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code/anim.py
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128
code/anim.py
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import numpy as np
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import plotly.graph_objects as go
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"""
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Interactive 3D Plotly animation of the scalar line integral
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∮_C f(x, y) ds with f(x, y) = x · y
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along the unit circle C, plotted on the surface z = f(x,y).
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No files are written; the animation is displayed with interactive controls.
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"""
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# Parameters
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t_frame = np.linspace(0.0, 2 * np.pi, 120)
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# Scalar field definition
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def scalar_field(x, y):
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return x * y
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# Surface mesh for scalar field
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lin = np.linspace(-1.2, 1.2, 50)
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X, Y = np.meshgrid(lin, lin)
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Z = scalar_field(X, Y)
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# Compute running integral along circle
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# Midpoint rule
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mid_ts = (t_frame[1:] + t_frame[:-1]) / 2
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f_mid = scalar_field(np.cos(mid_ts), np.sin(mid_ts))
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dt = t_frame[1:] - t_frame[:-1]
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running = np.concatenate([[0], np.cumsum(f_mid * dt)])
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# Prepare static surface trace
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surface = go.Surface(
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x=X, y=Y, z=Z, colorscale="RdBu", cmin=-1, cmax=1, opacity=0.6, showscale=False
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)
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# Full path on surface
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curve_x = np.cos(t_frame)
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curve_y = np.sin(t_frame)
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curve_z = scalar_field(curve_x, curve_y)
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path = go.Scatter3d(
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x=curve_x, y=curve_y, z=curve_z, mode="lines", line=dict(color="black", width=2)
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)
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# Initial marker at t=0
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t0 = t_frame[0]
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marker = go.Scatter3d(
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x=[np.cos(t0)],
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y=[np.sin(t0)],
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z=[scalar_field(np.cos(t0), np.sin(t0))],
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mode="markers",
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marker=dict(size=5, color="black"),
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)
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# Build frames for animation
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frames = []
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for i, t in enumerate(t_frame):
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x0, y0 = np.cos(t), np.sin(t)
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z0 = scalar_field(x0, y0)
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frame = go.Frame(
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data=[marker.update(x=[x0], y=[y0], z=[z0])],
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layout=go.Layout(title_text=f"Running integral: {running[i]:.3f}"),
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name=f"frame{i}",
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)
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frames.append(frame)
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# Layout with sliders and buttons
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graph = go.Figure(
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data=[surface, path, marker],
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layout=go.Layout(
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title="Running integral: 0.000",
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scene=dict(
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xaxis=dict(range=[-1.3, 1.3]),
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yaxis=dict(range=[-1.3, 1.3]),
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zaxis=dict(range=[Z.min(), Z.max()]),
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),
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updatemenus=[
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dict(
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type="buttons",
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showactive=False,
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buttons=[
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dict(
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label="Play",
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method="animate",
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args=[
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None,
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dict(
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frame=dict(duration=50, redraw=True),
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fromcurrent=True,
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transition=dict(duration=0),
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),
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],
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)
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],
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)
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],
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),
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frames=frames,
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)
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# Add slider
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sliders = [
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dict(
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steps=[
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dict(
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method="animate",
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args=[
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[f.name],
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dict(
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mode="immediate",
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frame=dict(duration=50, redraw=True),
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transition=dict(duration=0),
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),
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],
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label=str(i),
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)
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for i, f in enumerate(frames)
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],
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transition=dict(duration=0),
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x=0,
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y=0,
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currentvalue=dict(prefix="Frame: "),
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len=1.0,
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)
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]
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graph.update_layout(sliders=sliders)
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graph.show()
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code/shadertoy.glsl
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code/shadertoy.glsl
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/* Pixel unit conversion function */
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vec2 pos(in float x, in float y) { return st + vec2(x * rx, y * rx); }
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vec2 pos(in float x) { return pos(x, x); }
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vec2 pos(in vec2 p) { return pos(p.x, p.y); }
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float size(in float x) { return x * rx; }
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vec2 size(in float x, in float y) { return vec2(x * rx, y * rx); }
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void mainImage( out vec4 fragColor, in vec2 fragCoord )
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{
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// Normalized pixel coordinates (from 0 to 1)
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vec2 uv = fragCoord/iResolution.xy;
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// Time varying pixel color
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vec3 col = 0.5 + 0.5*cos(iTime+uv.xyx+vec3(0,2,4));
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// Output to screen
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fragColor = vec4(col,1.0);
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}
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228
formulas/Formelsammlung.md
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formulas/Formelsammlung.md
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## Woche 1
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- Ich kenne die Begriffe Integral, Stammfunktion, lineare Substitution und deren wichtigste Eigenschaften.
