From 03de9b1decf6b4fe0167b5544ba0cd42f57f41b4 Mon Sep 17 00:00:00 2001 From: Luc Date: Sat, 23 Sep 2023 12:46:18 +0200 Subject: [PATCH] Added the wiki. --- docs/filosofie/start.md | 0 docs/index.md | 3 + docs/javascripts/mathjax.js | 16 ++ docs/natuurkunde/start.md | 3 + docs/programmeertalen/start.md | 0 docs/scheikunde/analyse-methoden.md | 10 + docs/scheikunde/basis/bindingstypen.md | 28 +++ docs/scheikunde/basis/chemisch-rekenen.md | 46 ++++ docs/scheikunde/basis/energie-diagram.jpg | Bin 0 -> 196888 bytes docs/scheikunde/basis/energie-omzetting.md | 37 +++ docs/scheikunde/basis/evenwichten.md | 25 ++ .../basis/fase-overgang-diagram.png | Bin 0 -> 69255 bytes .../scheikunde/basis/materialen-en-stoffen.md | 62 +++++ docs/scheikunde/basis/reacties.md | 17 ++ docs/scheikunde/basis/scheidingsmethoden.md | 28 +++ .../organische-chemie/cis-trans-isomerie.md | 19 ++ .../organische-chemie/cis-trans-voorbeeld.png | Bin 0 -> 23421 bytes docs/scheikunde/organische-chemie/isomerie.md | 8 + 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docs/wiskunde/calculus/transcendental-functions/exponential-and-logarithmic-functions.md create mode 100755 docs/wiskunde/calculus/transcendental-functions/inverse-functions.md create mode 100755 docs/wiskunde/start.md create mode 100755 mkdocs.yml diff --git a/docs/filosofie/start.md b/docs/filosofie/start.md new file mode 100755 index 0000000..e69de29 diff --git a/docs/index.md b/docs/index.md new file mode 100755 index 0000000..6a6168c --- /dev/null +++ b/docs/index.md @@ -0,0 +1,3 @@ +# Welkom + +Welkom op deze webpagina waarop ik mijn aantekeningen van verscheidene vakgebieden heb gedigitaliseerd. diff --git a/docs/javascripts/mathjax.js b/docs/javascripts/mathjax.js new file mode 100755 index 0000000..e648674 --- /dev/null +++ b/docs/javascripts/mathjax.js @@ -0,0 +1,16 @@ +window.MathJax = { + tex: { + inlineMath: [["\\(", "\\)"]], + displayMath: [["\\[", "\\]"]], + processEscapes: true, + processEnvironments: true + }, + options: { + ignoreHtmlClass: ".*|", + processHtmlClass: "arithmatex" + } + }; + + document$.subscribe(() => { + MathJax.typesetPromise() + }) \ No newline at end of file diff --git a/docs/natuurkunde/start.md b/docs/natuurkunde/start.md new file mode 100755 index 0000000..70519b3 --- /dev/null +++ b/docs/natuurkunde/start.md @@ -0,0 +1,3 @@ +# Natuurkunde + +Welkom op de natuurkunde pagina. \ No newline at end of file diff --git a/docs/programmeertalen/start.md b/docs/programmeertalen/start.md new file mode 100755 index 0000000..e69de29 diff --git a/docs/scheikunde/analyse-methoden.md b/docs/scheikunde/analyse-methoden.md new file mode 100755 index 0000000..9d5f1fc --- /dev/null +++ b/docs/scheikunde/analyse-methoden.md @@ -0,0 +1,10 @@ +# Analyse methoden + +## Titratie + +Titratie is een kwantitatieve analyse methode, waarbij de concentratie van oplossing wordt bepaald met een stof en molverhouding. + +Dit gebeurt door bij deze stof geleidelijk een andere oplossing, een reagens/titrant toe te voegen. De concentratie van de titrant is bekend. Er wordt net zolang titrant toegevoegd totdat alle te meten stof is omgezet. Hier wordt dan het equivalentie punt bereikt. Om dit punt te vinden is vaak handig om een indicator te gebruiken. + +Het verschil in de buret kan gebruikt worden om de concentratie van de oplossing te berekenen. + diff --git a/docs/scheikunde/basis/bindingstypen.md b/docs/scheikunde/basis/bindingstypen.md new file mode 100755 index 0000000..83c315c --- /dev/null +++ b/docs/scheikunde/basis/bindingstypen.md @@ -0,0 +1,28 @@ +# Bindingstypen + +## Van der Waals binding + +Moleculen hebben een tijdelijke positieve en negatieve kant die andere moleculen aantrekt. Deze binding is zwak. + +## Atoombinding + +Twee positieve atoomresten worden bijeengehouden door een gemeenschappelijk elektronen paar. Deze binding is zeer sterk. Wordt ook wel covalente binding genoemd. + +## Metaalbinding + +Vrij bewegende valentie elektronen houden positieve atoomresten bijeen. Deze binding is redelijk sterk tot zeer sterk afhankelijk van andere factoren. + +## Dipool-Dipool binding + +Moleculen hebben een permanente positieve en negatieve kant door verschillen in elektronegativiteit tussen de gebonden atomen. Hierdoor worden andere polaire moleculen aangetrokken. Deze binding is sterk. + +Dit geldt voor: $1.7 > \Delta \space elektronegativiteit > 0.4$. + +### Waterstofbruggen + +Moleculen vormen onderling waterstofbruggen als ze een NH of OH group hebben. Dit is een lokale vorm van een dipool-dipool binding. + +## Ionbinding + +Positieve ionen trekken negatieve ionen aan. Wanneer de $\Delta \space elektronegativiteit > 1.7$ dan vormt er een ion binding, een elektron wordt er dan gejat. + diff --git a/docs/scheikunde/basis/chemisch-rekenen.md b/docs/scheikunde/basis/chemisch-rekenen.md new file mode 100755 index 0000000..bd46ad7 --- /dev/null +++ b/docs/scheikunde/basis/chemisch-rekenen.md @@ -0,0 +1,46 @@ +# Chemisch rekenen + +## Constante van Avogadro + +De constante van Avogadro $N_A$ is een fysische constante die de verhouding tussen het aantal deeltjes en de hoeveelheid stof $n$ in mol aangeeft, + +$$n=\frac{N}{N_A}.$$ + +Initieel was de constante van Avogadro gedifinieerd als het aantal atomen in 12 gram koolstof-12. Vanaf de herdefinitie van de basiseenheden, waaronder de mol, is de constante een bepaald exact aantal $\mathrm{mol^{-1}}$ en hangt niet meer af van het koolstof-12-atoom. + +## Chemische hoeveelheid + +De chemische hoeveelheid is een grootheid met als eenheid mol en is evenredig met het aantal deeltjes $N$. De deeltjes kunnen moleculen, atomen, ionen of subatomaire deeltjes zijn, afhankelijk van de context. + +Tussen de massa $m$ van een hoeveelheid stof en de chemische hoeveelheid $n$ bestaat de volgende relatie, + +$$n=\frac{m}{M}.$$ + +Waarin $M$ staat voor de molaire massa van de stof. De molaire massa kan dan gedifinieerd worden als, + +$$M=N_A \cdot m_M,$$ + +waarin $m_M$ de massa van één deeltje van de stof. + +## Significantie (versimpeld) + +Deze significantie wordt toegepast als er niet met onzekerheid wordt gerekend. Hier geldt. + +* Voor een vermenigvuldiging; kleinste getal in vermenigvuldiging. +* Voor een summatie; som van kleinste getal. + +## Molariteit + +De molariteit is een maat vor de sterkte van een oplossing van een stof. Gedefinieerd als, + +$$Molariteit = \frac{n}{V}.$$ + +## Molair volume + +Het molair volume $V_m$ is het volume dat ingenomen wordt door één mol van zuivere stof. Het molaire volume van één mol ideaal gas is onder dezelfde druk en temperatuur altijd hetzelfde en kan met de ideale gaswet berekend worden: + +$$V_m = \frac{R \space T}{p}.$$ + +Hiermee kan het volume die een hoeveelheid gas in neemt berekend worden doormiddel van: + +$$V = n \cdot V_m.$$ diff --git a/docs/scheikunde/basis/energie-diagram.jpg b/docs/scheikunde/basis/energie-diagram.jpg new file mode 100755 index 0000000000000000000000000000000000000000..0926c6bacc25310fe49cc683cce4f911d77f4a16 GIT binary patch literal 196888 zcmeFa1ymJGv@d>Wq)|Eq0YOTT?hxq)X{5WAMoJKn?k-8`?v!qjZjhFiZulP%6@~Av zci;Q3ci(&W8`k3N+55Nm%7pKh z3=Po)fddAD3;;z20Ye74XawK^KmeeiV3&0RT&H`m&@d2?P~c#oAQxi*=pTaVuRq2A zlb??Ee=Blqyu4(oxjp>+muTX=1L!sYhzrWcDXqU`bjr=bPq~0hNN6cvhfCsS`r>-e zsQi+F@koe%@1_JugFT0PTrhvhRC9YWJn)wk9-2DrXR%4D5`RhQ@avUs6gxbfjHyJ! z1%eqT{OltDlCgfj8K01ajjuj73jhFp^nGTdpL@(Kj`dSA&<^B#3}NloOM`VVlDTOp zT6=5njWBhXKp(D9RdrW?t1AJDKD6&prh=$2_(jetCj_Vg0H~U)R1py}u~vsoe4qBM zF4g~R9jH(2x6%-L*AJrG_eN70 zt=^2koY-yDFmNX_Q)jCHRe5S=yCM^>%{Zu8QJ+C-JX|^ZbWd_Hsl?rN30K=;T-26Pb}xhW{KHb;`hE0_TF02j0}p#9skGjjW%yC0Cx;sd zif2o0`jp7|CY=VS-2WUU&A7K!&A3UG%{@79st!QNSlnimn({>Oc2!Z2d>Rlfpg

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Bindings-energie wordt vaak omgezet in warmte. + +## Definitie endotherm + +In een endotherme reactie zal de binding-energie stijgen. Warmte is vaak dan benodigt om de reactie in gang te zetten, deze warmte wordt dan omgezet in bindings-energie. + +## Energie-effect + +Het energie-effect is het verschil tussen de bindings-energie van de producten en de beginstoffen. + +$$\Delta E = E_{producten} - E_{beginstoffen}.$$ + +Het energie-effect is de energie die wordt omgezet in een reactie. + +## Energie diagram + +![Energie diagram](energie-diagram.jpg) + +## Reactie warmte + +De reactie warmte is alle energie die in warmte wordt omgezet. + +Voor de vergelijking: + +$$m\mathrm{A} + n\mathrm{B} \rightarrow p\mathrm{C} + q\mathrm{D},$$ + +$$\Delta E = (p\cdot E_V(C) + q\cdot E_V(D)) - (m\cdot E_V(A) + n\cdot E_V(B)).$$ + +Waarin $E_V$ de vormingswarmte van de stoffen is. + +### Verbrandingswarmte + +De reactie warmte van een verbrandings reactie. \ No newline at end of file diff --git a/docs/scheikunde/basis/evenwichten.md b/docs/scheikunde/basis/evenwichten.md new file mode 100755 index 0000000..3741619 --- /dev/null +++ b/docs/scheikunde/basis/evenwichten.md @@ -0,0 +1,25 @@ +# Evenwichten + +## Evenwichts reacties + +Er is een chemisch evenwicht als er evenveel begin als eind producten worden omgezet met een gelijke [reactiesnelheid](reactie-kinetiek.md), dit is dus een omkeerbare reactie. + +## Concentratie breuk + +De concentratie breuk $Q$ tussen de begin en eind producten is gedefinieerd als: + +$$Q=\frac{[C]^p [D]^q}{[A]^m [B]^n},$$ + +met als evenwichtsvergelijking: + +$$m\mathrm{A} + n\mathrm{B} \rightleftarrows p\mathrm{C} + q\mathrm{D}.