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img/elektronik_2/Kombiniertes_Kennlinienfeld_Transistor_2.svg
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img/elektronik_2/Kombiniertes_Kennlinienfeld_Transistor_2.svg
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<path
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x="385"
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<text
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xml:space="preserve"
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<g
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<text
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xml:space="preserve"><tspan
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<text
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<text
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<text
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<text
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xml:space="preserve"
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xml:space="preserve"
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style="font-size:12px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;text-align:start;line-height:100%;writing-mode:lr-tb;text-anchor:start;fill:#800080;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;display:inline;font-family:DejaVu Serif;-inkscape-font-specification:DejaVu Serif"
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xml:space="preserve"
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img/linalg_2/2026-02-23-211625_hyprshot.png
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@ -159,26 +159,6 @@ Definition
|
|||||||
== Folgen, Reihen & Grenzwerte
|
== Folgen, Reihen & Grenzwerte
|
||||||
|
|
||||||
|
|
||||||
== Integralrechnung
|
|
||||||
|
|
||||||
=== Standard-Integrale
|
|
||||||
#table(columns: (1fr, 0.4fr),
|
|
||||||
[$integral m dot "dx" eq m dot x plus q$], [Gilt für alle $m in RR$],
|
|
||||||
[$integral x^p dot "dx" eq frac(1, 1 plus p) dot x^(p plus 1) plus c$], [Gilt für alle $p in RR \\ {minus 1}$],
|
|
||||||
[$integral a^x dot "dx" eq frac(1, ln(a)) dot a^x plus c$], [Gilt für alle $a in attach(RR, tr: plus) \\ {1}$],
|
|
||||||
[$integral e^x dot "dx" eq frac(1, ln(e)) dot e^x plus c eq frac(1, 1) dot e^x plus c eq e^x + c$], [Es gilt],
|
|
||||||
[$integral frac(1, x) dot "dx" eq ln(abs(x)) plus c $], [Es gilt],
|
|
||||||
// [$attach(integral, tr: x_E, br: x_0) frac(1, x) dot "dx" eq ln(abs(x_E)) minus ln(abs(x_0)) eq ln(frac(abs(x_E), abs(x_0))) eq ln(abs(frac(x_E, x_0))) eq ln(frac(x_E, x_0))$], [Es gilt für $x_0, x_E in RR$ mit $"sgn"(x_E) eq "sgn"(x_0)$],
|
|
||||||
[$integral^(x_E)_(x_0) frac(1, x) dot "dx" eq ln(abs(x_E)) minus ln(abs(x_0)) eq ln(frac(abs(x_E), abs(x_0))) eq ln(abs(frac(x_E, x_0))) eq ln(frac(x_E, x_0))$], [Es gilt für $x_0, x_E in RR$ mit $"sgn"(x_E) eq "sgn"(x_0)$],
|
|
||||||
[$integral (m dot x plus q)^p dot "dx" eq frac(1, m dot (p plus 1)) dot (m dot x plus q)^(p plus 1) plus c$], [],
|
|
||||||
[$integral a^(m dot x plus q) dot "dx" eq frac(1, m dot ln(a)) dot a^(m dot x plus q) plus c$], [],
|
|
||||||
[$integral y_0 dot a^frac(x minus x_0, sum) dot "dx" eq frac(sum, ln(a)) dot y_0 dot a^frac(x minus x_0, sum) plus c$], [],
|
|
||||||
