetrisch, orthogonal, regulär (invertierbar), Bild (img) und Kern (ker) (4 | 5 | 6 | 7) + +- Bestimmen von Invertierbarkeit und Rang über Determinante oder Gauß-Verfahren (5 | 7) + +- Eigenschaften orthogonaler bzw. symmetrischer Matrizen (4 | 6) + + +--- + +#### **Aufgabe 5 – Weitere Aussagen zu Matrizen** + +- Produkte invertierbarer Matrizen und deren Invertierbarkeit (5) + +- Zusammenhang Matrixpotenz A³ – Eigenwerte λ³ (8) + +- Existenz unabhängiger Eigenvektoren zu verschiedenen Eigenwerten (8 | 10) + + +--- + +#### **Aufgabe 6 – Diverse Einzelaussagen** + +- Komplexe Zahlen: Konjugation, Potenzgleichungen, trigono­metrische Form (1 | 2 | 3) + +- Gradient und seine Orthogonalität zu Höhenlinien (9 | 10 | 12) + +- Lineare Unabhängigkeit & Basisbegriffe in Vektorräumen (9) + +- Definition einer linearen Abbildung (Homogenität & Additivität) (9) + + +--- + +#### **Aufgabe 7 – Gemischte Rechenaufgaben** + +a) **z⁴ = 16 e^{iπ}** lösen + +- Potenz- & Wurzelziehen komplexer Zahlen in Exponential-/Trig- & arithmetischer Form (3) + b) **A¹⁵** berechnen + +- Dreh- bzw. orthogonale 2×2-Matrizen erkennen, Potenzen mittels Diagonalisierung/Rotor-Eigenschaften bilden (6 | 8 | 10) + c) Integrationsreihenfolge vertauschen + +- Mehrfachintegrale, Gebietsskizze, Reihenfolgenwechsel (7) + d) Richtungsableitung an P(1, 2) in Richtung **v** + +- Gradient, Richtungsableitung, Normierung des Richtungsvektors (10) + e) Skalares Kurvenintegral längs **γ(t)** + +- Bogenlänge-Parameter t, Kurvenintegral eines Skalarfeldes (11) + + +--- + +#### **Aufgabe 8 – Lokale Extrema & Sattelpunkte** + +- Partielle Ableitungen, Gradient = 0 setzen (9) + +- Hesse-Matrix, Eigenwerte-Kriterium für Extrema/Sattel (10) + + +--- + +#### **Aufgabe 9 – Flächenschwerpunkt eines Trapezes** + +- Flächenschwerpunkt mittels Doppelintegration berechnen (5) + +- Anwendung von Mehrfachintegralen auf Masse/Schwerpunkt (7) + + +--- + +#### **Aufgabe 10 – Diagonalisierung einer 3×3-Matrix** + +- Charakteristisches Polynom, Eigenwerte, Eigenvektoren (8) + +- Bild, Kern, algebraische / geometrische Vielfachheit, Diagonalisierbarkeit & Transformationsmatrix (10) + + diff --git a/formulas/Formulas.pur b/formulas/Formulas.pur new file mode 100644 index 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Diagonalisierung und Transformationsmatrix +# 9. Schwerpunkt +1. **Gebiet $\mathcal A$ festlegen** + Wähle Integrationsgrenzen, z. B. + $$ + x\in[a,b],\quad y\in\bigl[y_{\mathrm{unten}}(x),\,y_{\mathrm{oben}}(x)\bigr] + $$ + oder umgekehrt + $$ + y\in[c,d],\quad x\in\bigl[x_{\mathrm{links}}(y),\,x_{\mathrm{rechts}}(y)\bigr]. + $$ + +2. **Dichtefunktion $\rho(x,y)$ angeben** + - Allgemein beliebig. + - Regelfall homogen: $\rho(x,y)\equiv 1$. + +3. **Gesamtmasse \(M\) berechnen** + $$ + M \;=\;\iint_{\mathcal A}\rho(x,y)\,dA + \;=\;\int_{x=a}^{b}\!\int_{y=y_{\mathrm{unten}}(x)}^{y_{\mathrm{oben}}(x)} + \rho(x,y)\,dy\,dx. + $$ + +4. **Erstes Moment zur \(y\)-Achse (\(x_S\))** + $$ + M_y + \;=\;\iint_{\mathcal A} x\,\rho(x,y)\,dA + \;=\;\int_{x=a}^{b}\!\int_{y=y_{\mathrm{unten}}(x)}^{y_{\mathrm{oben}}(x)} + x\,\rho(x,y)\,dy\,dx. + $$ + +5. **Erstes Moment zur \(x\)-Achse ($y_S$)** + $$ + M_x + \;=\;\iint_{\mathcal A} y\,\rho(x,y)\,dA + \;=\;\int_{x=a}^{b}\!\int_{y=y_{\mathrm{unten}}(x)}^{y_{\mathrm{oben}}(x)} + y\,\rho(x,y)\,dy\,dx. + $$ + +6. **Schwerpunktkoordinaten** + $$ + x_S \;=\;\frac{M_y}{M}, + \qquad + y_S \;=\;\frac{M_x}{M}. + $$ + +--- + +### Beispiel für $\rho(x,y)\equiv1$ und $0\le x\le2a,\;0\le y\le a+x$ + +- Gebiet: + $$ + x\in[0,2a],\quad y\in[0,\;a+x]. + $$ +- Gesamtmasse: + $$ + M = \int_{0}^{2a}\!\int_{0}^{a+x}1\,dy\,dx = 4a^2. + $$ +- Erste Momente: + $$ + M_y = \int_{0}^{2a}\!\int_{0}^{a+x} x\,dy\,dx, + \quad + M_x = \int_{0}^{2a}\!