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- Ich kenne die Begriffe komplexe Zahl, Realteil, Imaginärteil, Betrag und komplex konjugiert und deren wichtigste Eigenschaften.
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- Ich kenne die elementaren Rechenregeln für die komplexe Konjugation.
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- Ich kann die Methode der linearen Substitution anwenden, um bestimmte und unbestimmte Integrale von linear substituierten Elementarfunktionen zu berechnen.
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- Ich kann den Flächeninhalt zwischen einer Funktion und den Koordinatenachsen bestimmen.
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- Ich kann den Flächeninhalt zwischen zwei sich schneidenden Funktionen bestimmen.
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- Ich kann das Volumen von Rotationskörpern bestimmen.
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- Ich kann für jede komplexe Zahl ihren Realteil, Imaginärteil, Betrag und ihre komplex Konjugierte bestimmen.
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- Ich kann Summe, Differenz, Produkt und Quotient von komplexen Zahlen berechnen.
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- Ich kann einfache Brüche und Potenzen von komplexen Zahlen durch Anwenden der Rechenregeln vereinfachen.
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## Woche 2
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- Ich kenne die Begriffe Integral, Stammfunktion, Substitution, Trapezformel und deren wichtigste Eigenschaften.
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- Ich kenne die Begriffe Gaußsche Zahlenebene, arithmetische und trigonometrische Form einer komplexen Zahl und Arg-Funktion und deren Eigenschaften.
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- Ich kann die Methode der Substitution anwenden, um bestimmte und unbestimmte Integrale zu berechnen.
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- Ich kann bestimmte Integrale näherungsweise auf eine vorgegebene Anzahl Dezimalstellen mit Python/Numpy berechnen.
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- Ich kann komplexe Zahlen in der Gaußschen Zahlenebene darstellen.
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- Ich kann komplexe Zahlen von der arithmetischen in die trigonometrische Form und umgekehrt umwandeln.
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- Ich kann einfache Brüche und Potenzen von komplexen Zahlen durch Anwenden der Rechenregeln vereinfachen.
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- Ich kann quadratische Gleichungen mit reellen Koeffizienten lösen.
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## Woche 3
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- Ich kenne die Begriffe Integral, Stammfunktion, partielle Integration und deren wichtigste Eigenschaften.
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- Ich kenne die Begriffe Exponentialform einer komplexen Zahl, Eulersche Formel, Potenzgleichungen und deren Eigenschaften.
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- Ich kann die partielle Integration anwenden, um bestimmte und unbestimmte Integrale zu berechnen.
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- Ich kann unbestimmte Integrale mit Python/Sympy berechnen.
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- Ich kann komplexe Zahlen in der arithmetischen, trigonometrischen und Exponentialform darstellen und von einer in die andere Form umwandeln.
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- Ich kann die Grundrechenarten für die komplexen Zahlen anwenden.
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- Ich kann komplexe Zahlen sowohl potenzieren als auch aus ihnen die Wurzel ziehen.
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## Woche 4
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- Ich kenne den Begriff uneigentliches Integral und dessen wichtigste Eigenschaften.
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- Ich kenne die Begriffe Matrix, symmetrische/schiefsymmetrische Matrix, Einheitsmatrix, inverse Matrix, Transposition und deren wichtigste Eigenschaften.
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- Ich kann die Existenz eines uneigentlichen Integrals beurteilen und gegebenenfalls seinen Wert berechnen.
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- 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.
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## Woche 5
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- Ich kenne die Begriffe Bogenlänge, Mantelfläche von Rotationskörpern, Massenmittelpunkt, Schwerpunkt, Flächenschwerpunkt und deren wichtigste Eigenschaften.
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- Ich kenne die Begriffe Matrix, inverse Matrix, Drehmatrix, lineare Abbildung und deren wichtigste Eigenschaften.
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- Ich kenne die 2x2 Standardmatrizen und die dadurch beschriebenen linearen Abbildungen in 2D.
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- Ich kann mit Hilfe der Integration die Bogenlänge einer ebenen Kurve, die Mantelfläche, den Flächenschwerpunkt und das Volumen von Rotationskörpern berechnen.
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- Ich kann lineare Gleichungssysteme mit Hilfe von Matrizen darstellen.
