Zacharias/Analys-och-Linj-r-algibra
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

vault backup: 2026-01-26 11:53:05

Browse Source
This commit is contained in:
Zacharias Zellén
2026-01-26 11:53:05 +01:00
parent f253fe796d
commit 3084db1e72
2 changed files with 18 additions and 212 deletions
Download Patch File Download Diff File Expand all files Collapse all files

225
.obsidian/workspace.json vendored
View File

@@ -4,207 +4,11 @@
"type": "split", "type": "split",
"children": [ "children": [
{ {
"id": "277432b5491ac5a8", "id": "9b8c02200aa5b353",
"type": "tabs", "type": "tabs",
"children": [ "children": [
{ {
"id": "5d5c0fef64eecc2b", "id": "9ec7a3d1ef5d43cc",
"type": "leaf",
"state": {
"type": "markdown",
"state": {
"file": "Gräsvärde (1).md",
"mode": "source",
"source": false
},
"icon": "lucide-file",
"title": "Gräsvärde (1)"
}
},
{
"id": "ad6eb280b4b8718c",
"type": "leaf",
"state": {
"type": "markdown",
"state": {
"file": "Derivata.md",
"mode": "source",
"source": false
},
"icon": "lucide-file",
"title": "Derivata"
}
},
{
"id": "66704e0159322e3f",
"type": "leaf",
"state": {
"type": "markdown",
"state": {
"file": "Funktioner Forts.md",
"mode": "source",
"source": false
},
"icon": "lucide-file",
"title": "Funktioner Forts"
}
},
{
"id": "66f6bbd26d9be1ca",
"type": "leaf",
"state": {
"type": "markdown",
"state": {
"file": "ODE.md",
"mode": "source",
"source": false
},
"icon": "lucide-file",
"title": "ODE"
}
},
{
"id": "2de57915f68f42ea",
"type": "leaf",
"state": {
"type": "markdown",
"state": {
"file": "Maclaurin.md",
"mode": "source",
"source": false
},
"icon": "lucide-file",
"title": "Maclaurin"
}
},
{
"id": "54baa3edd65a7c5d",
"type": "leaf",
"state": {
"type": "markdown",
"state": {
"file": "Grafer.md",
"mode": "source",
"source": false
},
"icon": "lucide-file",
"title": "Grafer"
}
},
{
"id": "4eef5f8feb086f9e",
"type": "leaf",
"state": {
"type": "markdown",
"state": {
"file": "Primära Funktioner.md",
"mode": "source",
"source": false
},
"icon": "lucide-file",
"title": "Primära Funktioner"
}
},
{
"id": "be47d5ede3a9176b",
"type": "leaf",
"state": {
"type": "markdown",
"state": {
"file": "Derivata.md",
"mode": "source",
"source": false
},
"icon": "lucide-file",
"title": "Derivata"
}
},
{
"id": "68837a4aa2729e39",
"type": "leaf",
"state": {
"type": "markdown",
"state": {
"file": "Grafer.md",
"mode": "source",
"source": false
},
"icon": "lucide-file",
"title": "Grafer"
}
},
{
"id": "6226fb790fca274b",
"type": "leaf",
"state": {
"type": "markdown",
"state": {
"file": "Tenta Example.md",
"mode": "source",
"source": false
},
"icon": "lucide-file",
"title": "Tenta Example"
}
},
{
"id": "46f7f5c1bc5d29b2",
"type": "leaf",
"state": {
"type": "markdown",
"state": {
"file": "Differential.md",
"mode": "source",
"source": false
},
"icon": "lucide-file",
"title": "Differential"
}
},
{
"id": "d693f26373e8dc42",
"type": "leaf",
"state": {
"type": "markdown",
"state": {
"file": "Primära Funktioner.md",
"mode": "source",
"source": false
},
"icon": "lucide-file",
"title": "Primära Funktioner"
}
},
{
"id": "1b73858e17021be2",
"type": "leaf",
"state": {
"type": "markdown",
"state": {
"file": "Integraler.md",
"mode": "source",
"source": false
},
"icon": "lucide-file",
"title": "Integraler"
}
},
{
"id": "76c8d943958d45bf",
"type": "leaf",
"state": {
"type": "markdown",
"state": {
"file": "Funktioner.md",
"mode": "source",
"source": false
},
"icon": "lucide-file",
"title": "Funktioner"
}
},
{
"id": "30539ff6e6b3eb6a",
"type": "leaf", "type": "leaf",
"state": { "state": {
"type": "markdown", "type": "markdown",
@@ -217,8 +21,7 @@
"title": "Vektorer" "title": "Vektorer"
} }
} }
], ]
"currentTab": 14
} }
], ],
"direction": "vertical" "direction": "vertical"
@@ -250,7 +53,7 @@
"state": { "state": {
"type": "search", "type": "search",
"state": { "state": {
"query": "\\text{p", "query": "def",
"matchingCase": false, "matchingCase": false,
"explainSearch": false, "explainSearch": false,
"collapseAll": false, "collapseAll": false,
@@ -377,23 +180,23 @@
"obsidian-git:Open Git source control": false "obsidian-git:Open Git source control": false
} }
}, },
"active": "30539ff6e6b3eb6a", "active": "9ec7a3d1ef5d43cc",
"lastOpenFiles": [ "lastOpenFiles": [
"Funktioner.md", "Primära Funktioner.md",
"Vektorer.md", "ODE.md",
"Maclaurin.md",
"Komplexa tal.md",
"Integraler.md",
"Gräsvärde (1).md", "Gräsvärde (1).md",
"Grafer.md", "Grafer.md",
"Funktioner Forts.md", "Funktioner Forts.md",
"Maclaurin.md", "Funktioner.md",
"ODE.md",
"Primära Funktioner.md",
"Differential.md", "Differential.md",
"Integraler.md",
"Definitioner.md",
"Derivata.md", "Derivata.md",
"Int1.png", "Definitioner.md",
"Vektorer.md",
"Tenta Example.md", "Tenta Example.md",
"Komplexa tal.md", "Int1.png",
"Def_graf1.png", "Def_graf1.png",
"TE1.png", "TE1.png",
"Trigonometri.md", "Trigonometri.md",