\int_{0}^{a+x} y\,dy\,dx. + $$ +- Schwerpunkt: + $$ + x_S = \frac{M_y}{M} = \frac{7}{6}a, + \quad + y_S = \frac{M_x}{M} = \frac{13}{12}a. + $$ + + + +# 8. Extrema - Mehrdimensional + +| **Baustein** | **Was es ist / tut** | **Warum es wichtig ist** | +|--------------|----------------------|--------------------------| +| **1. Funktion** $f(x,y)$ | Die „Höhenlandschaft“, die untersucht wird. | Ohne Funktion keine Analyse. | +| **2. Gradient** $\nabla f=(f_x,f_y)$ | Erste partielle Ableitungen → zeigt stärksten Anstieg. | Setze $\nabla f = \mathbf 0$, um **kritische Punkte** zu finden. | +| **3. Gleichungssystem** $\;f_x=0,\;f_y=0\;$ | Zwei Gleichungen. | Liefert die Koordinaten aller kritischen Punkte. | +| **4. Hesse-Matrix** $$H=\begin{pmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{pmatrix}$$ | Matrix der zweiten Ableitungen. | Enthält Krümmungs-Info der Fläche. | +| **5. Determinante** $$D=\det H=f_{xx}f_{yy}-f_{xy}^{2}$$ | Eine Zahl pro Punkt. | Entscheidet schnell über Punktart:
• $D>0$ ⇒ Extremum
• $D<0$ ⇒ Sattel
• $D=0$ ⇒ Test unentschieden | +| **6. Vorzeichen von** $f_{xx}$ | Nur wenn $D>0$. | Unterscheidet Minimum ($f_{xx}>0$) vs. Maximum ($f_{xx}<0$). | +| **7. Funktionswert** $f(x_0,y_0)$ *(optional)* | Höhe des Punktes. | Rein informativ, z. B. tiefstes Minimum. | +| **8. Sonderfälle**
• Eigenwerte von $H$
• Lagrange-Multiplikatoren | Alternativen/Erweiterungen. | Eigenwerte geben gleiche Entscheidung; Lagrange bei Nebenbedingungen. | + +--- + +## Ablauf in Kurzform + +1. **Aufschreiben:** $f(x,y)$ +2. **Gradient bilden:** $f_x,\;f_y$ +3. **Kritische Punkte:** Löse $f_x=0,\;f_y=0$ → $\{P_i\}$ +4. **Hesse-Matrix:** $f_{xx},f_{xy},f_{yy}$ in jedem $P_i$ +5. **Klassifikation:** + * $D<0$ ⇒ Sattel + * $D>0$ & $f_{xx}>0$ ⇒ Minimum + * $D>0$ & $f_{xx}<0$ ⇒ Maximum + * $D=0$ ⇒ weiterer Test nötig +1. **(Optional) Funktionswerte** $f(P_i)$ einsetzen, um Höhen/Tiefen zu vergleichen. + + +# 7. e) Skalares Kurvenintegral +**Allgemeines Schema zur Berechnung eines skalaren Kurvenintegrals** + +1. **Parameterdarstellung** + $$\gamma(t) = \bigl(x(t),\,y(t)\bigr),\quad t\in[a,b]$$ + +2. **Einsetzen in die skalare Feldfunktion** + $$f\bigl(\gamma(t)\bigr) = f\bigl(x(t),\,y(t)\bigr)$$ + +3. **Berechnung des Bogenelements** + - Ableitung: + $$\gamma'(t) = \bigl(x'(t),\,y'(t)\bigr)$$ + - Norm (Geschwindigkeit): + $$\|\gamma'(t)\| = \sqrt{x'(t)^2 + y'(t)^2}$$ + +4. **Formulierung des Kurvenintegrals** + $$ + \int_{K} f\,\mathrm{d}s + = \int_{a}^{b} f\bigl(\gamma(t)\bigr)\,\|\gamma'(t)\|\,\mathrm{d}t + $$ + +5. **Auswertung des Integrals** + - Ggf. Substitution wählen, z. B. $u = g(t)$ + - $\mathrm{d}u$ bestimmen und $t\,\mathrm{d}t$ umschreiben + - Grenzen anpassen: + $$t=a\mapsto u=u(a),\quad t=b\mapsto u=u(b)$$ + - Stammfunktion finden und einsetzen + - Ergebnis vereinfachen + +6. **Interpretation** + - Physikalisch: Gesamtmasse, Arbeit, Energie, … + - Mathematisch: Konvergenz prüfen (bei offenen Endpunkten) + +## Richtungsableitung + +1. **Gradient berechnen** + $$ + \nabla f(x) + = + \begin{pmatrix} + \displaystyle \frac{\partial f}{\partial x_{1}}(x) \\[6pt] + \vdots \\[4pt] + \displaystyle \frac{\partial f}{\partial x_{n}}(x) + \end{pmatrix}. + $$ + +2. **Einheitsvektor bilden** + Sei $\vec v = (v_{1},\dots,v_{n})^\top\in\mathbb{R}^{n}$. Dann + $$ + \|\vec v\| = \sqrt{v_{1}^{2} + \dots + v_{n}^{2}}, + \qquad + \vec u = \frac{\vec v}{\|\vec v\|}. + $$ + +3. **Richtungsableitung** + $$ + D_{\vec v}f(P) + = \lim_{h\to 0}\frac{f\bigl(P + h\,\vec u\bigr) - f(P)}{h} + = \nabla f(P)\,\bullet\,\vec u + = \nabla f(P)\,\bullet\,\frac{\vec v}{\|\vec v\|}. + $$ + +**Kurzgefasst:** +$$ +D_{\vec v}f(P) += \nabla f(P)\,\bullet\,\frac{\vec v}{\|\vec v\|}. +$$ + + +# 7. d) Integrationsreihenfolge ändern + +| **Vorher** – Typ I | **Nachher** – Typ II | **Was wohin ?** | +|--------------------|----------------------|-----------------| +| $$\int_{x=a}^{b}\;\int_{y=l(x)}^{u(x)} f(x,y)\,dy\,dx$$ | $$\int_{y=l(a)}^{u(b)}\;\int_{x=l^{-1}(y)}^{u^{-1}(y)} f(x,y)\,dx\,dy$$ | 1. äußere Variable wechselt $x\to y$