[$integral A dot sin(omega dot t plus phi) dot "dx" eq minus frac(A, omega) dot cos(omega dot t plus phi) plus c$], [],
|
|
||||||
[$integral cos(x) dot "dx" eq sin(x)$], [],
|
|
||||||
[$integral sin(x) dot "dx" eq minus cos(x)$], [],
|
|
||||||
[$integral frac(1, 1 plus x^2) dot "dx" eq minus arctan(x)$], [],
|
|
||||||
[$integral frac(1, root(, 1 minus x^2)) dot "dx" eq minus arcsin(x)$], [],
|
|
||||||
)
|
|
||||||
|
|
||||||
|
|
||||||
// }}}
|
// }}}
|
||||||
|
|||||||
@ -0,0 +1,24 @@
|
|||||||
|
== Integralrechnung
|
||||||
|
|
||||||
|
=== Standard-Integrale
|
||||||
|
#table(columns: (1fr, 0.4fr),
|
||||||
|
[$integral m dot "dx" eq m dot x plus q$], [Gilt für alle $m in RR$],
|
||||||
|
[$integral x^p dot "dx" eq frac(1, 1 plus p) dot x^(p plus 1) plus c$], [Gilt für alle $p in RR \\ {minus 1}$],
|
||||||
|
[$integral a^x dot "dx" eq frac(1, ln(a)) dot a^x plus c$], [Gilt für alle $a in attach(RR, tr: plus) \\ {1}$],
|
||||||
|
[$integral a^(b dot x) dot "dx" eq frac(1, b dot ln(a)) dot a^x plus c$], [Gilt für alle $a in attach(RR, tr: plus) \\ {1}$],
|
||||||
|
[$integral e^x dot "dx" eq frac(1, ln(e)) dot e^x plus c eq frac(1, 1) dot e^x plus c eq e^x + c$], [Es gilt],
|
||||||
|
[$integral frac(1, x) dot "dx" eq ln(abs(x)) plus c $], [Es gilt],
|
||||||
|
[$integral frac(1, a x plus b) dot "dx" eq frac(1, a) dot ln(abs(a x plus b)) plus c $], [Es gilt],
|
||||||
|
// [$attach(integral, tr: x_E, br: x_0) frac(1, x) dot "dx" eq ln(abs(x_E)) minus ln(abs(x_0)) eq ln(frac(abs(x_E), abs(x_0))) eq ln(abs(frac(x_E, x_0))) eq ln(frac(x_E, x_0))$], [Es gilt für $x_0, x_E in RR$ mit $"sgn"(x_E) eq "sgn"(x_0)$],
|
||||||
|
[$integral^(x_E)_(x_0) frac(1, x) dot "dx" eq ln(abs(x_E)) minus ln(abs(x_0)) eq ln(frac(abs(x_E), abs(x_0))) eq ln(abs(frac(x_E, x_0))) eq ln(frac(x_E, x_0))$], [Es gilt für $x_0, x_E in RR$ mit $"sgn"(x_E) eq "sgn"(x_0)$],
|
||||||
|
[$integral (m dot x plus q)^p dot "dx" eq frac(1, m dot (p plus 1)) dot (m dot x plus q)^(p plus 1) plus c$], [],
|
||||||
|
[$integral a^(m dot x plus q) dot "dx" eq frac(1, m dot ln(a)) dot a^(m dot x plus q) plus c$], [],
|
||||||
|
[$integral y_0 dot a^frac(x minus x_0, sum) dot "dx" eq frac(sum, ln(a)) dot y_0 dot a^frac(x minus x_0, sum) plus c$], [],
|
||||||
|
[$integral A dot sin(omega dot t plus phi) dot "dx" eq minus frac(A, omega) dot cos(omega dot t plus phi) plus c$], [],
|
||||||
|
[$integral cos(x) dot "dx" eq sin(x)$], [],
|
||||||
|
[$integral sin(x) dot "dx" eq minus cos(x)$], [],
|
||||||
|
[$integral frac(1, 1 plus x^2) dot "dx" eq minus arctan(x)$], [],
|
||||||
|
[$integral frac(1, root(, 1 minus x^2)) dot "dx" eq minus arcsin(x)$], [],
|
||||||
|
)
|
||||||
|
|
||||||
|
lineare Kettenregel (innere Ableitung)
|
||||||
@ -213,6 +213,7 @@ Falls in einer gui umgebung gearbeitet wird gibt es dafür schaltflächen, aber
|
|||||||
[$abs(5)$], [```typ $abs(6)$ ```],
|
[$abs(5)$], [```typ $abs(6)$ ```],
|
||||||
[$ln(x)$], [```typ $ln(x)$ ```],
|
[$ln(x)$], [```typ $ln(x)$ ```],
|
||||||
[$integral$], [```typ $integral$ ```],
|
[$integral$], [```typ $integral$ ```],
|
||||||
|
[$bb(1)$], [```typ $bb(1)$ ```],
|
||||||
)
|
)
|
||||||
], [
|
], [
|
||||||
#table(columns: (1fr, 1fr),
|
#table(columns: (1fr, 1fr),
|
||||||
@ -265,6 +266,48 @@ table.cell(fill: lime)[lime],
|
|||||||
)
|
)
|
||||||
Alternativ kann auch einfach `rgb("#001f3f")` verwendet werden.