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- Ich kann bestimmen, ob eine Matrix invertierbar ist oder nicht und gegebenenfalls die inverse Matrix bestimmen.
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- Ich kann Verknüpfungen von linearen Abbildungen durch Matrix-Produkte ausdrücken.
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## Woche 6
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- Ich kenne die Begriffe Polarkoordinaten, Zylinderkoordinaten, Schwerpunkt, Flächenschwerpunkt und deren wichtigste Eigenschaften.
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- Ich kenne die Begriffe orthogonale Matrix, Drehmatrix, Spiegelmatrix und deren wichtigste Eigenschaften.
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- Ich kenne diejenigen der 2x2 Standardmatrizen, die orthogonal sind.
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- Ich kenne das Spaltenvektor Konstruktionsverfahren zur Bestimmung von Matrizen und kann dieses anwenden.
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- Ich kann mit Hilfe der Integration den Schwerpunkt homogener Flächen und von Rotationskörpern berechnen.
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- Ich kann kartesische Koordinaten in Polar- bzw. Zylinderkoordinaten umwandeln und umgekehrt.
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- Ich kann Dreh- und Spiegelmatrizen zur Lösung konkreter Fragestellungen anwenden.
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## Woche 7
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- Ich kenne die Begriffe Mehrfachintegral, Integrationsgebiet und deren wichtigste Eigenschaften.
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- Ich kenne die Begriffe Spur, Determinante, Regel von Sarrus und deren wichtigste Eigenschaften.
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- Ich kann - z. B. für die Vereinfachung von Zweifach- und Dreifachintegralen - kartesische Koordinaten in Polar- bzw. Zylinderkoordinaten umwandeln.
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- Ich kann Mehrfachintegrale auf einfachen Gebieten in 2D und 3D berechnen und die Integrationsreihenfolge vertauschen.
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- Ich kann Masse, Volumen und Schwerpunkt mittels Mehrfachintegralen bestimmen.
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- Ich kann die Eigenschaften einer Matrix anhand ihrer Spur und Determinante beurteilen.
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- Ich kann die Determinante quadratischer Matrizen in 2D und 3D berechnen.
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- Ich kann die Determinante einer quadratischen Matrix mit Hilfe des Gaußschen Verfahrens berechnen
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## Woche 8
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- Ich kenne die Begriffe Skalarfeld, Vektorfeld, Kurve, eindimensionale Schnittkurve, Höhenlinie, Niveaufläche, Niveaumenge und deren wichtigste Eigenschaften.
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- Ich kenne die Begriffe charakteristisches Polynom, Eigenwert, Eigenvektor, Eigenraum, Spektrum sowie deren wichtigste Eigenschaften.
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- Ich kann die natürliche Definitionsmenge und Wertemenge einer Funktion mehrerer Variabler bestimmen.
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- Ich kann Höhenlinien und Niveauflächen von Funktionen von zwei bzw. drei Variablen bestimmen und skizzieren.
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- Ich kann das charakteristische Polynom, die Eigenwerte und Eigenvektoren einer quadratischen Matrix berechnen.
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- Ich kann die Eigenschaften einer Matrix bzw. linearen Abbildung anhand ihrer Eigenwerte und Eigenvektoren beurteilen und umgekehrt.
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## Woche 9
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- Ich kenne die Begriffe partielle Ableitung, Tangentialebene, Gradient, totales Differential und deren wichtigste Eigenschaften.
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- Ich kenne die Begriffe Axiom, Skalarkörper, Vektorraum, Linearkombination, lineare Hülle, linear abhängig, linear unabhängig, erzeugend, Basis, Dimension und deren wichtigste Eigenschaften.
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- Ich kann die partiellen Ableitungen von Funktionen in mehreren Variablen berechnen.
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- Ich kann die Tangentialebene in einem Punkt an ein Skalarfeld bestimmen.
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- Ich kann den Gradienten und das totale Differential von Skalarfeldern bestimmen.
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- Ich kann beurteilen, ob gegebene Mengen eine Vektorraumstruktur bilden.
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- Ich kann beurteilen, ob die Vektoren einer Teilmenge von ℝ𝑛Rn linear abhängig, linear unabhängig oder erzeugend sind und ob sie eine Basis bilden.
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## Woche 10
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- Ich kenne die Begriffe Satz von Schwarz, Hesse-Matrix, Richtungsableitung, lokale/globale Extrema, Sattelpunkt, Extrema unter Nebenbedingungen, Lagrange´scher Multiplikator und deren wichtigste Eigenschaften.