2. **äußere Grenzen** = tiefster / höchster $y$-Wert des Gebiets (hier $l(a),u(b)$)
3. **innere Grenzen** = Umkehrfunktionen der alten Kurven $l,u$ (links / rechts) | +| *(Spezialfall $l(x)=0$)* | | | + +#### Idiotensicheres Beispiel +Gegeben +$$ +\int_{x=a}^{b}\;\int_{y=0}^{x^{2}} f(x,y)\,dy\,dx +$$ + +➜ Tauschen + +$$ +\int_{y=0}^{b^{2}}\;\int_{x=\sqrt{y}}^{b} f(x,y)\,dx\,dy +$$ + +* drag-and-drop-Mapping: + * $0$ bleibt $0$ (äußere unten) + * $x^{2}$ → Umkehr $\sqrt{y}$ (innere links) + * $a,b$ bleiben Konstanten, landen als innere/rechte Grenze ($x=b$) und sorgen für $y$-Max $=b^{2}$ + +### Matrixpotenz $A^{15}$ – idiotensicher + +1. **Faktor ausrechnen** + $$A=\tfrac12\begin{pmatrix}\sqrt3&-1\\[2pt]1&\sqrt3\end{pmatrix} + =\begin{pmatrix}\tfrac{\sqrt3}{2}&-\tfrac12\\[4pt]\tfrac12&\tfrac{\sqrt3}{2}\end{pmatrix}$$ + +2. **Erkennen: Rotationsmatrix** + Form $\bigl(\!\begin{smallmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{smallmatrix}\!\bigr)$ + ⇒ $\cos\theta=\sqrt3/2,\ \sin\theta=1/2 \;\Longrightarrow\; \theta=\pi/6$ (30°). + +3. **Potenzregel für Rotationsmatrizen** + $$A^n=\begin{pmatrix}\cos(n\theta)&-\sin(n\theta)\\\sin(n\theta)&\cos(n\theta)\end{pmatrix}$$ + +4. **Hier $n=15$** + $$n\theta=15\cdot\frac{\pi}{6}= \frac{5\pi}{2}=2\pi+\frac{\pi}{2}$$ + (Rotation um $450^\circ$ ≙ $90^\circ$). + +5. **Kosinus/Sinus auslesen** + $$\cos\!\bigl(\tfrac{\pi}{2}\bigr)=0,\qquad + \sin\!\bigl(\tfrac{\pi}{2}\bigr)=1.$$ + +6. **Ergebnis** + $$\boxed{A^{15}= \begin{pmatrix}0&-1\\[4pt]1&0\end{pmatrix}}$$ + + +# WIP 7. c Vertauschen Integrationsreihenfolge +# WIP 7. b) Matrix Potenz +Diagonalmatrix oder Drehmatrix. + +# 7. a) Komplexe Zahl Polarform gleichung +$$ +z^{\,n}=R\,e^{i\Phi},\qquad R>0,\;n\in\mathbb N +$$ + +| Schritt | Was tun? | Formel / Erläuterung | +|---------|-----------|----------------------| +| **1. Polar-/Exponentialform herstellen** | Schreibe die rechte Seite als $R\,e^{i\Phi}$ mit Betrag $R$ und (Haupt-)Argument $\Phi$.
Falls sie in kartesischer Form $a+ib$ vorliegt: $$R=\sqrt{a^{2}+b^{2}},\qquad \Phi=\operatorname{atan2}(b,a).$$ | +| **2. n-te-Wurzel-Formel anwenden** | Für $$z^{n}=R\,e^{i\Phi}$$ lauten alle Lösungen $$z_k=\sqrt[n]{R}\,e^{\,i(\Phi+2k\pi)/n},\qquad k=0,1,\dots,n-1.$$ | +| **3. Winkel berechnen** | Definiere $$\theta_k=\frac{\Phi+2k\pi}{n}.$$ Diese $n$ Winkel liegen äquidistant auf dem Kreis (Abstand $2\pi/n$). | +| **4. Arithmetische (kartesische) Form** | Nutze Euler $e^{i\theta_k}=\cos\theta_k+i\sin\theta_k$:
$$z_k=\sqrt[n]{R}\,(\cos\theta_k+i\sin\theta_k).$$ | +| **5. (Optional) Prüfen** | Einsetzen zeigt $$z_k^{\,n}=R\,e^{i\Phi}.$$ | + +#### Kurzform der Lösungsmengen-Formel +$$ +\boxed{\,z_k=\sqrt[n]{R}\,e^{\,i(\Phi+2k\pi)/n},\;k=0,\dots,n-1\,} +$$ + +**Merke** +1. Betrag der Wurzeln: $\sqrt[n]{R}$. +2. Winkel: $\theta_k=\dfrac{\Phi+2k\pi}{n}$. +3. $n$ Lösungen liegen gleichmäßig verteilt auf einem Kreis mit Radius $\sqrt[n]{R}$. + +# Vektorfelder + + + +# Matrizen Eigenschaften + + +--- +## Singular vs. Regulär +>Check: Deteminante berechnen +- **Singular** ⇔ $\det A=0$ → nicht invertierbar. +- **Regulär** ⇔ $\det A\neq0$ → Inverse $A^{-1}$ existiert. + +--- +## Diagonalmatrix +Matrix +$$ +D=\begin{pmatrix}4&0&0\\0&-2&0\\0&0&7\end{pmatrix} +$$ + +Eigenschaften +- **Besteht typischerweise aus den Eigenwerten** einer Matrix entlang der Hauptdiagonalen, wenn man $A=PDP^{-1}$ diagonalisiert. +- Kommutiert mit jeder anderen Diagonalmatrix +- **Leicht zu potenzieren/invertieren**: + $D^k=\operatorname{diag}(4^k,-2^k,7^k)$, + $D^{-1}=\operatorname{diag}\!