|
Alternativ kann auch einfach `rgb("#001f3f")` verwendet werden.
|
||||||
|
|
||||||
|
== Mathe
|
||||||
|
#table(columns: (2fr, 1fr),
|
||||||
|
[```typ $ underbrace(0 + 1 + dots.c + n, n + 1 "numbers") $```], [$ underbrace(0 + 1 + dots.c + n, n + 1 "numbers") $],
|
||||||
|
[```typ
|
||||||
|
$ f(x, y) := cases(
|
||||||
|
1 "if" (x dot y)/2 <= 0,
|
||||||
|
2 "if" x "is even",
|
||||||
|
3 "if" x in NN,
|
||||||
|
4 "else",
|
||||||
|
) $
|
||||||
|
```], [
|
||||||
|
$ f(x, y) := cases(
|
||||||
|
1 "if" (x dot y)/2 <= 0,
|
||||||
|
2 "if" x "is even",
|
||||||
|
3 "if" x in NN,
|
||||||
|
4 "else",
|
||||||
|
) $
|
||||||
|
],
|
||||||
|
[```typ
|
||||||
|
$ vec(a, b, c) dot vec(1, 2, 3)
|
||||||
|
= a + 2b + 3c $
|
||||||
|
```], [
|
||||||
|
$ vec(a, b, c) dot vec(1, 2, 3)
|
||||||
|
= a + 2b + 3c $
|
||||||
|
],
|
||||||
|
[```typ
|
||||||
|
$ mat(delim: "[",
|
||||||
|
1, 2, ..., 10;
|
||||||
|
2, 2, ..., 10;
|
||||||
|
dots.v, dots.v, dots.down, dots.v;
|
||||||
|
10, 10, ..., 10;
|
||||||
|
) $
|
||||||
|
```], [
|
||||||
|
$ mat(delim: "[",
|
||||||
|
1, 2, ..., 10;
|
||||||
|
2, 2, ..., 10;
|
||||||
|
dots.v, dots.v, dots.down, dots.v;
|
||||||
|
10, 10, ..., 10;
|
||||||
|
) $
|
||||||
|
],
|
||||||
|
)
|
||||||
|
|
||||||
== cetz
|
== cetz
|
||||||
importieren:
|
importieren:
|
||||||
```typ
|
```typ
|
||||||
@ -331,9 +374,7 @@ import cetz.draw: *
|
|||||||
content((0, 0), [1])
|
content((0, 0), [1])
|
||||||
content((0, -1), [Median = 4])
|
content((0, -1), [Median = 4])
|
||||||
content((0, -2), [$integral x dot "dt"$], anchor: "north-west")
|
content((0, -2), [$integral x dot "dt"$], anchor: "north-west")
|
||||||
})], [```typ
|
})],
|
||||||
|
|
||||||
```],[]
|
|
||||||
)
|
)
|
||||||
|
|
||||||
beispiel code
|
beispiel code
|
||||||
@ -342,16 +383,9 @@ beispiel code
|
|||||||
import cetz.draw: *
|
import cetz.draw: *
|
||||||
scale(0.5)
|
scale(0.5)
|
||||||
rect((0, 0), (5, 1), fill: blue)
|
rect((0, 0), (5, 1), fill: blue)
|
||||||
rect((0, 0), (5, 1), fill: rgb(0, 0, 255, 60))
|
|
||||||
grid((0, 0), (5, 1), step: 1)
|
|
||||||
circle((3, 5))
|
circle((3, 5))
|
||||||
circle((3, 5), radius: (1, 0.5))
|
|
||||||
line((0, 0), (3, 2))
|
line((0, 0), (3, 2))
|
||||||
line((6, 0), (6, 2), stroke: (dash: "dashed"))
|
|
||||||
line((3, 0), (6, 0), mark: (symbol: ">"), fill: blue, stroke: blue)
|
|
||||||
line((2.5, 0), (2.5, -0.8), mark: (end: ">"), fill: blue, stroke: blue)
|
|
||||||
content((0.5, 0.5), [1])
|
content((0.5, 0.5), [1])
|
||||||
content((2.5, -1), [Median = 4])
|
|
||||||
})
|
})
|
||||||
```
|