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- Ich kenne die Begriffe Bild, Kern, algebraische und geometrische Vielfachheit, ähnliche Matrix, Diagonalisierbarkeit einer Matrix und deren wichtigste Eigenschaften.
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- Ich kann die Hesse-Matrix von Skalarfeldern bestimmen.
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- Ich kann die Richtungsableitung berechnen.
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- Ich kann lokale Extrema und Sattelpunkte bestimmen.
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- Ich kann Extrema einer Funktion unter Nebenbedingungen bestimmen.
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- Ich kann Bild und Kern einer linearen Abbildung berechnen.
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- Ich kann bestimmen, ob eine Matrix diagonalisierbar ist oder nicht und die Diagonalmatrix angeben.
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## Woche 11
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- Ich kenne die Begriffe Kurve, Spur, Geschwindigkeits-/Tangentenvektor, Beschleunigungsvektor, Tangenteneinheitsvektor, Hauptnormalenvektor, Binormalenvektor, Parametrisierung, Bogenlänge, Kurven-/Linienintegral, Vektorfeld und deren wichtigste Eigenschaften.
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- Ich kann die Spur einer parametrisierten Kurve in 2D skizzieren.
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- Ich kann ein Vektorfeld in 2D skizzieren.
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- Ich kann die Bogenlänge einer Kurve berechnen.
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- Ich kann Linienintegrale berechnen.
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## Woche 12
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- Ich kenne die Begriffe Laplace Operator, Divergenz und Rotation eines Vektorfeldes, quellenfrei, wirbelfrei, konservativ, Potential-/Gradientenfeld und deren wichtigste Eigenschaften.
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- Ich kann die Rotation und Divergenz von Vektorfeldern bestimmen.
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- Ich kann bestimmen, ob ein Vektorfeld quellen- bzw. wirbelfrei ist.
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## Probeprüfung Themen
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---
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#### **Aufgabe 1 – Aussagen zu Integralen**
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- Begriff und Eigenschaften von Integral, Stammfunktion, (linearer) Substitution (1 | 2)
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- Anwendung der linearen Substitution auf bestimmte/unbestimmte Integrale (1 | 2)
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- Beurteilung & Berechnung eines uneigentlichen Integrals (4)
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- Zusammenhang zwischen Flächeninhalt in 2 D und (Doppel-)Integral (7)
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---
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#### **Aufgabe 2 – Aussagen über eine Funktion**
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- Begriff Skalarfeld, Niveau-/Höhenlinien, Niveaumengen (8)
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- Bestimmung des Funktionswertes in einem Punkt (grundlegend, 8)
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- Interpretation des Graphen einer Funktion zweier Variablen im 3 D-Raum (8)
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---
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#### **Aufgabe 3 – Aussagen über ein Vektorfeld**
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- Rotation, Divergenz, konservative (= wirbel- & quellenfreie) Vektorfelder, Potentialfeld (12)
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- Prüfen, ob ∇ × **v** = 0 bzw. ∇·**v** = 0 und Existenz eines Potentials (12)
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---
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#### **Aufgabe 4 – Aussagen über zwei Matrizen**
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- Begriffe Matrix, symmetrisch, orthogonal, regulär (invertierbar), Bild (img) und Kern (ker) (4 | 5 | 6 | 7)
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- Bestimmen von Invertierbarkeit und Rang über Determinante oder Gauß-Verfahren (5 | 7)
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- Eigenschaften orthogonaler bzw. symmetrischer Matrizen (4 | 6)
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---
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#### **Aufgabe 5 – Weitere Aussagen zu Matrizen**