\bigl(\tfrac14,-\tfrac12,\tfrac17\bigr)$ +- **Orthogonal** genau dann, wenn $|d_i|=1$ (dann $DD^\top=I$) +--- +## Orthogonale Matrix +Matrix +$$ +Q=\frac1{\sqrt2}\begin{pmatrix}1&-1\\1&1\end{pmatrix} +$$ + +Eigenschaften +- **$Q^\top Q=I$**, also $Q^{-1}=Q^\top$ +- **Determinante = ±1**, erhält Länge & Winkel + + + +--- +## Rotationsmatrix (Ebene) +Matrix +$$ +R(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix} +$$ + +Eigenschaften +- Orthogonal mit **$\det=1$** +- **Potenzregel**: $R(\theta)^n=R(n\theta)$ +--- +## Nullmatrix +Matrix +$$ +O_2=\begin{pmatrix}0&0\\0&0\end{pmatrix} +$$ +Eigenschaften +- **Additives Neutrum**: $A+O=A$ +- **Rang = 0**, **Spur = 0** + +--- + +## Einheits-/Identitätsmatrix +Matrix +$$ +I_2=\begin{pmatrix}1&0\\0&1\end{pmatrix} +$$ + +Eigenschaften +- **Multiplikatives Neutrum**: $AI_2=I_2A=A$ +- **Eigenwerte alle 1**, $I_2^\top=I_2^{-1}=I_2$ + +--- + +## Skalarmatrix +Matrix +$$ +2I_2=\begin{pmatrix}2&0\\0&2\end{pmatrix} +$$ + +Eigenschaften +- Spezialfall Diagonalmatrix, **alle Eigenwerte = 2** +- Orthogonal/unitär ⇔ $|2|=1$ (hier also **nicht** orthogonal) + +--- +## Symmetrische Matrix ( +Transformationsmatrix) +Matrix +$$ +S=\begin{pmatrix}2&3&-1\\3&5&0\\-1&0&4\end{pmatrix} +$$ + +Eigenschaften +- $S^\top=S$ +- **Reelle Eigenwerte**, **orthogonal diagonal­isierbar**: + $$S = Q\,D\,Q^\top$$ +- **Transformationsmatrix** $Q$: + $$Q=\begin{pmatrix}v_1 & v_2 & v_3\end{pmatrix},$$ + wobei $v_i$ die normierten Eigenvektoren von $S$ sind (Spalten von $Q$). + + +--- + +## Schiefsymmetrische Matrix +Matrix +$$ +K=\begin{pmatrix}0&-2&1\\2&0&4\\-1&-4&0\end{pmatrix} +$$ + +Eigenschaften +- $K^\top=-K$, **Diagonale = 0** +- **Spur = 0**, Eigenwerte rein imaginär oder 0 + + +--- + +## Projektionsmatrix (idempotent) +Matrix +$$ +P=\begin{pmatrix}1&0\\0&0\end{pmatrix} +$$ + +Eigenschaften +- **$P^2=P$**, projiziert auf $x$-Achse +- Eigenwerte **0 oder 1** + +--- + +## Involutorische Matrix +Matrix +$$ +J=\begin{pmatrix}0&1\\1&0\end{pmatrix} +$$ + +Eigenschaften +- **$J^2=I$**, also $J^{-1}=J$ +- Eigenwerte **±1** (tauscht Koordinaten­achsen) + +--- + +## Nilpotente Matrix +Matrix +$$ +N=\begin{pmatrix}0&1\\0&0\end{pmatrix} +\quad\text{mit}\quad N^2=O_2 +$$ + +Eigenschaften +- **Alle Eigenwerte = 0**, Spur = 0 +- $N^k=O$ für $k\ge2$ + + +--- + +## Reflexionsmatrix (Householder-Typ) +Matrix (Spiegelung an $x$-Achse) +$$ +H=\begin{pmatrix}1&0\\0&-1\end{pmatrix} +$$ + +Eigenschaften +- Orthogonal, **$H^2=I$**, $\det=-1$ +- Spiegelt $y\to -y$ + +--- + +## Obere Dreiecksmatrix +Matrix +$$ +U=\begin{pmatrix}3&-2&1\\0&5&4\\0&0&-1\end{pmatrix} +$$ + +Eigenschaften +- **$\det=\prod$ Diagonal = $3\cdot5\cdot(-1)=-15$** +- **Eigenwerte = Diagonalelemente** (hier 3, 5, −1) + +--- + +## Positiv definite Matrix +Matrix +$$ +A=\begin{pmatrix}2&1\\1&2\end{pmatrix} +$$ + +Eigenschaften +- **$x^\top A x>0$ für alle $x\neq0$** +- Alle Eigenwerte positiv (hier 1 und 3) +- **Cholesky-Zerlegung** existiert: $A=LL^\top$ + + + +# Formeln - Ableitung - Integration + +### Potenz- und Wurzelfunktionen + +| Funktion $f(x)$ | Stammfunktion $F(x)$ | | | | +| :--------------------------------: | :------------------------------------------------------------------------------------------------------------------------------------: | --- | --- | --- | +| $0$ | $0$ | | | | +| $k \quad (k \in \mathbb{R})$ | $kx$ | | | | +| $$x^n$$ | $$F(x)=\begin{cases} \frac{1}{n+1}x^{n+1} & , \text{if}\quad \,n \neq -1,\\ \ln\lvert x\rvert, & ,\text{if}\quad n = -1. \end{cases}$$ | | | | +| $nx^{n-1}$ | $x^n$ | | | | +| $x$ | $\tfrac12 x^2$ | | | | +| $2x$ | $x^2$ | | | | +| $x^2$ | $\tfrac13 x^3$ | | | | +| $3x^2$ | $x^3$ | | | | +| $\sqrt{x}$ | $\tfrac23 x^{\tfrac32}$ | | | | +| $\sqrt[n]{x}$ | $\displaystyle \frac{n}{n+1},\bigl(\sqrt[n]{x}\bigr)^{n+1}\quad(n\neq -1)$ | | | | +| $\frac1{\sqrt{x}}$ | $2\sqrt{x}$ | | | | +| $\frac{1}{n,(\sqrt[n]{x^{,n-1}})}$ | $\sqrt[n]{x}$ | | | | +| $-\frac{2}{x^3}$ | $\frac{1}{x^2}$ | | | | +| $-\frac{1}{x^2}$ | $\frac{1}{x}$ | | | | + +--- + +### Exponential- und Logarithmusfunktionen + +| Funktion $f(x)$ | Stammfunktion $F(x)$ | +| :-----------------------------------------------------: | :----------------------------------------------------------------: | +| $\mathrm e^x$ | $\mathrm e^x$ | +| $\mathrm e^{kx}$ | $\displaystyle \frac{1}{k},\mathrm e^{kx}$ | +| $a^x\ln a \quad (a>0)$ | $a^x$ | +| $a^x$ | $\displaystyle \frac{a^x}{\ln a}$ | +| $x^x,(1+\ln x)$ | $x^x \quad (x>0)$ | +| $\mathrm e^{x\ln\lvert x\rvert},(\ln\lvert x\rvert +1)$ | $\lvert x\rvert^x = \mathrm e^{x\ln\lvert x\rvert}\quad (x\neq0)$ | +| $\frac{1}{x}$ | $\ln\lvert x\rvert$ Sonderfall von $x^n$ für $n=-1$, siehe oben | +| $\ln x$ | $x\ln x - x$ | +| $x^n\ln x$ | $\frac{x^{n+1}}{n+1}\Bigl(\ln x - \frac{1}{n+1}\Bigr)\quad(n\ge0)$ | +| $u'(x),\ln u(x)$ | $u(x)\ln u(x) - u(x)$ | +| $\displaystyle \frac{1}{x},\ln^{n}x\quad(n\neq -1)$ | $\displaystyle \frac{1}{n+1},\ln^{n+1}x$ | +| $\displaystyle \frac{1}{x},\ln x^n\quad(n\neq 0)$ | $\displaystyle \frac{1}{2n},\ln^2 x^n = \frac{n}{2}\ln^2 x$ | +| $\frac{1}{x}\,\frac{1}{\ln a}$ | $\log_a x$ | +| $\frac{1}{x\ln x}$ | $\displaystyle \ln\lvert\ln x\rvert\quad(x>0, \,x\neq1)$ | +| $\log_a x$ | $\displaystyle \frac{1}{\ln a}\,(x\ln x - x)$ | + +--- + +## Trigonometrische Funktionen und Hyperbelfunktionen + +### Trigonometrische Funktionen + +| Funktion $f(x)$ | Stammfunktion $F(x)$ | | | +| :-------------------------------------: | :---------------------------------------------------------------------------------------: | --- | --- | +| $\sin x$ | $-\cos x$ | | | +| $\cos x$ | $\sin x$ | | | +| $\tan x = \frac{\sin x}{\cos x}$ | $\ln\bigl[\sec x\bigr]$ | | | +| $\cot x = \frac{\cos x}{\sin x}$ | $-\ln\bigl[\csc x\bigr]$ | | | +| $\sec x = \frac1{\cos x}$ | $\operatorname{artanh}[\sin x]$ | | | +| $\csc x = \frac1{\sin x}$ | $-\operatorname{artanh}[\cos x]$ | | | +| $\sec^2 x = 1+\tan^2 x$ | $\tan x$ | | | +| $-\csc^2 x = -\bigl(1+\cot^2 x\bigr)$ | $\cot x$ | | | +| $\sin^2 x$ | $\frac{1}{2}[x−sin⁡x\,cos⁡x]=\frac{1}{2}x− \frac{1}{4} sin⁡(2x)$ | | | +| $\cos^2 x$ | $\frac{1}{2}[x+sin⁡x\,cos⁡x]=\frac{1}{2}x+ \frac{1}{4} sin⁡(2x)$ | | | +| $\sin(kx)\cos(kx)$ | $-\displaystyle \frac{1}{4k}\cos(2kx)$ | | | +| $\sin(kx)\cos(kx)$ | $\displaystyle \frac{1}{2k}\sin^2(kx)$ | | | +| $\displaystyle \frac{\sin(ax)}{e^{bx}}$ | $\displaystyle \frac{a\,e^{bx} - a\cos(ax) - b\sin(ax)}{(a^2+b^2)\,e^{bx}}$ | | | +| $\displaystyle \frac{\cos(ax)}{e^{bx}}$ | $\displaystyle \frac{a\sin(ax) - b\cos(ax) + b\,e^{bx}}{(a^2+b^2)\,e^{bx}}$ | | | +| $\arcsin x$ | $x\arcsin x + \sqrt{1-x^2}$ | | | +| $\arccos x$ | $x\arccos x - \sqrt{1-x^2}$ | | | +| $\arctan x$ | $x\arctan x - \tfrac12\ln\bigl(1+x^2\bigr)$ | | | +| $\arccot x$ | $x\arccot x + \tfrac12\ln\bigl(1+x^2\bigr)$ | | | +| $\displaystyle \frac{1}{\sqrt{1-x^2}}$ | $\arcsin x$ | | | +| $\displaystyle \frac{-1}{\sqrt{1-x^2}}$ | $\arccos x$ | | | +| $\displaystyle \frac{1}{x^2+1}$ | $\arctan x$ | | | +| $\displaystyle -\frac{1}{x^2+1}$ | $\arccot x$ | | | +| $\displaystyle \frac{x^2}{x^2+1}$ | $x - \arctan x$ | | | +| $\displaystyle \frac{1}{(x^2+1)^2}$ | $\displaystyle \tfrac12\Bigl(\frac{x}{x^2+1} + \arctan x\Bigr)$ | | | +| $\sqrt{a^2 - x^2}$ | $\displaystyle \frac{a^2}{2}\arcsin\Bigl(\frac{x}{a}\Bigr) + \frac{x}{2}\sqrt{a^2 - x^2}$ | | | +| $\displaystyle \frac{1}{ax^2+bx+c}$ | $\frac{2}{\sqrt{4ac - b^2}}\;\arctan\!