```
|
||||||
|
|
||||||
|
|||||||
@ -74,3 +74,41 @@ cv::circle(img, cv::Point(10, 10), 20, cv::Scalar(0, 0, 128), 30);
|
|||||||
[cv::Scalar], [Farbe in BRG],
|
[cv::Scalar], [Farbe in BRG],
|
||||||
[30], [Linienbreite (-1 für geffülter Kreis],
|
[30], [Linienbreite (-1 für geffülter Kreis],
|
||||||
)
|
)
|
||||||
|
|
||||||
|
=== Bild einlesen
|
||||||
|
#table(columns: 1fr, [```cpp
|
||||||
|
std::string filename = "mond.png";
|
||||||
|
cv::Mat img = cv::imread(filename, cv::IMREAD_ANYCOLOR);
|
||||||
|
```])
|
||||||
|
|
||||||
|
=== Bild zu einem Graubild konvertieren
|
||||||
|
#table(columns: 1fr, [```cpp
|
||||||
|
cv::Mat img_grayray;
|
||||||
|
cv::cvtColor(img, img_gray, cv::COLOR_BGR2GRAY);
|
||||||
|
```])
|
||||||
|
|
||||||
|
=== Bild von Grau zu Farb
|
||||||
|
#table(columns: 1fr, [```cpp
|
||||||
|
cv::Mat img_color;
|
||||||
|
cv::cvtColor(img_gray, imgNotes, cv::COLOR_GRAY2BGR);
|
||||||
|
```])
|
||||||
|
|
||||||
|
=== Maximale und Minimale Helligkeit finden
|
||||||
|
#table(columns: 1fr, [```cpp
|
||||||
|
double minGray, maxGray;
|
||||||
|
cv::Point minLoc, maxLoc;
|
||||||
|
cv::minMaxLoc(imgGray, &minGray, &maxGray, &minLoc, &maxLoc);
|
||||||
|
|
||||||
|
// auf x oder y zugreifen:
|
||||||
|
std::cout << maxLoc.x << std::endl;
|
||||||
|
std::cout << maxLoc.y << std::endl;
|
||||||
|
```])
|
||||||
|
|
||||||
|
=== Spalten/Reihen umfärben
|
||||||
|
#table(columns: 1fr, [```cpp
|
||||||
|
imgGray.col(maxLoc.x).setTo(cv::Scalar(32));
|
||||||
|
imgGray.row(maxLoc.y).setTo(cv::Scalar(32));
|
||||||
|
```])
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|||||||
@ -1,10 +1,22 @@
|
|||||||
#import "@preview/zap:0.4.0"
|
#import "@preview/zap:0.4.0"
|
||||||
|
|
||||||
== Elektronik 2
|
== Elektronik 2
|
||||||
=== Feldeffekttransistor
|
=== Vierquadrantenkennlinienfeld
|
||||||
|
#image("../img/elektronik_2/Kombiniertes_Kennlinienfeld_Transistor_2.svg")
|
||||||
|
|
||||||
|
#table(columns: (120pt, 1fr),
|
||||||
|
[Ausgangskennlinienfeld \ (oben rechts)],
|
||||||
|
[Sobald $U_"CE"$ einen kleinen Schwellenwert überschreitet, bleibt der Strom $I_C$ fast konstant, egal wie weit man die Spannung erhöht. Der Transistor wirkt hier wie eine stromgesteuerte Stromquelle.],
|
||||||
|
[Stromsteuerkennlinie \ (oben links)], [#grid(columns: (100pt, 1fr),
|
||||||
|
[$ B eq frac(I_c, I_B) $], [Ein kleiner Strom an der Basis steuert also linear einen viel größeren Strom am Kollektor.])],
|
||||||
|
[Eingangskennlinie \ (unten links)],
|
||||||
|