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- Produkte invertierbarer Matrizen und deren Invertierbarkeit (5)
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- Zusammenhang Matrixpotenz A³ – Eigenwerte λ³ (8)
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- Existenz unabhängiger Eigenvektoren zu verschiedenen Eigenwerten (8 | 10)
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---
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#### **Aufgabe 6 – Diverse Einzelaussagen**
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- Komplexe Zahlen: Konjugation, Potenzgleichungen, trigonometrische Form (1 | 2 | 3)
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- Gradient und seine Orthogonalität zu Höhenlinien (9 | 10 | 12)
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- Lineare Unabhängigkeit & Basisbegriffe in Vektorräumen (9)
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- Definition einer linearen Abbildung (Homogenität & Additivität) (9)
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---
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#### **Aufgabe 7 – Gemischte Rechenaufgaben**
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a) **z⁴ = 16 e^{iπ}** lösen
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- Potenz- & Wurzelziehen komplexer Zahlen in Exponential-/Trig- & arithmetischer Form (3)
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b) **A¹⁵** berechnen
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- Dreh- bzw. orthogonale 2×2-Matrizen erkennen, Potenzen mittels Diagonalisierung/Rotor-Eigenschaften bilden (6 | 8 | 10)
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c) Integrationsreihenfolge vertauschen
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- Mehrfachintegrale, Gebietsskizze, Reihenfolgenwechsel (7)
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d) Richtungsableitung an P(1, 2) in Richtung **v**
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- Gradient, Richtungsableitung, Normierung des Richtungsvektors (10)
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e) Skalares Kurvenintegral längs **γ(t)**
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- Bogenlänge-Parameter t, Kurvenintegral eines Skalarfeldes (11)
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---
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#### **Aufgabe 8 – Lokale Extrema & Sattelpunkte**
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- Partielle Ableitungen, Gradient = 0 setzen (9)
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- Hesse-Matrix, Eigenwerte-Kriterium für Extrema/Sattel (10)
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---
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#### **Aufgabe 9 – Flächenschwerpunkt eines Trapezes**
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- Flächenschwerpunkt mittels Doppelintegration berechnen (5)
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- Anwendung von Mehrfachintegralen auf Masse/Schwerpunkt (7)
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---
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#### **Aufgabe 10 – Diagonalisierung einer 3×3-Matrix**
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- Charakteristisches Polynom, Eigenwerte, Eigenvektoren (8)
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|
||||
- Bild, Kern, algebraische / geometrische Vielfachheit, Diagonalisierbarkeit & Transformationsmatrix (10)
|
||||
|
||||
|
||||
BIN
formulas/Formulas.pur
Normal file
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formulas/Formulas.pur
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565
formulas/Schemas.md
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formulas/Schemas.md
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|
||||
# WIP Konzept - Uneigentliches Integral
|
||||
# WIP Konzept - Komplexe Zahlen
|
||||
|
||||
# 10. 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:<br>• $D>0$ ⇒ Extremum<br>• $D<0$ ⇒ Sattel<br>• $D=0$ ⇒ 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**<br>• Eigenwerte von $H$<br>• 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 $x\to y$ <br> 2. **äußere Grenzen** = tiefster / höchster $y$-Wert des Gebiets (hier $l(a),u(b)$) <br> 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 $\Phi$.<br>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 $e^{i\theta_k}=\cos\theta_k+i\sin\theta_k$:<br> $$z_k=\sqrt[n]{R}\,(\cos\theta_k+i\sin\theta_k).$$ |
|
||||
| **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 $F(x)$ | | |
|
||||
| :-------------------------------------: | :---------------------------------------------------------------------------------------: | --- | --- |