\Bigl(\frac{2ax + b}{\sqrt{4ac - b^2}}\Bigr)$ | | | + +### Hyperbelfunktionen + +| Funktion $f(x)$ | Stammfunktion $F(x)$ | +| :----------------------------------------------------: | :-------------------------------------------------------------------------------------------------------: | +| $\sinh x$ | $\cosh x$ | +| $\cosh x$ | $\sinh x$ | +| $\tanh x$ | $\ln\bigl[\cosh x\bigr]$ | +| $\coth x$ | $\ln\lvert\sinh x\rvert$ | +| $\operatorname{sech} x$ | $\operatorname{gd} x = \arctan[\sinh x]$ | +| $\operatorname{csch} x$ | $-\operatorname{arcoth}[\cosh x]$ | +| $\displaystyle \frac{1}{\cosh^2 x} = 1-\tanh^2 x$ | $\tanh x$ | +| $\displaystyle \frac{-1}{\sinh^2 x} = 1-\coth^2 x$ | $\coth x$ | +| $\operatorname{arsinh} x$ | $x\,\operatorname{arsinh} x - \sqrt{x^2+1}$ | +| $\operatorname{arcosh} x$ | $x\,\operatorname{arcosh} x - \sqrt{x^2-1}$ | +| $\operatorname{artanh} x$ | $x\,\operatorname{artanh} x + \tfrac12\ln\bigl(1-x^2\bigr)$ | +| $\operatorname{arcoth} x$ | $x\,\operatorname{arcoth} x + \tfrac12\ln\bigl(x^2-1\bigr)$ | +| $\displaystyle \frac{1}{\sqrt{x^2+1}}$ | $\operatorname{arsinh} x$ | +| $\displaystyle \frac{1}{\sqrt{x^2-1}}\quad(x>1)$ | $\operatorname{arcosh} x$ | +| $\sqrt{a^2 + x^2}$ | $\displaystyle \frac{a^2}{2}\,\operatorname{arsinh}\Bigl(\frac{x}{a}\Bigr) + \frac{x}{2}\sqrt{a^2 + x^2}$ | +| $\displaystyle \frac{1}{\sqrt{ax^2+bx+c}}$ | $\displaystyle \frac{1}{\sqrt{a}}\operatorname{arsinh}\Bigl(\frac{2ax + b}{\sqrt{4ac - b^2}}\Bigr)$ | +| $\displaystyle \frac{1}{1-x^2}\quad(\lvert x\rvert<1)$ | $\operatorname{artanh} x$ | +| $\displaystyle \frac{1}{1-x^2}\quad(\lvert x\rvert>1)$ | $\operatorname{arcoth} x$ | + +# Regeln - Ableiten +### Produktregel +$$u(x) \cdot v(x) \Longrightarrow u'v + u\,v'$$ +### Quotientenregel +$$\frac{u(x)}{v(x)} \Longrightarrow \frac{u'v + u\,v'}{v^2}$$ +### Kettenregel +Äußere Funktion abgeleitet, mal innere Funktion abgeleitet. +$$u[v(x)] \Longrightarrow u'[v(x)] \cdot v'(x)$$ + + +# Regeln - Integrieren +### Partielle Integration +>**Idee**: +>Ein Teil herausziehen, dass später etwas Integrierbares herauskommt. $u$ wird abgeleitet. $v$ wird integriert. +$$ +\int{u \, v} \Longrightarrow u\, V - \int{V\, u' \,dx} +$$ +### U-Substitution +Substituieren: +$$\int{\frac{1}{u'}u}\, du$$ +> **Wichtig:** +> 1/Ableitung von u, muss sich mit dem Variablen Faktor vorne kürzen. Also es darf bei $du$ kein $x$ mehr übrig bleiben. +> Beispiel: +> $$\int{\frac{1}{x}}2x\cdot u \, \, du \longrightarrow \int{2u} \,\,du$$ diff --git a/notes/Komplexe Zahlen.md b/notes/Komplexe Zahlen.md new file mode 100644 index 0000000..2a24588 --- /dev/null +++ b/notes/Komplexe Zahlen.md @@ -0,0 +1,54 @@ +# Komplexe Zahlen + +## Eulersche Gleichung & Formel +Die Eulersche Gleichung lautet: +$$e^{i\varphi} = \cos(\varphi) + i\sin(\varphi)$$ + +Daraus folgt die Darstellung einer komplexen Zahl in Exponentialform: +$$z = r \Bigl(\cos(\varphi) + i\sin(\varphi)\Bigr) = r \cdot e^{i\varphi}$$ + +> **Exponentialform:** +$$z = r \cdot e^{i(\varphi + 2\pi k)}$$ + +### Beispiel: Arithmetische in Exponentialform +Für die arithmetische Darstellung: +$$z = -2 + 2i$$ + +Berechnung des Betrags: +$$r = |z| = \sqrt{(-2)^2 + 2^2} = \sqrt{8}$$ + +Berechnung des Winkels: +$$\varphi = \arctan\left(\frac{2}{-2}\right) + \pi = \frac{3\pi}{4}$$ + +Somit ergibt sich: +$$z = \sqrt{8} \cdot e^{i\frac{3\pi}{4}}$$ + +## Multiplikation +Gegeben sind: +$$z_1 = 2 \cdot e^{i\frac{\pi}{6}}$$ +$$z_2 = 3 \cdot e^{i\frac{4\pi}{3}}$$ + +Multiplikation: +$$z_1 \cdot z_2 = 2 \cdot 3 \cdot e^{i\left(\frac{\pi}{6} + \frac{4\pi}{3}\right)} = 6 \cdot e^{i\frac{3\pi}{2}}$$ + +### Division +Gegeben: +$$z_1 = 2 \cdot e^{i\frac{\pi}{6}}$$ +$$z_2 = 3 \cdot e^{i\frac{4\pi}{3}}$$ + +Division: +$$\frac{z_1}{z_2} = \frac{2}{3} \cdot e^{i\left(\frac{\pi}{6} - \frac{4\pi}{3}\right)} = \frac{2}{3} \cdot