[Da die Basis-Emitter-Strecke physikalisch eine Diode ist, fließt erst ab einer Schwellenspannung (bei Silizium ca. 0,6V bis 0,7V) ein nennenswerter Strom. Danach steigt der Strom exponentiell an.],
|
||||||
|
[Spannungsrückwirkung \ (unten rechts)],
|
||||||
|
[Im Idealfall sollte die Ausgangsspannung keinen Einfluss auf den Eingang haben. In der Realität sieht man eine minimale Verschiebung der Kennlinien (der Early-Effekt spielt hier eine Rolle), was als "Rückwirkung" bezeichnet wird.],
|
||||||
|
)
|
||||||
|
|
||||||
=== Logikschaltungen
|
=== Logikschaltungen
|
||||||
=== Sinussignale
|
=== Sinussignale
|
||||||
=== Sinussignale
|
|
||||||
=== Passive Zweipole
|
=== Passive Zweipole
|
||||||
=== Aktive Zweipole (Versorgungen)
|
=== Aktive Zweipole (Versorgungen)
|
||||||
=== Passive Vierpole
|
=== Passive Vierpole
|
||||||
|
|||||||
@ -0,0 +1,156 @@
|
|||||||
|
== Matrix
|
||||||
|
Matrix wird mit *Zeile* X *Spalten* beschrieben. Das ist eine $2 crossmark 3 "-Matrix:" A eq mat(delim: "[", 2, 1, 3; 7, 5, 2)$
|
||||||
|
|
||||||
|
=== Addition und Subtraktion
|
||||||
|
#grid(columns: (1fr, 1fr), gutter: 40pt, [
|
||||||
|
$ A plus B :eq mat(delim: "[",
|
||||||
|
A^1_1 plus B^1_1, A^1_2 plus B^1_2, dots.h, A^1_n plus B^1_n;
|
||||||
|
A^2_1 plus B^2_1, A^2_2 plus B^2_2, dots.h, A^2_n plus B^2_n;
|
||||||
|
dots.v, dots.v, dots.down, dots.v;
|
||||||
|
A^m_1 plus B^m_1, A^m_2 plus B^m_2, dots.h, A^m_n plus B^m_n;
|
||||||
|
) $
|
||||||
|
], [
|
||||||
|
$ A minus B :eq mat(delim: "[",
|
||||||
|
A^1_1 minus B^1_1, A^1_2 minus B^1_2, dots.h, A^1_n minus B^1_n;
|
||||||
|
A^2_1 minus B^2_1, A^2_2 minus B^2_2, dots.h, A^2_n minus B^2_n;
|
||||||
|
dots.v, dots.v, dots.down, dots.v;
|
||||||
|
A^m_1 minus B^m_1, A^m_2 minus B^m_2, dots.h, A^m_n minus B^m_n;
|
||||||
|
) $
|
||||||
|
])
|
||||||
|
|
||||||
|
=== Multiplikation
|
||||||
|
$ a dot A :eq mat(delim: "[",
|
||||||
|
a dot A^1_1, a dot A^1_2, dots.h, a dot A^1_n;
|
||||||
|
a dot A^2_1, a dot A^2_2, dots.h, a dot A^2_n;
|
||||||
|
dots.v, dots.v, dots.down, dots.v;
|
||||||
|
a dot A^m_1, a dot A^m_2, dots.h, a dot A^m_n;
|
||||||
|
) $
|
||||||
|
|
||||||
|
=== Transposition
|
||||||
|
#grid(columns: (1fr, 1fr), gutter: 40pt, [
|
||||||
|
$ A^T :eq mat(delim: "[",
|
||||||
|
A^1_1, A^2_1, dots.h, A^m_1;
|
||||||
|
A^1_2, A^2_2, dots.h, A^m_2;
|
||||||
|
dots.v, dots.v, dots.down, dots.v;
|
||||||
|
A^1_n, A^2_n, dots.h, A^m_n;
|
||||||
|
) $
|
||||||
|
], [
|
||||||
|
Bsp.:
|
||||||
|
$ mat(delim: "[",
|
||||||
|
2, -1
|
||||||
|
)^T eq
|
||||||
|
mat(delim: "[",
|
||||||
|
2; -1
|
||||||
|
) $
|
||||||
|
$ mat(delim: "[",
|
||||||