5
Vektorer.md
View File

@@ -1,10 +1,13 @@
- **DEF** - **DEF**
- *I en rätviklig rektangle stämmer $\overrightarrow{AC}=\left(\overrightarrow{AB}+\overrightarrow{AD}\right)$* - *I en rätviklig rektangle stämmer $\overrightarrow{AC}=\left(\overrightarrow{AB}+\overrightarrow{AD}\right)$*
- $\overrightarrow{u}=\left(1,2,3\right)=\left(\begin{aligned}1\\2\\3\end{aligned}\right)=\left[\begin{aligned}1\\2\\3\end{aligned}\right]$ - $\overrightarrow{u}=\left(1,2,3\right)=\left(\begin{aligned}1\\2\\3\end{aligned}\right)=\left[\begin{aligned}1\\2\\3\end{aligned}\right]$
- *$\mid\mid{V}\mid\mid$ Är längden av $V$*
- **Exemple** - **Exemple**
- $$\begin{align}\text{Rektangeln }A,\;B,\;C,\;D\;\text{. Låt }E\text{ Vara punkten som delar diagonalen }\overline{AC}:\text{förhållandet }1:3\\\left(\text{dvs: }\overline{AE}:\overline{EC}=1:3\right)\\\text{Benämna }\overrightarrow{a}=\overrightarrow{AB},\overrightarrow{h}=\overrightarrow{AD}\text{ Uttryc vektor }\overrightarrow{c}=\overrightarrow{EC}\text{ i termer av }\overrightarrow{a}\text{ och }\overrightarrow{h}\\\\\text{Vart ligger punkten }E\:\text{? Hur kan vi uttrycka }\overrightarrow{c}\text{ med hjälp av }\overrightarrow{a}\text{ och }\overrightarrow{h}\:\text{?}\\\overrightarrow{c}=\overrightarrow{EC}=\frac34\overrightarrow{AC}=\frac34\left(\overrightarrow{AB}+\overrightarrow{BC}\right)=\frac34\left(\overrightarrow{a}+\overrightarrow{h}\right)=\frac34\overrightarrow{a}+\frac34\overrightarrow{h}\end{align}$$ - $$\begin{align}\text{Rektangeln }A,\;B,\;C,\;D\;\text{. Låt }E\text{ Vara punkten som delar diagonalen }\overline{AC}:\text{förhållandet }1:3\\\left(\text{dvs: }\overline{AE}:\overline{EC}=1:3\right)\\\text{Benämna }\overrightarrow{a}=\overrightarrow{AB},\overrightarrow{h}=\overrightarrow{AD}\text{ Uttryc vektor }\overrightarrow{c}=\overrightarrow{EC}\text{ i termer av }\overrightarrow{a}\text{ och }\overrightarrow{h}\\\\\text{Vart ligger punkten }E\:\text{? Hur kan vi uttrycka }\overrightarrow{c}\text{ med hjälp av }\overrightarrow{a}\text{ och }\overrightarrow{h}\:\text{?}\\\overrightarrow{c}=\overrightarrow{EC}=\frac34\overrightarrow{AC}=\frac34\left(\overrightarrow{AB}+\overrightarrow{BC}\right)=\frac34\left(\overrightarrow{a}+\overrightarrow{h}\right)=\frac34\overrightarrow{a}+\frac34\overrightarrow{h}\end{align}$$
- **Koordinatrummet $\mathbb{R}^m$** - **Koordinatrummet $\mathbb{R}^m$**
- *Mängden $\mathbb{R}^m\;\left(\text{där }m\in\mathbb{N}\right)$ består av koordinattpunkter av längden $m$ vars element är reella tal. Som skalärer tas $\mathbb{R}\text{(vanliga reela tal)}$* - *Mängden $\mathbb{R}^m\;\left(\text{där }m\in\mathbb{N}\right)$ består av koordinattpunkter av längden $m$ vars element är reella tal. Som skalärer tas $\mathbb{R}\text{(vanliga reela tal)}$*
- **Hur funkar $+$ och $\times$** - **Hur funkar $+$ och $\times$**