e^{-i\frac{7\pi}{6}} = \frac{2}{3} \cdot e^{i\frac{5\pi}{6}}$$ + +#### Umwandeln in Arithmetische Form +$$\frac{2}{3} \cdot e^{i\frac{5\pi}{6}} = \frac{2}{3} \cdot \Bigl(\cos\left(\frac{5\pi}{6}\right) + i \sin\left(\frac{5\pi}{6}\right)\Bigr)$$ + +**Bemerkungen:** +- $|e^{i\varphi}| =\sqrt{\cos^2(\varphi) + \sin^2(\varphi)} = 1$ +- $e^{i \ 0} = 1 = e^{i 2k\pi}$ +- $e^{i\pi} = -1$ +- $\sinh(i x) = \sin x$ +- $\cosh(i x) = \cos x$ + + +### Potenzieren + diff --git a/notes/Matrizen.md b/notes/Matrizen.md new file mode 100644 index 0000000..cd82804 --- /dev/null +++ b/notes/Matrizen.md @@ -0,0 +1,52 @@ + +**Menge der Matrizen**: +- $\mathbb{M}(m, n, \mathbb{R}) = \mathbb{R}^{m\times n}$ + +> **Anwendungen**: +> - Beschreibung linearer Gleichungssysteme +> - Lineare Abbildungen zwischen Vektorräumen +> - Transformationen in der Geometrie (z. B. Drehungen, Spiegelungen, Skalierungen) +> - Darstellung von Netzwerken oder Graphen +> - Datenrepräsentation in maschinellem Lernen und Statistik + +--- + +## Zeilen, Spaltenvektoren + +- Eine Matrix $A \in \mathbb{R}^{m \times n}$ besteht aus $m$ Zeilen und $n$ Spalten. +- Zeilenvektoren: $A_{i*} \in \mathbb{R}^{1 \times n}$ +- Spaltenvektoren: $A_{*j} \in \mathbb{R}^{m \times 1}$ +- Jeder Eintrag $a_{ij}$ steht an der Kreuzung von Zeile $i$ und Spalte $j$. + +--- + +## Transponieren + +- Zeilen- und Spaltenindex vertauschen: + +$$ A \longrightarrow A^T $$ +$$(A^T)^T = A$$ + +- Eigenschaften: + - $(A + B)^T = A^T + B^T$ + - $(\lambda A)^T = \lambda A^T$ für $\lambda \in \mathbb{R}$ + - $(AB)^T = B^T A^T$ + +--- + +## Matrizenmultiplikation + +- Definition: $(AB)_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}$ +- Voraussetzung: Spaltenanzahl von $A$ muss gleich der Zeilenanzahl von $B$ sein + +$$(A \in \mathbb{R}^{m \times n},\ B \in \mathbb{R}^{n \times p}) \Rightarrow AB \in \mathbb{R}^{m \times p}$$ + +- **Shape des Outputs**: $m_A \times n_B$ + +- **Eigenschaften**: + - Assoziativ: $A(BC) = (AB)C$ + + + + - Distributiv: $A(B + C) = AB + AC$ + - Im Allgemeinen **nicht kommutativ**: $AB \neq BA$ diff --git a/notes/Prompts.md b/notes/Prompts.md new file mode 100644 index 0000000..bbf46d3 --- /dev/null +++ b/notes/Prompts.md @@ -0,0 +1,6 @@ + +## Für Aufgaben +1. Wie berechnet man das? Schritt für schritt +2. Bitte einmal das Schema allgemein Formuliert. Also für aufgaben in dem Format. +3. Kannst du mir den Hintergrund erklären, was wir hier berechnen und wie man sich das ganze Vorstellen kann? +4. Ok, das Allgemeine Schema jetzt bitte als Markdown code, mit dollar als inline latex delimiter anstatt \( \) und zwei dollar für multiline/block anstatt \[ \] (Katex) diff --git a/notes/Vektorfelder.md b/notes/Vektorfelder.md new file mode 100644 index 0000000..e0709c0 --- /dev/null +++ b/notes/Vektorfelder.md @@ -0,0 +1,107 @@ +**Menge der Vektorfelder** +Für eine offene Teilmenge $U\subseteq \mathbb{R}^n$ bezeichnet +$$ + \mathfrak{X}(U) = \{\,F : U \to \mathbb{R}^n \mid F \text{ ist (stetig) differenzierbar}\}. +$$ + +> **Anwendungen** +> - Beschreibung von Strömungsfeldern in Fluiddynamik +> - Elektromagnetische Felder (Maxwell-Gleichungen) +> - Kraftfelder in der Mechanik +> - Computergrafik: Beleuchtung und Verformung +> - Kontinuumsmechanik und Materialwissenschaften + +--- + +## Definition und Notation + +Ein Vektorfeld $F\in\mathfrak{X}(U)$ schreibt man komponentenweise: +$$ + F(x) = \bigl(F_1(x),\,F_2(x),\,\dots,\,F_n(x)\bigr),\quad x\in U. +$$ +Jeder $F_i$ ist eine skalare Funktion $F_i:U\to\mathbb{R}$. +Für $n=3$ oft: +$$ + F = F_x\,\mathbf{i} + F_y\,\mathbf{j} + F_z\,\mathbf{k}. +$$ + +--- + +## Lineare Operationen + +- **Addition**: + $$ + (F + G)(x) = F(x) + G(x). + $$ +- **Skalarmultiplikation** ($\lambda\in\mathbb{R}$): + $$ + (\lambda F)(x) = \lambda\,F(x). + $$ +$\mathfrak{X}(U)$ ist damit ein Vektorraum. + +--- + +## Differentialoperatoren + +- **Gradient** (für Skalarfeld $\phi:U\to\mathbb{R}$): + $$ + \nabla\phi = \bigl(\tfrac{\partial\phi}{\partial x_1},\dots,\tfrac{\partial\phi}{\partial x_n}\bigr). + $$ +- **Divergenz** (für $F\in\mathfrak{X}(U)$): + $$ + \mathrm{div}\,F = \nabla\cdot F + = \sum_{i=1}^n \frac{\partial F_i}{\partial x_i}. + $$ +- **Rotation** / **Wirbel** ($\mathbb{R}^3$): + $$ + \nabla\times F + = \bigl(\tfrac{\partial F_z}{\partial y}-\tfrac{\partial F_y}{\partial z},\; + \tfrac{\partial F_x}{\partial z}-\tfrac{\partial F_z}{\partial x},\; + \tfrac{\partial F_y}{\partial x}-\tfrac{\partial F_x}{\partial y}\bigr). + $$ +- **Laplacian** (auf Skalarfeld $\phi$): + $$ + \Delta \phi = \nabla\cdot(\nabla\phi) + = \sum_{i=1}^n \frac{\partial^2 \phi}{\partial x_i^2}. + $$ + +--- + +## Integrale und klassische Sätze + +- **Linien­integral** entlang einer Kurve $C$ mit Parametrisierung $\gamma(t)$: + $$ + \int_C F\cdot \mathrm{d}r + = \int_a^b F\bigl(\gamma(t)\bigr)\cdot \gamma'(t)\,\mathrm{d}t. + $$ +- **Fluss** durch eine Fläche $S$: + $$ + \Phi = \iint_S F\cdot \mathrm{d}S + = \iint_D F\bigl(\mathbf{r}(u,v)\bigr)\cdot\bigl(\mathbf{r}_u\times\mathbf{r}_v\bigr)\,\mathrm{d}u\,\mathrm{d}v. + $$ +- **Green’scher Satz** ($\mathbb{R}^2$): + $$ + \oint_{\partial D} P\,\mathrm{d}x + Q\,\mathrm{d}y + = \iint_D \bigl(\tfrac{\partial Q}{\partial x} - \tfrac{\partial P}{\partial y}\bigr)\,\mathrm{d}A. + $$ +- **Divergenzsatz** (Gauß): + $$ + \iint_{\partial V} F\cdot \mathrm{d}S + = \iiint_V \mathrm{div}\,F\,\mathrm{d}V. + $$ +- **Stokes’ Theorem** ($\mathbb{R}^3$): + $$ + \oint_{\partial S} F\cdot \mathrm{d}r + = \iint_S (\nabla\times F)\cdot \mathrm{d}S. + $$ + +--- + +## Spezielle Feldtypen + +- **Konservatives Feld**: $\exists\,\phi$ mit $F=\nabla\phi$ + $\Longrightarrow$ curl $F=0$ und Linienintegral nur von Endpunkten abhängig. +- **Solenoidales Feld**: $\mathrm{div}\,F=0$ + $\Longrightarrow$ kein Netto-Fluss durch geschlossene Oberfläche. +- **Potentialfeld**: Synonym zu konservativem Feld in Mechanik/Elektrostatik. +- **Wirbelfreies Feld**: curl $F=0$. diff --git a/notes/Zusammenfassung.md b/notes/Zusammenfassung.md new file mode 100644 index 0000000..caee4e9 --- /dev/null +++ b/notes/Zusammenfassung.md @@ -0,0 +1,27 @@ +# 1.1 Gleitkommaarithmetik + +- Reele Zahlen in der Informatik +- Hauptproblem: **Auslöschung** + +**Symbolik**: +- Simpy +- Wolfram Alpha +- Brüche + +# 1.2 Fixpunkt Iteration + +**Fixpunkt**: +Funktionen: +>Ein Fixpunkt ist ein punkt an dem x = y + +**Stabilisator (i.e. Fixpunkt einer rekursiven Folge)**: +> In einer Folge würde dieser *einrasten*, das heisst das nächste Folgeglied ist gleich dem aktuellen. Sie steckt fest. +$$f(x) = x$$ + +**Aussagen**: +- Grenzwert $\implies$ Fixpunkt. +- Fixpunkt $\centernot\implies$ Grenzwert. + +- Kein Stabilisator? $\implies$ Divergiert +- Stabilisator $\centernot\implies$ Konvergent + diff --git a/pyproject.toml b/pyproject.toml new file mode 100644 index 0000000..a0807f9 --- /dev/null +++ b/pyproject.toml @@ -0,0 +1,11 @@ +[project] +name = "cds402-mathematics-ii" +version = "0.1.0" +requires-python = ">=3.12" +dependencies = [ + "imageio>=2.37.0", + "matplotlib>=3.10.3", + "numpy>=2.2.5", + "pandas>=2.2.3", + "plotly>=6.1.0", +] diff --git a/uv.lock b/uv.lock new file mode 100644 index 0000000..ea513eb --- /dev/null +++ b/uv.lock @@ -0,0 +1,392 @@ +version = 1 +revision = 1 +requires-python = ">=3.12" + +[[package]] +name = "cds402-mathematics-ii" +version = "0.1.0" +source = { virtual = "." } +dependencies = [ + { name = "imageio" }, + { name = "matplotlib" }, + { name = "numpy" }, + { name = "pandas" }, + { name = "plotly" }, +] + 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