|
a, b, c; d, e, f
|
||||||
|
)^T eq
|
||||||
|
mat(delim: "[",
|
||||||
|
a, d; b, e; c, f
|
||||||
|
) $
|
||||||
|
])
|
||||||
|
|
||||||
|
=== Matrix-Multiplikation
|
||||||
|
$ M eq A dot B $
|
||||||
|
|
||||||
|
#image("../img/linalg_2/2026-02-23-211625_hyprshot.png")
|
||||||
|
// $
|
||||||
|
// #let hi(c, body) = box(fill: c, inset: 2pt, radius: 2pt)[#body]
|
||||||
|
//
|
||||||
|
// mat(delim: "[",
|
||||||
|
// #box(fill: yellow, inset: 2pt, radius: 2pt)[1],
|
||||||
|
// #box(fill: green, inset: 2pt, radius: 2pt)[2],;
|
||||||
|
// #box(fill: aqua, inset: 2pt, radius: 2pt)[3],
|
||||||
|
// #box(fill: orange, inset: 2pt, radius: 2pt)[4]
|
||||||
|
// )
|
||||||
|
// dot
|
||||||
|
// mat(delim: "[",
|
||||||
|
// #box(fill: red, inset: 2pt, radius: 2pt)[5],
|
||||||
|
// #box(fill: purple, inset: 2pt, radius: 2pt)[6],;
|
||||||
|
// #box(fill: blue, inset: 2pt, radius: 2pt)[7],
|
||||||
|
// #box(fill: lime, inset: 2pt, radius: 2pt)[8]
|
||||||
|
// )
|
||||||
|
// eq
|
||||||
|
// mat(delim: "[",
|
||||||
|
// #hi(yellow, 1) dot #hi(orange, 5) plus #hi(green, 2) dot #hi(blue, 7),
|
||||||
|
// #hi(yellow, 1) dot #hi(orange, 5) plus #hi(green, 2) dot #hi(blue, 7),
|
||||||
|
// )
|
||||||
|
// $
|
||||||
|
|
||||||
|
- $mat(delim: "[",
|
||||||
|
1, 2; 3, 4
|
||||||
|
) dot mat(delim: "[",
|
||||||
|
5; 6
|
||||||
|
) eq
|
||||||
|
mat(delim: "[",
|
||||||
|
1 dot 5 plus 2 dot 6; 3 dot 5 plus 4 dot 6
|
||||||
|
) eq mat(delim: "[",
|
||||||
|
17; 39
|
||||||
|
) $
|
||||||
|
|
||||||
|
- $mat(delim: "[",
|
||||||
|
1, 2; 3, 4
|
||||||
|
) dot mat(delim: "[",
|
||||||
|
5, 6
|
||||||
|
) eq
|
||||||
|
mat(delim: "[",
|
||||||
|
1 dot 5 plus 3 dot 6, 2 dot 5 plus 4 dot 6
|
||||||
|
) eq mat(delim: "[",
|
||||||
|
23, 34
|
||||||
|
) $
|
||||||
|
|
||||||
|
- $mat(delim: "[",
|
||||||
|
1, 2
|
||||||
|
) dot mat(delim: "[",
|
||||||
|
3; 4
|
||||||
|
) eq
|
||||||
|
mat(delim: "[",
|
||||||
|
1 dot 3 plus 2 dot 4
|
||||||
|
) eq mat(delim: "[",
|
||||||
|
11
|
||||||
|
) $
|
||||||
|
|
||||||
|
- $mat(delim: "[",
|
||||||
|
1; 2
|
||||||
|
) dot mat(delim: "[",
|
||||||
|
3, 4
|
||||||
|
) eq
|
||||||
|
mat(delim: "[",
|
||||||
|
1 dot 3, 1 dot 4; 2 dot 3, 2 dot 4
|
||||||
|
) eq mat(delim: "[",
|
||||||
|
3, 4; 6, 8
|
||||||
|
) $
|
||||||
|
|
||||||
|
=== Symmetrische Matrix
|
||||||
|
ist: #box(stroke: 1pt + red, inset: (x: 1em, y: 0.5em), [$A^T eq A$]) dann ist die Matrix Symmetrische. \
|
||||||
|
z.b.: $mat(delim: "[",
|
||||||
|
1, 2; 2, 3
|
||||||
|
)$
|
||||||
|
|
||||||
|
=== Schiefsymmetrische Matrix
|