$$ + +Wanneer $Q$ voldoet aan de evenwichtsvoorwaarde $Q=K$ met $K$ de evenwichts constante dan is er chemisch evenwicht. + +## Beïnvloeding van evenwichten + +* Een toename van de temperatuur zal leiden voor een voorkeur van de endotherme richting. De reactie producten van de endotherme richting zullen dan toenemen. +* Een toename van de concentratie van bijvoorbeeld $[\mathrm{A}]$ zal leiden tot de toename van $[\mathrm{C}]$ en $[\mathrm{D}]$, het evenwicht verschuift dus naar de rechter kant. +* Een toename van de volume waarin de deeltjes in worden behouden zal leiden tot een verschuiving van het evenwicht naar de kant met de meeste deeltjes. + +De insteltijd van een evenwicht kan worden verkort door een katalysator. \ No newline at end of file diff --git a/docs/scheikunde/basis/fase-overgang-diagram.png b/docs/scheikunde/basis/fase-overgang-diagram.png new file mode 100755 index 0000000000000000000000000000000000000000..59e6e551ccd69e94c3464580075489909f1dc11a GIT binary patch literal 69255 zcmYIv1yoyG*KKf0(c(^_xVsd0in~j3cXy|_yB2qMcXyWp#ogUsp!fUVfH5G+NzUGT z*_?B&1j|T?z`o@R7HM%{s{qm zOrR$!CkX(!k^lhS{s6!u@G0+o0KkC`065eE063BW0CelLW?4?)3y`|vB7%Uo_n)k` 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z-+EiWOMi#d{S~}@K1Xw0;C9aF%uP48Wp7L4ggF3~)``p;eOUqzd(jQK`K1$ZhO>c- z#;elAC$I_*jgDd7m)pk0hQP>KnnH($BA_KB(6ZxW1cDJ|B8$o*f&Sy*!3W|Y!7~tvM-YJ(etF^pDlP;BDyu``v@J$i-8#5K~6Xf5sCoDHJSv&_c ziNC(RZS5F!?iCX)*8TPCNC^$S(KG#T*SVIQUCMTNb29P+>3 zQ~~l-jh}s6%Z!x6G4m%fV3p=6rbl<4^sz&uu_!poNKm>OvKOnmJu}Acqoq>|Y_T40B*Kyk=(Uj5#{1rVPfd zJJNr*9S=Xm9Blu5KoldmX}+kIb%Q`4GG|eBXJZ3r6K*3%6W|w!nUR@=j**p)iBXx6 wl^gi5aL_O^ax*e6$-l?^9|zdj8C#fo{J#$Y34gmcSmHl2WtpD?W0Z{`&8~^|S literal 0 HcmV?d00001 diff --git a/docs/scheikunde/basis/materialen-en-stoffen.md b/docs/scheikunde/basis/materialen-en-stoffen.md new file mode 100755 index 0000000..8de6fae --- /dev/null +++ b/docs/scheikunde/basis/materialen-en-stoffen.md @@ -0,0 +1,62 @@ +# Materialen + +## Definitie grondstof: + +Grondstoffen zijn nodig voor producten die niet uit de natuur komen. + +## Definitie synthetisch: + +Een product die niet natuurlijk is zoals plastic, ijzer en kunstof. + +## Definitie composiet: + +Een samenstelling tussen twee materialen zoals gewapend beton of gewapend glas. + +## Definitie legering: + +Een samenstelling van bij elkaar gesmolten metalen zoals brons, soldeer en messing. + +--- + +# Stoffen + +## Definitie zuivere stof: + +Één stof die maar uit een soort moleculen bestaat. Suiker, koper*, aluminium*. + +## Definitie mengsel: + +Meerdere soorten stoffen door elkaar. + +## Definitie heterogeen mengsel: + +Een mengsel waar de verschillende stoffen zichtbaar zijn. Vormen van heterogene mengsel zijn: + +### Suspensie: + +Een heterogeen mengsel waarbij in een vloeistof deeltjes in een vaste fase aanwezig zijn. + +### Emulsie: + +Een heterogeen mengsel waarbij in een vloeistof druppels van een andere vloeistof aanwezig zijn. + +### Rook: + +Een heterogeen mengsel van een vaste stof in een gas. + +### Nevel: + +Een heterogeen mengsel waarbij vloeistof druppels in een gas aanwezig zijn. + +### Schuim: + +Een heterogeen mengsel waarbij gas opgesloten zit in een vaste stof. + +## Definitie homogeen mengsel: + +Een mengsel waar de verschillende stoffen niet zichtbaar zijn. + +## Fase overgang diagram + +![Fase overgang diagram](fase-overgang-diagram.png) + diff --git a/docs/scheikunde/basis/reacties.md b/docs/scheikunde/basis/reacties.md new file mode 100755 index 0000000..8a01f13 --- /dev/null +++ b/docs/scheikunde/basis/reacties.md @@ -0,0 +1,17 @@ +# Reacties + +Een fase overgang is geen reactie, de moleculen blijven intact, de interactie tussen de moleculen verandert alleen. + +## Chemische reactie + +Bij een chemische reactie verdwijnen de beginstoffen en ontstaan er reactie producten. + +De volgende reacties behoren onder chemische reacties: + +### Verbrandings reactie + +Bij een verbrandings reactie is er sprake van een reactie tussen zuurstof en een brandbare stof. + +### Ontledings reactie + +Bij een ontledingsreactie is er een sprake van maar één beginstof en meerdere reactieproducten. \ No newline at end of file diff --git a/docs/scheikunde/basis/scheidingsmethoden.md b/docs/scheikunde/basis/scheidingsmethoden.md new file mode 100755 index 0000000..790b55a --- /dev/null +++ b/docs/scheikunde/basis/scheidingsmethoden.md @@ -0,0 +1,28 @@ +# Scheidingsmethoden + +Alle scheidingsmethoden gaan uit van standaard omstandigheden omtrent, zwaartekracht en druk. + +## Filtreren + +Filtreren is op basis van deeltjesgrootte, vaak zijn suspensies filtreerbaar. De residu blijft dan achter in de filter en het filtraat is het gefilterde eindproduct. + +## Bezinken + +Bezinken is op basis van dichtheid, de zwaarste stoffen zinken. Centrifugeren is een snellere variant van bezinken. + +## Indampen + +Indampen is op basis van kookpunt voor oplossingen. + +## Extractie + +Extractie berust op het verscil in oplosbaarheid. + +## Adsorptie + +Adsorptie berust op het verchil in aanhechtings vermogen. Deeltjes in de vloeistof kunnen hechten aan het adsorptiemiddel. + +## Destillatie + +Destillatie berust op het verschil in kookpunt van twee of meer vloeistoffen in mengsel. Dit mengsel heeft dan een kooktraject. + diff --git a/docs/scheikunde/organische-chemie/cis-trans-isomerie.md b/docs/scheikunde/organische-chemie/cis-trans-isomerie.md new file mode 100755 index 0000000..3e30a5d --- /dev/null +++ b/docs/scheikunde/organische-chemie/cis-trans-isomerie.md @@ -0,0 +1,19 @@ +# Cis-trans isomerie + +Cis-trans isomerie is een vorm van stereo isomerie, waarbij de isomeren dezelfde structuur formule delen maar een andere ruimtelijke structuur hebben. Bij cis-trans isomerie is er sprake van een molecuul met een dubbele, drievoudige binding of cyclisch, het molecuul kan niet roteren (is star) ten opzichte van deze binding. + +Dit vereist twee verschillende zijgroepen voor beide atomen die via de starre binding met elkaar gebonden zijn. + +## Cis of trans? + +Het molecuul is cis geconfigureerd als de zijgroepen met hoogste atoomnummer gelijk geconfigureerd is (de lijn tussen beide niet kruist met de starre binding). Hierbij betekent cis 'aan dezelfde kant' in het Latijn. + +Het molecuul is trans geconfigureerd als de zijgroepen met hoogste atoomnummer schuin van elkaar staan (de lijn tussen beide kruist met de starre binding). Hierbij betekent trans 'aan de andere kant' in het Latijn. + +## Cis-trans eigenschapsverschillen + +Door de andere ruimtelijke bouw verandert het contact oppervlak van het molecuul, de zijgroepen hebben namelijk een ander formaat. Hierdoor kan de van der Waals binding tussen beide isomeren verschillen. Verderrest kan ook de reactiviteit verschillen door een verschil in polariteit. 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Hiermee zijn isomeren dus stoffen met dezelfde molecuulformule, maar een andere structuurformule. + +## Soorten isomerie + +* **Strucuurisomeren**: zelfde molecuulformule maar een andere structuurformule, zoals bijvoorbeeld butaan en isobutaan, beide $\mathrm{C_4H_{10}}$ +* **Stereo-isomeren**: zelfde structuur formule maar verschillende ruimtelijke structuren. 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zyDBx!|N8LqeR{ipEkB!1{m0)Q+7>$G&nNe8dv4_CFSo?Xx4&NNe{Of<_@5J&wrvgh zJFymRn^%5*@#|$3{(8Mw{J&O0zVCXm_1Z1RiwCdWV!Q}D?HA)k&~3jOFOHn{tMTH% zZNC~X(v9}3@gm)5zZx&njrObY|0vz?ZncZ&(k6=*fbaiv3~13h^zv56y%s(9$%0LK M$I0QS-Mo$e2c05FSO5S3 literal 0 HcmV?d00001 diff --git a/docs/scheikunde/organische-chemie/lewis-theorie.md b/docs/scheikunde/organische-chemie/lewis-theorie.md new file mode 100755 index 0000000..32ef89f --- /dev/null +++ b/docs/scheikunde/organische-chemie/lewis-theorie.md @@ -0,0 +1,19 @@ +# Lewis structuren + +## Octetregel + +De regel dat atomen op een zodanige manier proberen te combineren dat ze elk acht elektronen in hun valentieschil hebben, zodat ze dezelfde elektronenconfiguratie krijgen als een edelgas. Met uitzondering van waterstof, lithium en beryllium. De 18-elektronenregel gaat boven de octetregel bij overgangsmetalen. + +## Formele lading + +De lading die aan een atoom in een molecuul of samengesteld ion wordt toegekend met als uitgangspunt dat elektronen in een binding gelijkwaardig worden gedeeld door de atomen, zonder rekening te houden met verschil in elektronegativiteit. + +Lewis structuren met de minste formele lading zijn het stabielste. + +## Mesomere grensstructuren + +Alle mogelijke lewis-structuren, hier verplaatsen alleen de elektronen tussen de mogelijke structuren. Meer mesomeren betekent stabielere deeltjes. + +## Voorbeeld Lewis structuur + +![lewis-structuur-voorbeeld](lewis-structuur-voorbeeld.png) \ No newline at end of file diff --git a/docs/scheikunde/organische-chemie/organische-verbindingen.md