- $$\begin{align}\overrightarrow{u}=\left(u_1,u_2,u_3,\dots,u_m\right)\in\mathbb{R}^m\\\overrightarrow{v}=\left(v_1,v_2,\dots,y_m\right)\in\mathbb{R}^m\\\lambda\in\mathbb{R}\\\\\overrightarrow{u}+\overrightarrow{v}=\left(u_1+v_1,u_2+v_2,\dots,\lambda u_m\right)\\\text{OBS: }\left(1,2\right)+\left(3,4,5\right)\Rightarrow\text{Inte Definierat}\\\\\overrightarrow{u}\times\overrightarrow{v}=\left(u_1v_1,u_2v_2\dots,u_mv_m\right)\\\begin{aligned}\overrightarrow{u}=\left(1,2,0\right)\\\overrightarrow{v}=\left(0,0,-2\right)\end{aligned}\Rightarrow\overrightarrow{u}\times\overrightarrow{v}=\left(1\times0,2\times0,0\times\left(-2\right)\right)=\left(0,0,0\right)\\\text{Man kan i normala fall inte multiplecera vektorer!}\end{align}$$ - $$\begin{align}\overrightarrow{u}=\left(u_1,u_2,u_3,\dots,u_m\right)\in\mathbb{R}^m\\\overrightarrow{v}=\left(v_1,v_2,\dots,y_m\right)\in\mathbb{R}^m\\\lambda\in\mathbb{R}\\\\\overrightarrow{u}+\overrightarrow{v}=\left(u_1+v_1,u_2+v_2,\dots,\lambda u_m\right)\\\text{OBS: }\left(1,2\right)+\left(3,4,5\right)\Rightarrow\text{Inte Definierat}\\\\\overrightarrow{u}\times\overrightarrow{v}=\left(u_1v_1,u_2v_2\dots,u_mv_m\right)\\\begin{aligned}\overrightarrow{u}=\left(1,2,0\right)\\\overrightarrow{v}=\left(0,0,-2\right)\end{aligned}\Rightarrow\overrightarrow{u}\times\overrightarrow{v}=\left(1\times0,2\times0,0\times\left(-2\right)\right)=\left(0,0,0\right)\\\text{Man kan i normala fall inte multiplecera vektorer!}\end{align}$$
- - $$\overrightarrow{u},\overrightarrow{v}\in\mathbb{R}^3\Rightarrow\overrightarrow{n},\overrightarrow{n}\in\mathbb{R}^3$$
- **Sats**: *Låt $\overrightarrow{m},\:\overrightarrow{n}\in\mathbb{R}^3$. Då gäller det att: $\mid\mid\overrightarrow{m}\times\overrightarrow{n}\mid\mid=\mid\mid\overrightarrow{m}\mid\mid\times\mid\mid\overrightarrow{n}\mid\mid\times\sin(\theta)$. (Där $\theta$ är vinkeln mellan $\overrightarrow{m}$ och $\overrightarrow{n}$). (Jämför: $<\overrightarrow{u},\:\overrightarrow{v}>=\mid\mid\overrightarrow{u}\mid\mid\times\mid\mid\overrightarrow{v}\mid\mid\times\cos(\theta)$)*
- **Prof**: *Vi börjar med: $$\begin{aligned}\mid\mid\overrightarrow{u}\times\overrightarrow{v}\mid\mid^2=<\overrightarrow{u}\times\overrightarrow{v},\:\overrightarrow{u}\times\overrightarrow{v}>\stackrel{\text{(I)}}{=}<\overrightarrow{u},\;\overrightarrow{v}\times\left(\overrightarrow{u}\times\overrightarrow{v}\right)>\stackrel{\text{(II)}}{=}<\overrightarrow{u},\;<\overrightarrow{v},\;\overrightarrow{v}>\overrightarrow{u}-<\overrightarrow{v},\;\overrightarrow{u}>\overrightarrow{v}>\\=<\overrightarrow{u},\;<\overrightarrow{v},\;\overrightarrow{v}>\overrightarrow{u}>-<\overrightarrow{u},\;<\overrightarrow{v},\;\overrightarrow{u}>\overrightarrow{v}>\\=<\overrightarrow{v},\;\overrightarrow{v}><\overrightarrow{u},\;\overrightarrow{u}>-<\overrightarrow{u},\;\overrightarrow{v}><\overrightarrow{v},\;\overrightarrow{u}>\\=\mid\mid\overrightarrow{u}\mid\mid^2\times\mid\mid\overrightarrow{v}\mid\mid^2-\left(\mid\mid\overrightarrow{u}\mid\mid\times\mid\mid\overrightarrow{v}\mid\mid\times\cos(\theta)\right)^2\\=\mid\mid\overrightarrow{u}\mid\mid^2\times\mid\mid\overrightarrow{v}\mid\mid^2\times\left(1-\cos^2(\theta)\right)\\=\mid\mid\overrightarrow{u}\mid\mid^2\times\mid\mid\overrightarrow{v}\mid\mid^2\times\sin^2(\theta)\end{aligned}$$*