|
||||
| $\sin x$ | $-\cos x$ | | |
|
||||
| $\cos x$ | $\sin x$ | | |
|
||||
| $\tan x = \frac{\sin x}{\cos x}$ | $\ln\bigl[\sec x\bigr]$ | | |
|
||||
| $\cot x = \frac{\cos x}{\sin x}$ | $-\ln\bigl[\csc x\bigr]$ | | |
|
||||
| $\sec x = \frac1{\cos x}$ | $\operatorname{artanh}[\sin x]$ | | |
|
||||
| $\csc x = \frac1{\sin x}$ | $-\operatorname{artanh}[\cos x]$ | | |
|
||||
| $\sec^2 x = 1+\tan^2 x$ | $\tan x$ | | |
|
||||
| $-\csc^2 x = -\bigl(1+\cot^2 x\bigr)$ | $\cot x$ | | |
|
||||
| $\sin^2 x$ | $\frac{1}{2}[x−sinx\,cosx]=\frac{1}{2}x− \frac{1}{4} sin(2x)$ | | |
|
||||
| $\cos^2 x$ | $\frac{1}{2}[x+sinx\,cosx]=\frac{1}{2}x+ \frac{1}{4} sin(2x)$ | | |
|
||||
| $\sin(kx)\cos(kx)$ | $-\displaystyle \frac{1}{4k}\cos(2kx)$ | | |
|
||||
| $\sin(kx)\cos(kx)$ | $\displaystyle \frac{1}{2k}\sin^2(kx)$ | | |
|
||||
| $\displaystyle \frac{\sin(ax)}{e^{bx}}$ | $\displaystyle \frac{a\,e^{bx} - a\cos(ax) - b\sin(ax)}{(a^2+b^2)\,e^{bx}}$ | | |
|
||||
| $\displaystyle \frac{\cos(ax)}{e^{bx}}$ | $\displaystyle \frac{a\sin(ax) - b\cos(ax) + b\,e^{bx}}{(a^2+b^2)\,e^{bx}}$ | | |
|
||||
| $\arcsin x$ | $x\arcsin x + \sqrt{1-x^2}$ | | |
|
||||
| $\arccos x$ | $x\arccos x - \sqrt{1-x^2}$ | | |
|
||||
| $\arctan x$ | $x\arctan x - \tfrac12\ln\bigl(1+x^2\bigr)$ | | |
|
||||
| $\arccot x$ | $x\arccot x + \tfrac12\ln\bigl(1+x^2\bigr)$ | | |
|
||||
| $\displaystyle \frac{1}{\sqrt{1-x^2}}$ | $\arcsin x$ | | |
|
||||
| $\displaystyle \frac{-1}{\sqrt{1-x^2}}$ | $\arccos x$ | | |
|
||||
| $\displaystyle \frac{1}{x^2+1}$ | $\arctan x$ | | |
|
||||
| $\displaystyle -\frac{1}{x^2+1}$ | $\arccot x$ | | |
|
||||
| $\displaystyle \frac{x^2}{x^2+1}$ | $x - \arctan x$ | | |
|
||||
| $\displaystyle \frac{1}{(x^2+1)^2}$ | $\displaystyle \tfrac12\Bigl(\frac{x}{x^2+1} + \arctan x\Bigr)$ | | |
|
||||
| $\sqrt{a^2 - x^2}$ | $\displaystyle \frac{a^2}{2}\arcsin\Bigl(\frac{x}{a}\Bigr) + \frac{x}{2}\sqrt{a^2 - x^2}$ | | |
|
||||
| $\displaystyle \frac{1}{ax^2+bx+c}$ | $\frac{2}{\sqrt{4ac - b^2}}\;\arctan\!\Bigl(\frac{2ax + b}{\sqrt{4ac - b^2}}\Bigr)$ | | |
|
||||
|
||||
### 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:
|
||||
> $$\int{\frac{1}{x}}2x\cdot u \, \, du \longrightarrow \int{2u} \,\,du$$
|
||||
54
notes/Komplexe Zahlen.md
Normal file
54
notes/Komplexe Zahlen.md
Normal file
@ -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
|
||||
|
||||
52
notes/Matrizen.md
Normal file
52
notes/Matrizen.md
Normal file
@ -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**: $AB \neq BA$
|
||||
6
notes/Prompts.md
Normal file
6
notes/Prompts.md
Normal file
@ -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)
|
||||
107
notes/Vektorfelder.md
Normal file
107
notes/Vektorfelder.md
Normal file
@ -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 $F=0$.
|
||||
27
notes/Zusammenfassung.md
Normal file
27
notes/Zusammenfassung.md
Normal file
@ -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
|
||||
|
||||
11
pyproject.toml
Normal file
11
pyproject.toml
Normal file
@ -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",
|
||||
]
|
||||
392
uv.lock
generated
Normal file
392
uv.lock
generated
Normal file
@ -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" },
|
||||
]
|
||||
|
||||
[package.metadata]
|
||||
requires-dist = [
|
||||
{ name = "imageio", specifier = ">=2.37.0" },
|
||||
{ name = "matplotlib", specifier = ">=3.10.3" },
|
||||
{ name = "numpy", specifier = ">=2.2.5" },
|
||||
{ name = "pandas", specifier = ">=2.2.3" },
|
||||
{ name = "plotly", specifier = ">=6.1.0" },
|
||||
]
|
||||
|
||||
[[package]]
|
||||
name = "contourpy"
|
||||
version = "1.3.2"
|
||||
source = { registry = "https://pypi.org/simple" }
|
||||
dependencies = [
|
||||
{ name = "numpy" },
|
||||
]
|
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
]
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||||
|
||||
[[package]]
|
||||
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|
||||
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||||
source = { registry = "https://pypi.org/simple" }
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||||
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||||
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||||
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||||
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||||
|
||||
[[package]]
|
||||
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|
||||
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||||
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||||
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Reference in New Issue
Block a user