b/docs/scheikunde/organische-chemie/organische-verbindingen.md new file mode 100755 index 0000000..f12037c --- /dev/null +++ b/docs/scheikunde/organische-chemie/organische-verbindingen.md @@ -0,0 +1,37 @@ +# Organische verbindingen + +* Alkanen $\rightarrow$ verzadigd: enkele binding tussen koolstof atomen; $\mathrm{C_N H_{2N+2}}$. +* Alkenen $\rightarrow$ onverzadigd: dubbele binding tussen koolstof atomen; $\mathrm{C_N H_{2N}}$. +* Alkynen $\rightarrow$ drievoudige binding tussen koolstof atomen; $\mathrm{C_N H_{N+1}}$. +* Cyclo-alkanen $\rightarrow$ cyclische enkele binding tussen koolstof atomen; $\mathrm{C_N H_{2N}}$. + +## Alkylgroepen + +* Methyl, ethyl, propyl, ... + +## Systematische naamgeving + +1. Zoek langste keten. +2. Zoek vanaf de langste keten naar alle groepen, noteer de plaats. +3. Alfabetiseren van substituenten. +4. Laagste nummering. + +## Aldehyden + +Dubbel gebonden :O aan het einde van de keten. $\rightarrow$ Achter: -al, en voor: oxo-. Omtrent eigenschappen zijn aldehyden minder hydrofiel hebben geen zure eigenschappen en zijn vluchtiger dan carbonzuren. + +## Ketonen + +Dubbel gebonden ö in het middel van de keten. $\rightarrow$ Achter: -on, en voor: oxo-. Omtrent eigenschappen zijn aldehyden minder hydrofiel hebben geen zure eigenschappen en zijn vluchtiger dan carbonzuren. + +## Ethers + +-O- atoom tussen twee koolstofgroepen. Omtrent naamgeving; de korste koolstofverbinding is de karakteristieke groep: alkoxy. + +## Esters + +Dubbelgebonden :O en enkele -O die met de rest van de keten verbindt vormt een ester. Ook wel het zuurrest(ion). Omtrent naamgeving, deel met dubbelgebonden :O; alkanoaat en de groep gebonden met de enkele -O; alkyl. $\rightarrow$ alkylalkanoaat. Esters zijn over het algemeen erg reukzinnig en lossen slecht op in water. + +## Benzeenring + +Gedislokaliseerde dubbelbindingen. $\rightarrow$ Achter: -benzeen, en voor: fenyl-. \ No newline at end of file diff --git a/docs/scheikunde/organische-chemie/reacties.md b/docs/scheikunde/organische-chemie/reacties.md new file mode 100755 index 0000000..05c6921 --- /dev/null +++ b/docs/scheikunde/organische-chemie/reacties.md @@ -0,0 +1,33 @@ +# Reacties + +## Substitutiereactie + +Bij een substitutiereactie wordt een atoomgroep van een koolwaterstof vervangen door een andere atoomgroep. Deze reactie is endotherm en is moeilijk te sturen welke atoomgroepen worden vervangen waardoor een mengsel van allerlei verbindingen ontstaan. + +## Additiereactie + +Bij een additiereactie kunnen door een reactieve dubbele of drievoudige binding moleculen binden. Deze reactie is exotherm. Reactieve cycloalkanen kunnen ook aderen, hierbij springt de ring open. + +## Eleminatiereactie + +Additie reacties zijn in sommige gevallen omkeerbaar. + +## Condensatiereactie + +Een condensatie reactie is een evenwichtsreactie waaruit ethers, esters en amiden kunnen ontstaan. + +## Ethervorming + +Een ethervorming is een zuurgekatalyseerde reactie tussen 2 alcoholen, hierdoor vormt er een symmetrische ether. Een basische omgeving is benodigt om assymetrische ethers te vormen. + +## Estervorming + +Een estervorming is een zuurgekatalyseerde reactie tussen een carbonzuur en een alcohol. De reactiesnelheid van deze reactie is hoger bij een lager pH. + +## Amidevorming + +Bij een amidevorming reageren een alkaanzuur en alkaanaminen met elkaar zodat er een amide groep ontstaat. Eiwitten worden door deze reactie gesynthetiseerd. + +## Hydrolysereactie + +Een hydrolyse reactie si de splitsing van een chemische binding onder opname van één molecuul water. \ No newline at end of file diff --git a/docs/scheikunde/organische-chemie/spiegelbeeld-isomerie.md b/docs/scheikunde/organische-chemie/spiegelbeeld-isomerie.md new file mode 100755 index 0000000..514b2c9 --- /dev/null +++ b/docs/scheikunde/organische-chemie/spiegelbeeld-isomerie.md @@ -0,0 +1,17 @@ +# Spiegelbeeld isomerie + +Spiegelbeeld of optische isomerie is een vorm van stereo isomerie waarbij de isomeren elkaars spiegelbeeld zijn. Wanneer er geen inwendig spiegelvlak door de structuur bestaat. + +Een spiegelbeeld isomeer vergt een chiraal-(koolstof)atoom. Dit is een atoom met vier verschillende zijgroepen. Hierbij verkrijgt men $2^n$ spiegelbeeld isomeren. + +## Aanduiding spiegelbeeld isomeren + +Spiegelbeeld isomeren worden aangeduid met -D of -L afhankelijk van de draairichting van gepolariseerd licht. -D is hierbij rechtsdraaiend en -L is link draaiend. Spiegelbeeld isomeren worden ook met -R en -S aangeduid, hierbij staat -R voor rechtsdraaiend en -S voor linksdraaiend. Als er geen 'overall' draaing is, moet de concentratie gelijk zijn, dan spreekt men van racemisch mengsel. + +## Spiegelbeeld eigenschapsverschillen + +Wat niet verschilt tussen de spiegelbeeld isomeren is de polariteit en onderlinge van der Waals bindingen. Waardoor het smelt- en kookpunt van spiegelbield isomeren identiek is. De reactiveit is voor symmetrische systemen identiek. 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binding verklaart aan de hand van elektrostatische repulse tussen valentie-elektronen. De theorie gaat ervan uit dat de atomen in een molecuul zich rond één centraal atoom bevinden. + +## Omringingsgetal + +Het omringingsgetal ook wel sterisch getal van een atoom is de som van vrije- en gemeenschappelijke elektronenparen. Waarbij alle soorten binding tellen als één. + +| Omringingsgetal | Moleculair-geometrische basisconfiguratie | +| --- | --- | +| 2 | Lineair | +| 3 | Trigonaal planair | +| 4 | Tetraëdrisch | +| 5 | Trigonaal bipiramidaal | +| 6 | Octaëdrisch | + +## Dipoolmoleculen + +Een molecuul wordt een dipoolmolecuul genoemd als de onderlinge binding tussen de atomen dipool-dipool bindingen zijn en er geen samenval van partiële lading is. Zoals bij $\mathrm{CO_2}$ het geval is. Om dit te beoordelen is de VSEPR theorie benodigt. + +## Afwijkende tetraëderhoek + +De afwijkende tetraëderhoek van bijvoorbeeld $\mathrm{H_2O}$ komt uit feit dat vrije elektronen elkaar meer afstoten dan gemeenschappelijke elektronenparen. [Waarom?]() + +## Voorbeeld VSEPR theorie + +Het omringingsgetal van het koolstof ($\mathrm{C}$) atoom van chloormethaan ($\mathrm{CH_3Cl}$) is 4. 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mode 100755 index 0000000..f1289c5 --- /dev/null +++ b/docs/scheikunde/reactie-kinetiek.md @@ -0,0 +1,18 @@ +# Reactie kinetiek + +## Definitie reactiesnelheid + +In het homogene verdunde geval kan de reactiesnelheid gedefinieërd worden als: + +$$s = -\frac{d[A]}{m\cdot dt} = -\frac{d[B]}{n\cdot dt} = \frac{d[C]}{p\cdot dt} = \frac{d[D]}{q\cdot dt}.$$ + +Voor de vergelijking: + +$$m\mathrm{A} + n\mathrm{B} \rightleftarrows p\mathrm{C} + q\mathrm{D}.$$ + +## Botsende deeltjes model + +* Een stijging in temperatuur duidt op meer beweging en dus meer kans op botsingen tussen de deeltjes. +* Een stijging in concentratie duidt op meer deeltjes en dus meer kans op botsingen tussen de deeltjes. +* Een fijnere verdeling impliceert meer contact oppervlak en dus meer kans op botsingen. + diff --git a/docs/scheikunde/redox-chemie.md b/docs/scheikunde/redox-chemie.md new file mode 100755 index 0000000..31d2908 --- /dev/null +++ b/docs/scheikunde/redox-chemie.md @@ -0,0 +1,58 @@ +# Redox chemie + +## Reductor + +Een reductor ofwel elektronendonor is een chemische stof die in een redoxreactie elektronen afstaat aan een oxidator. + +## Oxidator + +Een oxidator ofwel elektronenacceptor is een chemische stof die in een redoxreactie elektronen opneemt afkomstig van een reductor. + +## Redoxreactie + +Een redoxreactie is een reactie tussen atomen, moleculen en/of ionen waarbij elektronen worden uitgewisseld, tussen de reductor ofwel elektronendonor en oxidator ofwel elektronenacceptor. Het onderscheid tussen een redoxreactie en een zuur-basereactie is in het feit dat bij een zuur-basereactie expliciet een elektronenpaar wordt uitgewisseld en bij een redoxreactie één of meerdere elektronen "losse". + +### Condities standaard elektrodepotentiaal + +* Als $\Delta V \leq -0.3\mathrm{V}$; geen redox reactie. +* Als $-0.3\mathrm{V} \geq \Delta V \leq 0.3\mathrm{V}$; evenwichtsreactie. +* Als $\Delta V \geq 0.3\mathrm{V}$; aflopende reactie. + +### Voorbeeld redoxreactie + +Opgedeeld in twee halfreacties. + +$$\mathrm{Oxidator}: \mathrm{F_2 + 2e^- \rightarrow 2F^-} \quad V_{ox} = +2.87\mathrm{V},$$ + +$$\mathrm{Reductor}: \mathrm{Li \rightarrow Li^+ + e^-} \quad V_{red} = -3.04\mathrm{V},$$ + +$$\mathrm{Totaal}: \mathrm{F_2 + 2 Li \rightarrow 2 Li^+ + 2 F^-} \quad \Delta V = 5.91\mathrm{V}.$$ + +### Stappenplan zelf opzetten van een halfreactie + +1. Voorlopige vergelijking, +2. atomen balans behalve H en O, +3. atomen balans O. +4. atomenbalans H en ladingsbalans, +5. Milieu (pH heeft invloed op de elektrodepotentiaal), +6. Halfreactie. + +## Berekening elektrode potentiaal + +Vergelijking van nernst voor de halfreactie: + +$$red \leftrightarrow ox + n e^-,$$ + +$$V = V^{\circ} + \frac{RT}{nF} \space \mathrm{ln\frac{[ox]}{[red]}}.