||||||
|
ist: #box(stroke: 1pt + red, inset: (x: 1em, y: 0.5em), [$A^T eq minus A$]) dann ist die Matrix Symmetrische. \
|
||||||
|
z.b.: $mat(delim: "[",
|
||||||
|
1, -2; 2, 0
|
||||||
|
)$
|
||||||
|
oder
|
||||||
|
$mat(delim: "[",
|
||||||
|
0, -1, 2; 1, 0, -3; -2, 3, 0
|
||||||
|
)$
|
||||||
|
|
||||||
|
=== Nullmatrix
|
||||||
|
$0 eq mat(delim: "[",
|
||||||
|
0, 0, dots.h, 0;
|
||||||
|
0, 0, dots.h, 0;
|
||||||
|
dots.v, dots.v, dots.down, dots.v;
|
||||||
|
0, 0, dots.h, 0;
|
||||||
|
)$
|
||||||
|
|
||||||
|
=== Einheitsmatrix
|
||||||
|
$bb(1) eq mat(delim: "[",
|
||||||
|
1, 0, dots.h, 0;
|
||||||
|
0, 1, dots.h, 0;
|
||||||
|
dots.v, dots.v, dots.down, dots.v;
|
||||||
|
0, 0, dots.h, 1;
|
||||||
|
)$
|
||||||
|
|
||||||
|
|
||||||
119
src/physik_2.typ
119
src/physik_2.typ
@ -87,7 +87,126 @@ $ frac(gamma_E, gamma_B) eq frac(1, epsilon_0 dot mu_0) eq c^2 $
|
|||||||
$gamma_E eq 1$ \
|
$gamma_E eq 1$ \
|
||||||
$gamma_B eq 10^(-7) frac(N, A^2)$
|
$gamma_B eq 10^(-7) frac(N, A^2)$
|
||||||
|
|
||||||
|
=== E-Feld
|
||||||
|
$bold(F)_E eq Q dot E$ #h(20pt) $[E] eq frac(N, C)$
|
||||||
|
|
||||||
|
=== B-Feld
|
||||||
|
$bold(F)_B eq Q dot v crossmark B$ #h(20pt) $[B] eq frac(N, A dot m) eq T "(Tesla)"$
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
#pagebreak()
|
||||||
|
#pagebreak()
|
||||||
|
#pagebreak()
|
||||||
|
|
||||||
|
|
||||||
== Schwingungen und Wellen
|
== Schwingungen und Wellen
|
||||||
|
Periodendauer $T [s]$ \
|
||||||
|
Frequenz $f eq frac(1, T)$ #h(20pt) $[f] eq "Hz"$
|
||||||
|
|
||||||
|
|
||||||
|
=== Ort-Zeit Funktion
|
||||||
|
$x(t) eq A dot cos(omega t plus delta)$ \
|
||||||
|
// Phase $omega t dot delta$ \
|
||||||
|
// Phasenwinkel $delta$ Wert von $delta$ bei $t eq 0$ \
|
||||||
|
// Kreisfrequenz $omega$ \
|
||||||
|
// $x(t) eq x(t plus T)$ wir setzen $delta eq 0$ \
|
||||||
|
// $A cos(omega t) &eq A dot cos(omega (t plus T)) \
|
||||||
|
// &eq A dot cos(omega t plus omega T)$ \
|
||||||
|
// da $cos 2 pi$ periodisch $omega T eq 2 pi$
|
||||||
|
|
||||||
|
// $omega eq frac(2 pi, T) eq frac(2 pi, frac(1, f)) eq 2 pi f$ \
|
||||||
|
// $[omega] eq frac(1, s)$
|
||||||
|
$omega eq frac(2 pi, T) eq 2 pi f$ #h(20pt) $[omega] eq frac(1, s)$
|
||||||
|
|
||||||
|
Geschwindigkeit: \
|
||||||
|
// $V_x &eq frac(d x (t), d t) \
|
||||||
|
// &eq - A omega sin(omega t plus delta)$
|
||||||
|
$V_x eq - A omega sin(omega t plus delta)$
|
||||||
|
|
||||||
|
Beschleunigung: \
|
||||||
|
// $a_x &eq frac(d v_x, d t) eq frac(d^2 x(t), d t^2) \
|
||||||
|
// &eq - A omega^2 cos(omega t plus delta) \