$$ + +Met $F = N_A \cdot e$. + +## Stroomsterkte cel + +$$n \cdot N_A \cdot e = Q = n \cdot F,$$ + +$$I = \frac{Q}{t}$$ + +## Corosie + +Het verweren van materialen door een chemische reactie. \ No newline at end of file diff --git a/docs/scheikunde/start.md b/docs/scheikunde/start.md new file mode 100755 index 0000000..933b9a1 --- /dev/null +++ b/docs/scheikunde/start.md @@ -0,0 +1,3 @@ +# Scheikunde + +Welkom op de scheikunde pagina, hier bewaar ik mijn aantekeningen van scheikunde. Dit zijn voornamelijk aantekeningen van vwo scheikunde en aantekeningen van een paar boeken die ik na die tijd gelezen heb. Tijdens mijn verdere studies ben ik niet heel veel meer scheikunde tegengekomen helaas. \ No newline at end of file diff --git a/docs/scheikunde/zuren-basen.md b/docs/scheikunde/zuren-basen.md new file mode 100755 index 0000000..35457e1 --- /dev/null +++ b/docs/scheikunde/zuren-basen.md @@ -0,0 +1,67 @@ +# Zuren en basen + +## Zuur + +Een zuur per definitie van Lewis is een molecuul of ionaire verbinding dat een elektronenpaar kan accepteren, een lewiszuur wordt dit ook wel genoemd. Oorspronkelijk werd de definitie beperkt tot stoffen die een waterstofion konden afstaan in water, een arrheniuszuur. Later werd dit uitgebreid met het feit dat zuren niet per se in water hoeven te reageren. Daarmee is een zuur een molecuul dat een proton kan doneren, een brønstedzuur. + +Per definitie van de pH schaal is een zuur altijd onder pH 7. + +## Basen + +Een lewisbase is een molecuul of ion dat over vrije elektronen beschikt die gebruikt kunnen worden om een chemische binding met een lewiszuur te vormen. Ofwel een lewisbase is een elektronenpaardonor. De oorspronkelijke definities zijn ook van toepassing op basen. + +Per definitie van de pH schaal is een base altijd boven pH 7. + +## Onderscheid zwakke en sterke zuren en basen + +Een zwak zuur of base kan niet volledig in water ioniseren, de gevormde ionen blijven in evenwicht met het moleculaire zuur. Een sterk zuur of base kan wel volledig in water ioniseren. Deze onderscheiding is alleen van toepassing wanneer de zuren en basen in oplossing zitten. + +## Zuur-basereactie + +Een zuur-basereactie is een evenwichts reactie tussen een zuur en een base. Die als volgt kan worden weergegeven: + +$$\mathrm{zuur + base \rightleftarrows zout + water}.$$ + +Of er al dan niet een praktisch meetbaar evenwicht optreedt, hangt af van de aard van het zuur (sterk of zwak) en base (sterk of zwak). + +Opstelling van een zuur-basereactie gaat als volgt: +1. Het inventariseren van de deeltjes, +2. Sterkste zuur en base, +3. Opstelling van de reactievergelijking. + + +## Evenwichtsvoorwaarde + +Evenwichtsreacties treden op wanneer er wordt uitgegaan van zwakke zuren en basen. Hiervoor moet een evenwichtsvoorwaarde opgesteld worden met evenwichtsconstante ofwel zuurconstante $K_z$: + +$$K_z = \mathrm{\frac{[Z^-][H_3O^+]}{[HZ]}}.$$ + +## pH en pOH berekening + +De pH, een maat voor de zuurgraad van een waterige oplossing is formeel gedefinieerd als: + +$$\mathrm{pH = -log(\gamma[H_3O^+])},$$ + +met $\mathrm{[H_3O^+]}$ de concentratie $\mathrm{H_3O^+}$ is en $\gamma$ de chemische activiteit van $\mathrm{H_3O^+}$. + +De pOH kan gedefinieerd worden als: + +$$\mathrm{pOH} = \mathrm{p}K_W - \mathrm{pH},$$ + +met $\mathrm{p}K_W$ de autoprotolyse van water waar $K_W = \mathrm{[H_3O^+][OH^-]} \approx 14$ bij standaard omstandigheden. + +## Definitie amfolyt + +Amfolyten kunnen zowel als base als zuur reageren, $\mathrm{H_2O}$ is hier een amfolyt. + +## Definitie meerwaardige zuren/basen + +Meerwaardige zuren en basen kunnen meer dan één elektronenpaar accepteren/doneren. + +## Buffer oplossing + +Een buffer oplossing is een sterk evenwicht waarbij de pH nauwelijks verandert. Hierbij is sprake van de aanwezigheid van een zwak zuur en zijn geconjugeerde base in oplossing. + +De elektronenpaar uitwisseling wordt opgevangen door zuur en geconjugeerde base die in evenwicht zijn: + +$$\mathrm{HZ + H_2O \rightleftarrows Z^- + H_3O^+}.$$ \ No newline at end of file diff --git a/docs/wiskunde/calculus/concavity-and-inflections.md b/docs/wiskunde/calculus/concavity-and-inflections.md new file mode 100755 index 0000000..8e8cc9e --- /dev/null +++ b/docs/wiskunde/calculus/concavity-and-inflections.md @@ -0,0 +1,22 @@ +# Concavity and inflections + +## Concave up + +A function $f$ is **concave up** on an open differentiable interval $I$ if the derivative $f'$ is an increasing function on $I$, then $f'' > 0$. Obtaining tangent line above the graph. + +## Concave dowm + +A function $f$ is **concave down** on an open and differentiable interval $I$ if the derivative is a decreasing function on $I$, then $f'' < 0$. Obtaining tangent lines below the graph. + +## Inflection points + +The function $f$ has an inflection point at $x_0$ if + +1. the tangent line in $(x_0, f(x_0))$ exists, and +2. the concavity of $f$ is opposite on opposite sides of $x_0$. + +If $f$ has an inflection point at $x_0$ and $f''(x_0)$ exists, then $f''(x_0) = 0$ + +## The second derivative test + +... \ No newline at end of file diff --git a/docs/wiskunde/calculus/continuity.md b/docs/wiskunde/calculus/continuity.md new file mode 100755 index 0000000..c47f14e --- /dev/null +++ b/docs/wiskunde/calculus/continuity.md @@ -0,0 +1,53 @@ +# Continuity + +Continuity is a local property. A function $f$ is continuous at an interior point $c$ of its domain if + +$$\lim_{x \to c} f(x) = f(c).$$ + +If either $\lim_{x \to c} f(x)$ fails to exist or it exists but is not equal to $f(c)$, then $f$ is discontinuous at $c$. + +## Right and left continuity + +$f$ is **right continuous** at $c$ thereby having a left endpoint $c$ of its domain if + +$$\lim_{x \downarrow c} f(x) = f(c)$$ + +and **left continuous** thereby having a right endpoint $c$ if + +$$\lim_{x \uparrow c} f(x) = f(c).$$ + +## Continuity on an interval + +$f$ is continuous on the interval $I$ if and only if $f$ is continuous in each point of $I$. In endpoints left/right continuity is sufficient. + +$f$ is called a continuous function if and only if $f$ is continuous on its domain. + +## Discontinuity + +A discontinuity is removable if and only if the limit exists otherwise the discontinuity is non-removable. + +## Combining continuous functions + +If the functions $f$ and $g$ are both defined on an interval containing $c$ and both are continuous at $c$, then the following functions are also continuous at $c$: + +* the sum $f + g$ and the difference $f - g$; +* the product $f g$; +* the constant multiple $k f$, where $k$ is any number; +* the quotient $\frac{f}{g}$, provided $g(c) \neq 0$; and +* the *n*th root $(f(x))^{\frac{1}{n}}$, provided $f(c) > 0$ if $n$ is even. + +This may be proved using the various [limit rules](limits.md/#limit-rules). + +## The extreme value theorem + +If $f(x)$ is continuous on the closed, bounded interval $[a,b]$, then there exists numbers $p$ and $q$ in $[a,b]$ such that $\forall x \in [a,b]$, + +$$f(p) \leq f(x) \leq f(q).$$ + +Thus, $f$ has the absolute minimum value $m=f(p)$, taken on at the point $p$, and the absolute maximum value $M=f(q)$, taken on at the point $q$. This follows from the consequence of the completeness property of the real numbers. + +## The intermediate value theorem + +If $f(x)$ is continuous on the interval $[a,b]$ and if $s$ is a number between $f(a)$ and $f(b)$, then there exists a number $c$ in $[a,b]$ such that $f(c)=s$. This follows from the consequence of the completeness property of the real numbers. + +In particular, a continuous function defined on a closed interval takes on all values between its minimum value $m$ and its maximum value $M$, so its range is also a closed interval, $[m,M]$. \ No newline at end of file diff --git a/docs/wiskunde/calculus/differentation.md b/docs/wiskunde/calculus/differentation.md new file mode 100755 index 0000000..efe7563 --- /dev/null +++ b/docs/wiskunde/calculus/differentation.md @@ -0,0 +1,231 @@ +# Differentation + +## The slope of a curve + +The slope $a$ of a curve $C$ at a point $p$ is the slope of the tangent line to $C$ at $P$ if such a tangent line exists. In particular, the slope of the graph of $y=f(x)$ at the point $x_0$ is + +$$ +\lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} = a. +$$ + +### Normal line + +If a curve $C$ has a tangent line $L$ at point $p$, then the straight line $N$ through $P$ perpendicular to $L$ is called the **normal** to $C$ at $P$. The slope of the normal $s$ is the negative reciprocal of the slope of the curve $a$, that is + +$$ +s = \frac{-1}{a} +$$ + +## Derivative + +The **derivative** of a function $f$ is another function $f'$ defined by + +$$ +f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} +$$ + +at all points $x$ for which the limits exists. If $f'(x)$ exists, then $f$ is **differentiable** at $x$. + +## Differentiability implies continuity + +If $f$ is differentiable at $x$, then $f$ is continuous at $x$. + +**Proof:** Since $f$ is differentiable at $x$ + +$$ +\lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = f'(x) +$$ + +must exist. Then, using the [limit rules](limits.md/#limit-rules) + +$$ +\lim_{h \to 0} f(x + h) - f(x) = \lim_{h \to 0} (\frac{f(x + h) - f(x)}{h}) (h) = (f'(x)) (0) = 0 +$$ + +This is equivalent to $\lim_{h \to 0} f(x + h) = f(x)$, which says that $f$ is continuous at $x$. + +## Differentation rules + +* **Differentation of a sum:** $(f + g)'(x) = f'(x) + g'(x)$. + * **Proof:** Follows from the [limit rules](limits.md/#limit-rules) + +$$ +\begin{array}{ll} +(f + g)'(x) &= \lim_{h \to 0} \frac{(f + g)(x + h) - (f + g)(x)}{h}, \\ + &= \lim_{h \to 0} (\frac{f(x + h) - f(x)}{h} + \frac{g(x + h) - g(x)}{h}), \\ + &= f'(x) + g'(x). +\end{array} +$$ + +* **Differentation of a constant multiple:** $(C f)'(x) = C f'(x)$. + * **Proof:** Follows from the [limit rules](limits.md/#limit-rules) + +$$ +\begin{array}{ll} +(C f)'(x) &= \lim_{h \to 0} \frac{C f(x + h) - C f(x)}{h}, \\ + &= C \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}, \\ + &= C f'(x). +\end{array} +$$ + +* **Differentation of a product:** $(f g)'(x) = f'(x) g(x) + f(x) g'(x)$. + * **Proof:** Follows from the [limit rules](limits.md/#limit-rules) + +$$ +\begin{array}{ll} +(f g)'(x) &= \lim_{h \to 0} \frac{f(x+h) g(x+h) - f(x) g(x)}{h}, \\ + &= \lim_{h \to 0} (\frac{f(x+h) - f(x)}{h} g(x+h) + f(x) \frac{g(x+h) - g(x)}{h}), \\ + &= f'(x) g(x) + f(x) g'(x). +\end{array} +$$ + +* **Differentation of the reciprocal:** $(\frac{1}{f})'(x) = \frac{-f'(x)}{(f(x))^2}$. + * **Proof:** Follows from the [limit rules](limits.md/#limit-rules) + +$$ +\begin{array}{ll} +(\frac{1}{f})'(x) &= \lim_{h \to 0} \frac{\frac{1}{f(x+h)} - \frac{1}{f(x)}}{h}, \\ + &= \lim_{h \to 0} \frac{f(x) - f(x+h)}{h f(x+h) f(x)}, \\ + &= \lim_{h \to 0} (\frac{-1}{f(x+h) f(x)}) \frac{f(x+h) - f(x)}{h}, \\ + &= \frac{-1}{(f(x))^2} f'(x). +\end{array} +$$ + +* **Differentation of a quotient:** $(\frac{f}{g})'(x) = \frac{f'(x) g(x) - f(x) g'(x)}{(g(x))^2}$. + * **Proof:** Follows from the product and reciprocal rule + +$$ +\begin{array}{ll} +(\frac{f}{g})'(x) &= (f \frac{1}{g})'(x), \\ + &= f'(x) \frac{1}{g(x)} + f(x) (- \frac{g'(x)}{(g(x))^2}), \\ + &= \frac{f'(x) g(x) - f(x) g'(x)}{(g(x))^2}. +\end{array} +$$ + +* **Differentation of a composite:** $(f \circ g)'(x) = f'(g(x)) g'(x)$. + * **Proof:** Follows from the [limit rules](limits.md/#limit-rules) + +$$ +\begin{array}{ll} +(f \circ g)'(x) &= \lim_{h \to 0} \frac{f(g(x+h)) - f(g(x))}{h} \quad \mathrm{let} \space h = a - x, \\ + &= \lim_{a \to x} \frac{f(g(a)) - f(g(x))}{a - x}, \\ + &= \lim_{a \to x} (\frac{f(g(a)) - f(g(x))}{g(a) - g(x)}) (\frac{g(a) - g(x)}{a -x}), \\ + &= f'(g(x)) g'(x). +\end{array} +$$ + +## The derivative of the sine and cosine function + +The derivative of the sine function is the cosine function $\frac{d}{dx} \sin x = \cos x$. + +**Proof:** using the definition of the derivative, the addition formula for the sine and the [limit rules](limits.md/#limit-rules) + +$$ +\begin{array}{ll} +\frac{d}{dx} \sin x &= \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h}, \\ + &= \lim_{h \to 0} \frac{\sin x \cos h + \cos x \sin h}{h}, \\ + &= \lim_{h \to 0} (\sin x (\frac{\cos h - 1}{h}) + \cos x (\frac{\sin h}{h})), \\ + &= (\sin x) \cdot (0) + (\cos x) \cdot (1) = \cos x. +\end{array} +$$ + +The derivative of the cosine function is the negative of the sine function $\frac{d}{dx} \cos x = -\sin x$. + +**Proof:** using the derivative of the sine and the composite (chain) rule + +$$ +\begin{array}{ll} +\frac{d}{dx} \cos x &= \frac{d}{dx} \sin (\frac{\pi}{2} - x), \\ + &= (-1) \cos (\frac{\pi}{2} - x) = - \sin x. +\end{array} +$$ + +## Implicit differentation + +Implicit equations; equations that cannot be solved may still be differentiated by implicit differentation. + +**Example:** $x y^2 + y = 4 x$ + +$$ +\begin{array}{ll} +\frac{dy}{dx}(x y^2 + y = 4 x) &\implies (y^2 + 2 x y \frac{dy}{dx} + \frac{dy}{dx} = 4), \\ + &\implies (\frac{dy}{dx} = \frac{f- y^2}{1 + 2 x y}). +\end{array} +$$ + +## Rolle's theorem + +Suppose that the function $g$ is continuous on the closed and bounded interval $[a,b]$ and is differentiable in the open interval $(a,b)$. If $g(a) = g(b)$ then there exists a point $c$ in the open interval $(a,b)$ such that $g'(c) = 0$. + +**Proof:** By the [extereme value theorem](continuity.md/#the-extreme-value-theorem) $g$ attains its maximum and its minimum in $[a,b]$, if these are both attained at the endpoints of $[a,b]$, then $g$ is constant on $[a.b]$ and so the derivative of $g$ is zero at every point in $(a,b)$. + +Suppose then that the maximum is obtained at an interior point $c$ of $(a,b)$. For a real $h$ such that $c + h$ is in $[a,b]$, the value $g(c + h)$ is smaller or equal to $g(c)$ because $g$ attains its maximum at $c$. + +Therefore, for every $h>0$, + +$$\frac{g(c + h) - g(c)}{h} \leq 0,$$ + +hence, + +$$\lim_{h \downarrow 0} \frac{g(c + h) - g(c)}{h} \leq 0.$$ + +Similarly, for every $h < 0$ + +$$\lim_{h \uparrow 0} \frac{g(c + h) - g(c)}{h} \geq 0.$$ + +Thereby obtaining, + +$$\lim_{h \to 0} \frac{g(c + h) - g(c)}{h} = 0 = g'(c)$$ + +The proof for a minimum value at $c$ is similar. + +## Mean-value theorem + +Suppose that the function $f$ is continuous on the closed and bounded interval $[a,b]$ and is differentiable in the open interval $(a,b)$. Then there exists a point $c$ in the open interval $(a,b)$ such that + +$$ +\frac{f(b) - f(a)}{b - a} = f'(c). +$$ + +**Proof:** Define $g(x) = f(x) - r x$, where $r$ is a constant. Since $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, the same is true for $g$. Now $r$ is chosen such that $g$ satisfies the conditions of [Rolle's theorem](differentation.md/#rolles-theorem). Namely + +$$ +\begin{array}{ll} +g(a) = g(b) &\iff f(a) - ra = f(b) - rb \\ + &\iff r(b - a) = f(b) - f(a) \\ + &\iff r = \frac{f(b) - f(a)}{b - a} +\end{array} +$$ + +By [Rolle's theorem](differentation.md/#rolles-theorem), since $g$ is differentiable and $g(a) = g(b)$, there is some $c$ in $(a,b)$ for which $g'(c) = 0$, and it follows from the equality $g(x) = f(x) - rx$ that, + +$$ +\begin{array}{ll} +g'(x) &= f'(x) - r\\ +g'(c) &= 0 \\ +g'(c) &= f'(c) - r = 0 \implies f'(c) = r = \frac{f(b) - f(a)}{b - a} +\end{array} +$$ + +## Generalized Mean-value theorem + +If the functions $f$ and $g$ are both continuous on $[a,b]$ and differentiable on $(a,b)$ and if $g'(x) \neq 0$ for every $x$ between $(a,b)$. Then there exists a $c \in (a,b)$ such that + +$$ +\frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(c)}{g'(c)}. +$$ + +**Proof:** Let $h(x) = (f(b) - f(a))(g(x) - g(a)) - (g(b) - g(a))(f(x) - f(a))$. + +Applying [Rolle's theorem](differentation.md/#rolles-theorem), since $h$ is differentiable and $h(a) = h(b)$, there is some $c$ in $(a,b)$ for which $h'(c) = 0$ + +$$ +h'(c) = (f(b) - f(a))g'(c) - (g(b) - g(a))f'(c) = 0, +$$ + +$$ +\begin{array}{ll} +\implies (f(b) - f(a))g'(c) = (g(b) - g(a))f'(c), \\ +\implies \frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(c)}{g'(c)}. +\end{array} +$$ \ No newline at end of file diff --git a/docs/wiskunde/calculus/extremes-values.md b/docs/wiskunde/calculus/extremes-values.md new file mode 100755 index 0000000..256b246 --- /dev/null +++ b/docs/wiskunde/calculus/extremes-values.md @@ -0,0 +1,47 @@ +# Extreme values + +## Absolute extreme values + +Function $f$ has an **absolute maximum value** $f(x_0)$ at the point $x_0$ in its domain if $f(x) \leq f(x_0)$ holds ofr every $x$ in the domain of $f$. + +Similarly, $f$ has an **absolute minimum value** $f(x_1)$ at the point $x_1$ in its domain if $f(x) \geq f(x_1)$ holds for every $x$ in the domain of $f$. + +## Local extreme values + +Function $f$ has an **local maximum value** $f(x_0)$ at the point $x_0$ in its domain provided there exists a number $h > 0$ such that $f(x) \leq f(x_0)$ whenever $x$ is in the domain of $f$ and $|x - x_0| < h$. + +Similarly, $f$ has an **local minimum value** $f(x_1)$ at the point $x_1$ in its domain provided there exists a number $h > 0$ such that $f(x) \geq f(x_1)$ whenever $x$ is in the domain of $f$ and $|x - x_1| < h$. + +## Critical points + +A critical point is a point $x \in \mathrm{Dom}(f)$ where $f'(x) =0$. + +## Singular points + +A singular point is a point $x \in \mathrm{Dom}(f)$ where $f'(x)$ is not defined. + +## Endpoints + +An endpoint $x \in \mathrm{Dom}(f)$ that does not belong to any open interval contained in $\mathrm{Dom}(f)$ + +## Locating extreme values + +If the function $f$ is defined on an interval $I$ and has a local maxima or minima in $I$ then the point must be either a critical point of $f$, a singular point of $f$ or an endpoint of $I$. + +**Proof:** + +Suppose that $f$ has a local maximum value at $x_0$ and that $x_0$ is neither an endpoint of the domain of $f$ nor a singular point of $f$. Then for some $h > 0$, $f(x)$ is defined on the open interval $(x_0 - h, x_0 + h)$ and has an absolute maximum at $x_0$. Also, $f'(x_0) exists, following from [Rolle's theorem](differentation.md#rolles-theorem). + +## The first derivative test + +### Example + +Find the local and absolute extreme values of $f(x) = x^4 - 2x^2 -3$ on the interval $[-2,2]$. + +$$f'(x) = 4x^3 - 4x = 4x(x^2 - 1) = 4x(x - 1)(x + 1)$$ + +| $x$ | $-2$| $-1$| $0$ | $1$ | $2$ | +| --- | --- | --- | --- | --- | --- | +| $f'$| |- 0 +|+ 0 -|- 0 +| | +| $f$ | max | min | max | min | max | +| | EP | CP | CP | CP | EP | \ No newline at end of file diff --git a/docs/wiskunde/calculus/first-order-differential-equations.md b/docs/wiskunde/calculus/first-order-differential-equations.md new file mode 100755 index 0000000..3ac664c --- /dev/null +++ b/docs/wiskunde/calculus/first-order-differential-equations.md @@ -0,0 +1,66 @@ +# First-order differential equations + +## First order linear differential equations + +A first-order **linear** differential equation is one of the type + +$$ +\frac{dy}{dx} + p(x) y = q(x). +$$ + +where $p(x)$ and $q(x)$ are given functions, which may be assumed to be continuous. The equation is called **nonhomogeneous** unless $q(x)$ is dentically zero. The corresponding **homogeneous** equation + +$$ +\frac{dy}{dx} + p(x)y = 0, +$$ + +is separable and so is easily solved. By seperation of variables + +$$ +\frac{dy}{dx} = -p(x)y \implies \int \frac{1}{y}dy = \int p(x)dx. +$$ + +Though, pay attention to absolute values. + +There are two methods for solving nonhomogeneous equations. + +### Integration factor + +The first method is by using an integrating factor. Let $\mu(x)$ be an antiderivative of $p(x)$ and multiply the equation by $e^{\mu(x)}$. + +$$ +\begin{array}{ll} +e^{\mu(x)} \frac{dy}{dx} + e^{\mu(x)} p(x) y = e^{\mu(x)} q(x) &\implies \frac{d}{dx}(e^{\mu(x)} y) = q(x) e^{\mu(x)}, \\ +&\implies e^{\mu(x)} y = \int q(x) e^{\mu(x)}dx, \\ +&\implies y(x) = e^{-\mu(x)} \int q(x) e^{\mu(x)}dx. +\end{array} +$$ + +### Variation of the constant + +The second method is by using a variation of a constant. Let $\mu(x)$ be an antiderivative of $p(x)$ and solve the homegeneous equation + +$$ +\frac{dy}{dx} + p(x)y = 0 \implies y(x) = k e^{-\mu(x)}. +$$ + +Try: + +$$ +y'(x) + p(x) y(x) = q(x) = k'(x) e^{\mu(x)}, +$$ + +thus $k'(x) = q(x) e^{\mu(x)}$. + +#### Example + +Solve $\frac{dy}{dx} + 2xy = x$ with $y(0) = 3$. + + +First solving the homogeneous equation + +$$ +\begin{array}{ll} +\frac{dy}{dx} + 2xy = 0 &\implies \int \frac{1}{y} dy = -2\int xdx +\end{array} +$$ \ No newline at end of file diff --git a/docs/wiskunde/calculus/improper-integrals.md b/docs/wiskunde/calculus/improper-integrals.md new file mode 100755 index 0000000..92b57ce --- /dev/null +++ b/docs/wiskunde/calculus/improper-integrals.md @@ -0,0 +1,151 @@ +# Improper integrals + +Proper integrals are [definite integrals](integration.md/#the-definite-integral) where the integrand $f$ is *continuous* on a *closed, finite* interval $[a,b]$. For positive $f$ it corresponds to the area of a **bounded region** of the plane, a region contained inside some disk of finite radius with centre at the origin. To extend the definite integral by allowing for two possibilities excluded in the situation described above. + +1. There may exist $a=-\infty$ or $b=\infty$ or both. +2. $f$ may be unbounded as $x$ approaches $a$ or $b$ or both. + +Integrals satisfying 1. are called **improper integrals of type I.** and integrals satisfying 2. are called **improper integrals of type II**. + +## Improper integrals of type I + +If $f$ is continuous on $[a,\infty]$, the improper integral of $f$ over $[a,\infty]$ is defined as a limit of proper integrals: + +$$ +\int_a^\infty f(x)dx = \lim_{R \to \infty} \int_a^R f(x)dx. +$$ + +Similarly, if $f$ is continuous on $[-\infty,b]$, then the improper integrals is defined as: + +$$ +\int_{-\infty}^b f(x)dx = \lim_{R \to -\infty} \int_R^b f(x)dx. +$$ + +In either case, if the limit exists, the improper integral **converges**. If the limit does not exist, the improper integral **diverges**. If the limit is $\infty$ or $-\infty$, the proper integral **diverges to (negative) infinity**. + +## Improper integrals of type II + +If $f$ is continuous on the interval $(a,b]$ and is possibly unbounded near $a$, the improper integral may be defined as: + +$$ +\int_a^b f(x)dx = \lim_{c \downarrow a} \int_c^b f(x)dx. +$$ + +Similarly, if $f$ is continuous on $[a,b)$ and is possibly unbounded near $b$, the improper integral may be defined as: + +$$ +\int_a^b f(x)dx = \lim_{c \uparrow b} \int_a^c f(x)dx. +$$ + +These improper integrals may also converge, diverge or diverge to (negative) infinity. + +## p-integrals + +Summerizing the behaviour of improper integrals of types I and II for powers of $x$ if $0 < a< \infty$, then: + +1. +$$ +\int_a^\infty x^{-p}dx = + \begin{cases} + \text{converges to } \frac{a^{1-p}}{p-1} \quad \text{if } p > 1, \\ + \text{diverges to } \infty \quad \text{if } p \leq 1. + \end{cases} +$$ + + +2. +$$ +\int_0^a x^{-p}dx = + \begin{cases} + \text{converges to } \frac{a^{1-p}}{1-p} \quad \text{if } p < 1, \\ + \text{diverges to } \infty \quad \text{if } p \geq 1. + \end{cases} +$$ + +**Proof of 1:** + +For $p=1$: + +$$ +\int_a^\infty x^{-1}dx = \lim_{R \to \infty} \int_a^R x^{-1}dx = \lim_{R \to \infty} (\ln R - \ln a) = \infty. +$$ + +For $p < 1$: + +$$ +\begin{array}{ll} +\int_a^\infty x^{-p}dx &= \lim_{R \to \infty} \int_a^R x^{-p}dx, \\ + &= \lim_{R \to \infty} [\frac{x^{-p+1}}{-p+1}]_a^R, \\ + &= \lim_{R \to \infty} \frac{R^{1-p}-a^{1-p}}{1-p} = \infty. +\end{array} +$$ + +For $p > 1$ + +$$ +\begin{array}{ll} +\int_a^\infty x^{-p}dx &= \lim_{R \to \infty} \int_a^R x^{-p}dx, \\ + &= \lim_{R \to \infty} [\frac{x^{-p+1}}{-p+1}]_a^R, \\ + &= \lim_{R \to \infty} \frac{a^{-(p-1)}-R^{-(p-1)}}{p-1} = \frac{a^{1-p}}{p-1}. +\end{array} +$$ + +**Proof of 2:** + +For $p=1$: + +$$ +\int_0^a x^{-1}dx = \lim_{c \space\downarrow\space 0} \int_c^a x^{-1}dx = \lim_{c \space\downarrow\space 0} (\ln a - \ln c) = \infty. +$$ + +For $p > 1$ + +$$ +\begin{array}{ll} +\int_0^a x^{-p}dx &= \lim_{c \space\downarrow\space 0} \int_c^a x^{-p}dx, \\ + &= \lim_{c \space\downarrow\space 0} [\frac{x^{-p+1}}{-p+1}]_c^a, \\ + &= \lim_{c \space\downarrow\space 0} \frac{c^{-(p-1)} - a^{-(p-1)}}{p-1} = \infty. +\end{array} +$$ + +For $p < 1$: + +$$ +\begin{array}{ll} +\int_0^a x^{-p}dx &= \lim_{c \space\downarrow\space 0} \int_c^a x^{-p} dx, \\ + &= \lim_{c \space\downarrow\space 0} [\frac{x^{-p+1}}{-p+1}]_c^a, \\ + &= \lim_{c \space\downarrow\space 0} \frac{a^{1-p}-c^{1-p}}{1-p} = \frac{a^{1-p}}{1-p}. +\end{array} +$$ + +## Comparison theorem for integrals + +Let $-\infty \leq a < b \leq \infty$, and suppose that functions $f$ and $g$ are continuous on the interval $(a,b)$ and satisfy $0 \leq f(x) \leq g(x)$. If $\int_a^b g(x)dx$ converges, then so does $\int_a^b f(x)dx$, and: + +$$ +\int_a^b f(x)dx \leq \int_a^b g(x)dx. +$$ + +Equivalently, if $\int_a^b f(x)dx$ diverges to $\infty$, then so does $\int_a^b g(x)dx$. + +**Proof:** Since both integrands are nonnegative, there are only two possibilities for each integral: it can either converge to a nonnegative number or diverge to $\infty$. Since $f(x) \leq g(x)$ on $(a,b)$, it follows by the [properties of the definite integral](integration.md/#properties) that if $a < r < s < b$, then: + +$$ +\int_r^s f(x)dx \leq \int_r^s g(x)dx. +$$ + +By taking limits as $r \space\downarrow\space a$ and $s \space\uparrow\space b$. + +### To prove convergence + +To find a function $g$, such that + +1. $\forall x \in [a,\infty], \space 0 \leq f(x) \leq g(x)$. +2. $\int_0^\infty g(x)dx$ is convergent. + +### To prove divergence + +To find a function $f$ such that + +1. $\forall x \in [a,\infty], \space g(x) \geq f(x) \geq 0$. +2. $\int_0^\infty f(x)dx$ is divergent. \ No newline at end of file diff --git a/docs/wiskunde/calculus/integration-techniques.md b/docs/wiskunde/calculus/integration-techniques.md new file mode 100755 index 0000000..6c886fa --- /dev/null +++ b/docs/wiskunde/calculus/integration-techniques.md @@ -0,0 +1,90 @@ +# Integration techniques + +## Elementary integrals + +$$ +\int \frac{1}{a^2 + x^2} dx = \frac{1}{a} \arctan(\frac{x}{a}) + C +$$ + +$$ +\int \frac{1}{\sqrt{a^2-x^2}} dx = \arcsin(\frac{x}{a}) + C +$$ + +## Linearity of the integral + +$$ +\int Af(x) + Bg(x)dx = A\int f(x)dx + B\int g(x)dx +$$ + +**Proof:** is missing. + +## Substitution + +Suppose that $g$ is a differentiable on $[a,b]$, that satisfies $g(a)=A$ and $g(b)=B$. Also suppose that $f$ is continuous on the range of $g$, then + +let $u = g(x)$ then $du = g'(x)dx$, + +$$ +\int_a^b f(g(x))g'(x)dx = \int_A^B f(u)du. +$$ + +## Inverse substitution + +Inverse substitutions appear to make the integral more complicated, thereby this strategy must act as last resort. Substituting $x=g(u)$ in the integral + +$$ +\int_a^b f(x)dx, +$$ + +leads to the integral + +$$ +\int_{x=a}^{x=b} f(g(u))g'(u)du. +$$ + +## Integration by parts + +Suppose $U(x)$ and $V(x)$ are two differentiable functions. According to the [product rule](differentation.md/#differentation-rules), + +$$ +\frac{d}{dx}(U(x)V(x)) = U(x) \frac{dV}{dx} + V(x) \frac{dU}{dx}. +$$ + +Integrating both sides of this equation and transposing terms + +$$ +\int U(x) \frac{dV}{dx} dx = U(x)V(x) - \int V(x) \frac{dU}{dx} dx, +$$ + +obtaining: + +$$ +\int U dV = U V - \int V dU. +$$ + +For definite integrals that is: + +$$ +\int_a^b f'(x)g(x)dx = [f(x)g(x)]_a^b - \int_a^b f(x)g'(x)dx. +$$ + +## Integration of rational functions + +Let $P(x)$ and $Q(x)$ be polynomial functions with real coefficients. Forming a rational function, $\frac{P(x)}{Q(x)}$. Let $\frac{P(x)}{Q(x)}$ be a **strictly proper rational function**, that is; $\mathrm{deg}(P(x)) < \mathrm{deg}(Q(x))$. If the function is not it can be possibly made into a **strictly proper rational function** by using **long division**. + +Then, $Q(x)$ can be factored into the product of a constant $K$, real linear factors of the form $x-a_i$, and real quadratic factors of the form $x^2+b_ix + c_i having no real roots. + +The rational function can be expressed as a sum of partial fractions. Corresponding to each factor $(x-a)^m$ of $Q(x)$ the decomposition contains a sum of fractions of the form + +$$ +\frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + ... + \frac{A_m}{(x-a)^m}. +$$ + +Corresponding to each factor $(x^2+bx+c)^n$ of $Q(x)$ the decomposition contains a sum of fractions of the form + +$$ +\frac{B_1x+C_1}{x^2+bx+c} + \frac{B_2x+C_2}{(x^2+bx+c)^2} + ... + \frac{B_nx+C_n}{(x^2+bx+c)^n}. +$$ + +The constant $A_1,A_2,...,A_m,B_1,B_2,...,B_n,C_1,C_2,....,C_n$ can be determined by adding up the fractions in the decomposition and equating the coefficients of like powers of $x$ in the numerator of the sum those in $P(x)$. + diff --git a/docs/wiskunde/calculus/integration.md b/docs/wiskunde/calculus/integration.md new file mode 100755 index 0000000..e81b26b --- /dev/null +++ b/docs/wiskunde/calculus/integration.md @@ -0,0 +1,191 @@ +# Integration + +## Sigma notation + +if $m$ and $n$ are integers with $m \leq n$, and if $f$ is a function defined as $f: \{m,m+1,...,n\} \to \mathbb{R}$, the symbol $\sum_{i=m}^{n} f(i)$ represents the sum of the values of $f$ at those integers: + +$$ +\sum_{i=m}^{n} f(i) = f(m) + f(m+1) + f(m+2) + ... + f(n). +$$ + +The explicit sum appearing on the right side of this equation is the **expansion** of the sum represented in sigma notation on the left side. + +## Partitions + +Let $P$ be a finite set of points arranged in order between $a$ and $b$ on the real line + +$$ +P = {x_0, x_1, ... , x_{n-1}, x_n}, +$$ + +where $a = x_0 < x_1 < ... < x_{n-1} < x_n = b$. Such a set $P$ is called a **partition** of $[a,b]$; it divides $[a,b]$ into $n$ subintervals of which the *i*th is $[x_{i-1},x_i]. The length of the *i*th subinterval of $P$ is + +$$ +\Delta x_i = x_i - x_{i-1} \quad \mathrm{for} \space 1 \leq i \leq n, +$$ + +Then, the **norm** of the partition $P$ is defined as + +$$ +\parallel P \parallel = \max_{1 \leq i \leq n} \Delta x_i +$$ + +If the function $f$ is continuous on the interval $[a,b]$, it is continuous on each subinterval $[x_{i-1},x_i]$, and has a maximum $u_i$ and minimum $l_i$ on each subinterval by the [Extreme value theorem](continuity.md/#the-extreme-value-theorem) such that + +$$ +f(l_i) \leq f(x) \leq f(u_i) \quad \forall x \in [x_{i-1},x_i]/ +$$ + +## Upper and lower Riemann sums + +The **lower Riemann sum**, $L(f,P)$, and the **upper Riemann sum**, $U(f,P)$, for the function $f$ amd the partition $P$ are defined by: + +$$ +\begin{array}{ll} +L(f,P) &= f(l_1)\Delta x_1 + f(l_2)\Delta x_2 + ... + f(l_n)\Delta x_n \\ + &= \sum_{i=1}^n f(l_i)\Delta x_i, \\ +U(f,P) &= f(u_1)\Delta x_1 + f(u_2)\Delta x_2 + ... + f(u_n)\Delta x_n \\ + &= \sum_{i=1}^n f(u_i)\Delta x_i. +\end{array} +$$ + +**Theorem:** for any partitions $P$, $Q$ on $[a,b]$ all lower Riemann sums are smaller than or equal to any upper Riemann sums: + +$$ +L(f,P) \leq U(f,Q). +$$ + +**Proof:** let $P$, $Q$ be partitions on $[a,b]$, suppose $L(f,P) \leq U(f,Q)$, define $R = P \cup Q$, $R$ is a refinement of $P$, $Q$. Then, + +$$ +L(f,P) \leq L(f,R) \leq U(f,R) \leq U(f,Q). +$$ + +## The definite integral + +Suppose there exists exactly one number $I \in \mathbb{R}$ such that for every partition $P$ of $[a,b]$: + +$$ +L(f,P) \leq I \leq U(f,P). +$$ + +Then the function $f$ is integrable on $[a,b]$ and $I$ is called the definite integral + +$$ +I = \int_a^b f(x) dx. +$$ + +**Theorem:** suppose that a function $f$ is bounded on the interval $[a,b]$, then $f$ is integrable on $[a,b]$ if and only if $\forall \varepsilon > 0$ there exists a partition $P$ of $[a,b]$ such that + +$$ +U(f,P) - L(f,P) < \varepsilon. +$$ + +**Proof:** let $a,b \in \mathbb{R}$, $\forall \varepsilon > 0$ there is $|a-b| < \varepsilon$ then $a=b$. + +**Theorem:** if $f$ is continuous on the interval $[a,b]$, then $f$ is integrable on $[a,b]$. + +**Proof:** is missing... + +### Properties + +* If $a \leq b$ and $f(x) \leq g(x) \space \forall x \in [a,b]$: + +$$ +\int_a^b f(x)dx \leq \int_a^b g(x)dx. +$$ + +**Proof:** is missing... + +* The **triangle inequality** for sums extends to definite integrals. If $a \leq b$, then + +$$ +|\int_a^b f(x)dx| \leq \int_a^b |f(x)|dx. +$$ + +**Proof:** is missing... + +* Integral of an odd function $f(-x) = -f(x)$: + +$$ +\int_{-a}^a f(x)dx = 0. +$$ + +**Proof:** is missing... + +* Integral of an even function $f(-x) = f(x)$: + +$$ +\int_{-a}^a f(x)dx = 2\int_0^a f(x)dx. +$$ + +**Proof:** is missing... + +## The Mean-value theorem for integrals + +If the function $f$ is continuous on $[a,b]$ then there exists a point $c$ in $[a,b]$ such that + +$$ +\int_a^b f(x)dx = (b-a)f(c) +$$ + +**Proof:** $\forall x \in [a,b]$, + +let $m \leq f(x) \leq M$, + +$$m(b-a)=\int_a^b mdx \leq \int_a^b f(x)dx \leq \int_a^b Mdx = M(b-a),$$ + +$$m \leq \frac{1}{b-a} \int_a^b f(x)dx \leq M,$$ + +According to [the intermediate value theorem](continuity.md/#the-intermediate-value-theorem) there exists a $c \in [a,b]$ such that + +$$\frac{1}{b-a} \int_a^b f(x)dx = f(c)$$ + +## Piecewise continuous functions + +Let $c_0 < c_1 < ... < c_n$ be a finite set of points on the real line. A function $f$ defined on $[c_0,c_n]$ except possibly at some of the points $c_i$, $(0 \leq i \leq n)$, is called piecewise continuous on that interval if for each $i$ $(1 \leq i \leq b)$ there exists a function $F_i$ continuous on the *closed* interval $[c_{i-1},c_i]$ such that + +$$ +f(x) = F_i(x). +$$ + +In this case, te integral of $f$ from $c_0$ to $c_n$ is defined to be + +$$ +\int_{c_0}^{c_n} f(x)dx = \sum_{i=1}^n \int_{c_i-1}^{c_i} F_i(x)dx. +$$ + +## The fundamental theorem of calculus + +Suppose that the function $f$ is continuous on an interval $I$ containing the point $a$. + +**Part I.** Let the function $F$ be defined on $I$ by + +$$ +F(x) = \int_a^x f(t)dt. +$$ + +Then $F$ is differentiable on $I$, and $F'(x) = f(x)$ there. Thus, $F$ is an antiderivative of $f$ on $I$: + +$$ +\frac{d}{dx} \int_a^x f(t)dt = f(x). +$$ + +**Part II.** If $G(x)$ is *any* antiderivative of $f(x)$ on $I$, so that $G'(x) = f(x)$ on $I$, then for any $b$ in $I$ there is + +$$ +\int_a^b f(x)dx = G(b) - G(a). +$$ + +**Proof:** using the definitions of the derivative + +$$ +\begin{array}{ll} +F'(x) &= \lim_{h \to 0} \frac{F(x+h)-F(x)}{h}, \\ + &= \lim_{h \to 0} \frac{1}{h}(\int_a^{x+h} f(t)dt - \int_a^x f(t)dt), \\ + &= \lim_{h \to 0} \frac{1}{h} \int_x^{x+h} f(t)dt, \\ + &= \lim_{h \to 0} hf(c) \quad \mathrm{for \space some} \space c \in [x,x+h], \\ + &= \lim_{c \to x} f(c), \\ + &= f(x). +\end{array} +$$ \ No newline at end of file diff --git a/docs/wiskunde/calculus/limits.md b/docs/wiskunde/calculus/limits.md new file mode 100755 index 0000000..61a6e4b --- /dev/null +++ b/docs/wiskunde/calculus/limits.md @@ -0,0 +1,90 @@ +# Limits + +If $f(x)$ is defined for all $x$ near a, except possibly at a itself, and if it can be ensured that $f(x)$ is as close to $L$ by taking $x$ close enough to $a$, but not equal to $a$. Then $f$ approaches the **limit** $L$ as $x$ approaches $a$: + +$$ +\lim_{x \to a} f(x) = L +$$ + +## One-sided limits + +If $f(x)$ is defined on some interval $(b,a)$ extending to the left of $x=a$, and if it can be ensured that $f(x)$ is as close to $L$ by taking $x$ to the left of $a$ and close enough to $a$, then $f(x) has **left limit** $L$ at $x=a$ and: + +$$ +\lim_{x \uparrow a} f(x) = L. +$$ + +If $f(x)$ is defined on some interval $(b,a)$ extending to the right of $x=a$ and if it can be ensured that $f(x)$ is as close to $L$ by taking $x$ to the right of $a$ and close enough to $a$, then $f(x) has **right limit** $L$ at $x=a$ and: + +$$ +\lim_{x \downarrow a} f(x) = L. +$$ + +## Limits at infinity + +If $f(x)$ is defined on an interval $(a,\infty)$ and if it can be ensured that $f(x)$ is as close to $L$ by taking $x$ large enough, then $f(x)$ **approaches the limit $L$ as $x$ approaches infinity** and + +$$ +\lim_{x \to \infty} f(x) = L +$$ + +## Limit rules + +If $\lim_{x \to a} f(x) = L$, $\lim_{x \to a} g(x) = M$, and $k$ is a constant then, + +* **Limit of a sum:** $\lim_{x \to a}[f(x) + g(x)] = L + M$. +* **Limit of a difference:** $\lim_{x \to a}[f(x) - g(x)] = L - M$. +* **Limit of a multiple:** $\lim_{x \to a}k f(x) = k L$. +* **Limit of a product:** $\lim_{x \to a}f(x) g(x) = L M$. +* **Limit of a quotient:** $\lim_{x \to a}\frac{f(x)}{g(x)} = \frac{L}{M}$, if $M \neq 0$. +* **Limit of a power:** $\lim_{x \to a}[f(x)]^\frac{m}{n} = L^{\frac{m}{n}}$. + +## Formal definition of a limit + +The limit $\lim_{x \to a} f(x) = L$ means, + +$$ +\forall \varepsilon > 0, \exists \delta \space \mathrm{,s.t.,} \space +\forall x \in \mathbb{R}, \space 0<|x-a|<\delta \implies |f(x) - L| < \varepsilon +$$ + +For one-sided, infinite and limits at infinity there are similar formal definitions. + +### Example + +Applying the formal definition of a limit for $\lim_{x \to 4}\sqrt{2x + 1}$ + +* Given $\varepsilon > 0$ +* Choose $\delta = \frac{\varepsilon}{2}$ +* Suppose $0 < |x - 4| < \delta$ +* Check $|\sqrt{2x + 1} - 3|$ + +$$ +\begin{array}{ll} +|\sqrt{2x + 1} - 3| &= |\frac{(\sqrt{2x + 1} - 3)(\sqrt{2x + 1} + 3)}{\sqrt{2x + 1} + 3}|\\ + &= \frac{2|x - 4|}{\sqrt{2x + 1} + 3}\\ + &< 2|x-4|\\ + &< 2\delta = \varepsilon +\end{array} +$$ + +## Squeeze Theorem + +Suppose that $f(x) \leq g(x) \leq h(x)$ holds for all $x$ in some open interval containing $a$, except possibly at $x=a$ itself. Suppose also that + +$$\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L.