|
||||||
|
// &eq minus omega^2 dot x(t)$
|
||||||
|
$a_x &eq - A omega^2 cos(omega t plus delta) \
|
||||||
|
&eq minus omega^2 dot x(t)$
|
||||||
|
|
||||||
|
// Amplitude von:\
|
||||||
|
// $v_x : A omega$ \
|
||||||
|
// $a_x : A omega^2$ \
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
=== Newtonssches Gesetz (Bewegungsgleichung)
|
||||||
|
// $m dot a eq sum F$ \
|
||||||
|
// $m dot a_x eq minus k dot x arrow a_x eq minus frac(k, m) dot x$ \
|
||||||
|
// oder \
|
||||||
|
// $m dot a_x eq minus omega^2 dot x dot m$ \
|
||||||
|
// gleichsetzen der gleichungen: \
|
||||||
|
// $minus k dot x eq minus m dot omega^2 dot x$ \
|
||||||
|
// $k eq m dot omega^2$ \
|
||||||
|
// $omega eq sqrt(frac(k, m))$ \
|
||||||
|
// mit: \
|
||||||
|
// $omega eq frac(s pi, T)$ $arrow$ $T eq 2 pi sqrt(frac(m,k))$ \
|
||||||
|
$omega eq sqrt(frac(k, m))$ \
|
||||||
|
|
||||||
|
Dehnung der Feder: \
|
||||||
|
$k dot Delta l eq m dot g$
|
||||||
|
|
||||||
|
// = pp S 21
|
||||||
|
// $F eq minus k dot x$ \
|
||||||
|
// $E_"pot" eq minus integral_0^x F d s eq minus integral_0^x (minus k s ) d s eq frac(1, 2) k x^2$ $(c eq 0)$
|
||||||
|
//
|
||||||
|
// $E_"pot" (x eq 0) eq 0$ Bei Gleichgewichtslage \
|
||||||
|
// $E_"pot" (x = A) eq frac(1, 2) k A^2)$ Maximeirt ...
|
||||||
|
// $E_"ges" &eq E_"pot" plus E_"kin" eq frac(1, 2) k x^2 plus frac(1, 2) m v^2 \
|
||||||
|
// &eq frac(1, 2) k (A cos(omega t plus delta))^2 plus frac(1, 2) m (minus A omega sin(omega t plus delta))^2 \
|
||||||
|
// &eq frac(1, 2) k A^2 cos^2(omega t plus delta) plus frac(1, 2) m A^2 omega^2 sin^2(omega t plus delta) \
|
||||||
|
// &eq frac(1, 2) k A^2 underbrace((cos^2(omega t plus delta) plus sin^2omega t plus delta), eq 1) \
|
||||||
|
// &eq frac(1, 2) k A^2 $
|
||||||
|
|
||||||
|
=== Gleichgewichtslage
|
||||||
|
$E_"pot" (x) eq 0$ Bei Gleichgewichtslage \
|
||||||
|
|
||||||
|
|
||||||
|
// = pp S 27
|
||||||
|
// Federkraft von m: \
|
||||||
|
// $F eq minus k y$ \
|
||||||
|
// Gewichtskraft von m: \
|
||||||
|
// $F eq m dot x$ \
|
||||||
|
// Newtown 2: \
|
||||||
|
// $m dot a_y eq minus k y plus m dot g$ \
|
||||||
|
// Einführung neue Koordinate: \
|
||||||
|
// $y^' eq y minus y_0$
|
||||||
|
//
|
||||||
|
//
|
||||||
|
// $sum f eq -(y^' plus y_0) dot k plus m dot g$ \
|
||||||
|
// mit $k y_0 eq m dot g$:
|
||||||
|
// $sum F eq - y^' dot k$
|
||||||
|
//
|
||||||
|
// $ y^'(t) eq A dot cos(omega t plus delta) $
|
||||||
|
// masse m schwingt um Gleichgewichtslage $y_0$
|
||||||
|
|||||||
Loading…
x
Reference in New Issue
Block a user