$$ + +Then $\lim_{x \to a} g(x) = L$ also. Similar statements hold for left and right limits. + +### Example + +Applying squeeze theorem on $\lim_{x \to 0} x^2 \cos(\frac{1}{x})$. + +$$ +\begin{array}{ll} +\forall x \neq 0\\ +-1 \leq \cos(\frac{1}{x}) \leq 1 \implies -x^2 \leq x^2 \cos(\frac{1}{x}) \leq x^2\\ +\mathrm{Since,} \space \lim_{x \to 0} x^2 = \lim_{x \to 0} -x^2 = 0\\ +\lim_{x \to 0} x^2 \cos(\frac{1}{x}) = 0 +\end{array} +$$ \ No newline at end of file diff --git a/docs/wiskunde/calculus/sets-and-numbers.md b/docs/wiskunde/calculus/sets-and-numbers.md new file mode 100755 index 0000000..cd698ea --- /dev/null +++ b/docs/wiskunde/calculus/sets-and-numbers.md @@ -0,0 +1,19 @@ +# Sets and numbers + +A collection of well defined objects. Such as the set of all even numbers + +$$ +V = \{x \in \mathbb{N} | x = 2n \space \mathrm{for} \space n \in \mathbb{N}\}. +$$ + +## Interception and union + +$$ +\mathrm{Let} \ +V = \{x \in \mathbb{R} | 2 \leq x < 4\} = [2 ; 4), \ +W = \{x \in \mathbb{R} | \pi < x < 2\pi\} = (\pi ; 2\pi). +$$ + +$$\mathrm{Interception: } V \cap W = (\pi ; 4).$$ + +$$\mathrm{Union: } V \cup W = [2 ; 2\pi).$$ \ No newline at end of file diff --git a/docs/wiskunde/calculus/taylor-polynomials.md b/docs/wiskunde/calculus/taylor-polynomials.md new file mode 100755 index 0000000..d9481ab --- /dev/null +++ b/docs/wiskunde/calculus/taylor-polynomials.md @@ -0,0 +1,197 @@ +# Taylor polynomials + +## Linearization + +A function $f(x)$ about $x = a$ may be linearized into + +$$ +P_1(x) = f(a) + f'(a)(x-a), +$$ + +obtaining a polynomial that matches the value and derivative of $f$ at $x = a$. + +## Taylor's theorem + +Even better approximations of $f(x)$ can be obtained by using higher degree polynomials if $f^{n+1}(t)$ exists for all $t$ in an interval containing $a$ and $x$. Thereby matching more derivatives at $x = a$, + +$$ +P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2+ ... + \frac{f^{(n)}(a)}{n!}(x-a)^n. +$$ + +Then the error $E_n(x) = f(x) - P_n(x)$ in the approximation $f(x) \approx P_n(x)$ is given by + +$$ +E_n(a) = \frac{f^{(n+1)}(s)}{(n+1)!}(x-a)^{n+1}, +$$ + +where $s$ is some number between $a$ and $x$. The resulting formula + +$$ +f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + ... + \frac{f^{(n)}(a)}{n!}(x-a)^n + \frac{f^{(n+1)}(s)}{(n+1)!}(x-a)^{n+1}, +$$ + +for some $s$ between $a$ and $x$, is called **Taylor's formula with Lagrange remainder**; the Lagrange is the error term $E_n(x)$. + +**Proof:** + +Observe that the case $n=0$ of Taylor's formula, namely + +$$ +f(x) = P_0(x) + E_0(x) = f(a) + \frac{f'(s)}{1!}(x-a), +$$ + +is just the [Mean-value theorem](differentation.md#mean-value-theorem) for some $s$ between $a$ and $x$ + +$$ +\frac{f(x) - f(a)}{x-a} = f'(s). +$$ + +Using induction to prove for $n > 0$. Suppose $n = k-1$ where $k \geq 1$ is an integer, then + +$$ +E_{k-1}(x) = \frac{f^{(k)}(s)}{k!}(x-a)^k, +$$ + +where $s$ is some number between $a$ and $x$. Consider the next higher case: $n=k$. Applying the [Generalized Mean-value theorem](differentation.md/#generalized-mean-value-theorem) to the functions $E_k(t)$ and $(t-a)^{k+1}$ on $[a,x]$. Since $E_k(a)=0$, a number $u$ in $(a,x)$ is obtained such that + +$$ +\frac{E_k(x) - E_k(a)}{(x-a^{k+1}) - (a-a)^{k+1}}= \frac{E_k(x)}{(x-a)^{k+1}} = \frac{E_k'(u)}{(k+1)(u - a)^k}. +$$ + +Since + +$$ +\begin{array}{ll} +E_k'(u)&=\frac{d}{dx}(f(x)-f(a)-f'(a)(x-a)-\frac{f''(a)}{2!}(t-a)^2-...-\frac{f^{(k)}(a)}{k!}(t-a)^k)|_{x=u} \\ + &= f'(u) - f'(a) - f''(a)(u-a)-...-\frac{f^{(k)}(a)}{(k-1)!}(u-a)^{k-1} +\end{array} +$$ + +is just $E_{k-1}(u)$ for the function $f'$ instead of $f$. By the induction assumption it is equal to + +$$ +\frac{(f')^{(k)}(s)}{k!}(u-a)^k = \frac{f^{(k+1)}(s)}{k!}(u-a)^k +$$ + +for some $s$ between $a$ and $u$. Therefore, + +$$ +E_k(x) = \frac{f^{(k+1)}(s)}{(k+1)!}(x-a)^{k+1} +$$ + +## Big-O notation + +$f(x) = O(u(x))$ for $x \to a$ if and only if there exists a $k > 0$ such that + +$$ +|f(x)| \leq k|u(x)| +$$ + +For all $x$ in the open interval around $x=a$. + +The following properties follow from the definition: + +1. If $f(x) = O(u(x))$ as $x \to a$, then $Cf(x) = O(u(x))$ as $x \to a$ for any value of the constant $C$. +2. If $f(x) = O(u(x))$ as $x \to a$ and $g(x) = O(u(x))$ as $x \to a$, then $f(x) \pm g(x) = O(u(x))$ as $x \to a$. +3. If $f(x) = O((x-a)^ku(x))$ as $x \to a$, then $\frac{f(x)}{(x-a)^k} = O(u(x))$ as $x \to a$ for any constant $k$. + +If $f(x) = Q_n(x) + O((x-a)^{n+1})$ as $x \to a$, where $Q_n$ is a polynomial of degree at most $n$, then $Q_n(x) = P_n(x)$. + +**Proof:** Follows from the properties of the big-O notation + +Let $P_n$ be the Taylor polynomial, then properties 1 and 2 of big-O imply +that $R_n(x) = Q_n(x) - P_n(x) = O((x - a)^{n+1})$ as $x \to a$. It must be shown that $R_n(x)$ is identically zero so that $Q_n(x) = P_n(x)$ for all $x$. $R_n(x)$ may be written in the form + +$$ +R_n(x) = c_0 + c_1(x-a) + c_2(x-a)^2 + ... + c_n(x-a)^n +$$ + +If $R_n(x)$ is not identically zero, then there is a smallest coefficient $c_k$ $k \leq n$, such that $c_k \neq 0$, but $c_j = 0$ for $0 \leq j \leq k -1$ + +$$ +R_n(x) = (x-a)^k(c_k + c_{k+1}(x-a) + ... + c_n(x-a)^{n-k}). +$$ + +Therefore, + +$$ +\lim_{x \to a} \frac{R_n(x)}{(x-a)^k} = c_k \neq 0. +$$ + +However, by property 3 + +$$ +\frac{R_n(x)}{(x-a)^k} = O((x-a)^{n+1-k}). +$$ + +Since $n+1-k > 0$, $\frac{R_n(x)}{(x-a)^k} \to 0$ as $x \to a$. This contradiction shows that $R_n(x)$ must be +identically zero. + +## Maclaurin formulas + +Some Maclaurin formulas with errors in big-O notation. These may be used in constructing Taylor polynomials from compsite functions. As $x \to 0$ + +1. $$\frac{1}{1-x} = 1 + x + ... + x^n + O(x^{n+1}),$$ +2. $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - ... + (-1)^{n-1}\frac{x^n}{n} + O(x^{n+1}),$$ +3. $$e^x = 1 + x + \frac{x^2}{2!} + ... + \frac{x^n}{n!} + O(x^{n+1}),$$ +4. $$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - ... + (-1)^n\frac{x^{2n+1}}{(2n+1)!} + O(x^{2n+3}),$$ +5. $$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - ... + (-1)^n\frac{x^{2n}}{(2n)!} + O(x^{2n+1}),$$ +6. $$\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - ... + (-1)^n\frac{x^{2n+1}}{2n+1} + O(x^{2n+3}).$$ + +### Example + +Construct $P_4(x)$ for $f(x) = e^{\sin x}$ around $x=0$. + +$$ +e^{\sin x} \approx 1 + (x - \frac{x^3}{3!} + \frac{x^5}{5!}) + \frac{1}{2!}(x - \frac{x^3}{3!} + \frac{x^5}{5!})^2 + \frac{1}{3!}(x - \frac{x^3}{3!} + \frac{x^5}{5!})^3 +$$ + +$$ +\begin{array}{ll} +P_4(x) &= 1 + x \frac{1}{2}x^2 + (-\frac{1}{6} + \frac{1}{6})x^3 + (-\frac{1}{6} + \frac{1}{4!})x^4 + O(x^5), \\ + &= 1 + x \frac{1}{2}x^2 - \frac{1}{8}x^4 + O(x^5). +\end{array} +$$ + +## Evaluating limits with Taylor polynomials + +Taylor and Macluarin polynomials provide a method for evaluating limits of indeterminate forms. + +### Example + +Determine the limit $\lim_{x \to 0} \frac{x \arctan x - \ln(1+x^2)}{x \sin x - x^2}$. + +$$ +\begin{array}{ll} +x \sin x - x^2 \approx x^2 - \frac{x^4}{6} + O(x^6) - x^2 = - \frac{x^4}{6} + O(x^6) \\ +x \arctan x - \ln(1+x^2) \approx x^2 - \frac{x^4}{3} + O(x^6) - x^2 + \frac{x^4}{2} + O(x^6) = \frac{x^4}{6} + O(x^6) +\end{array} +$$ + +$$ +\lim_{x \to 0} \frac{\frac{x^4}{6} + O(x^6)}{- \frac{x^4}{6} + O(x^6)} = -1 +$$ + +## L'Hostpital's rule + +Suppose the function $f$ and $g$ are differentiable on the interval $(a,b)$, and $g'(x) \neq 0$ there. Also suppose that $\lim_{x \downarrow a} f(x) = \lim_{x \downarrow a} g(x) = 0$ then + +$$ +\lim_{x \downarrow a} \frac{f(x)}{g(x)} = \lim_{x \downarrow a} \frac{f'(x)}{g'(x)} = L. +$$ + +The outcome is exactly the same as using Taylor polynomials. + +**Proof:** using Taylor polynomials around $x = a$. + +$$ +\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x-a)^2 + O((x-a)^3)}{g(a) + g'(a)(x-a) + \frac{g''(a)}{2}(x-a)^2 + O((x-a)^3)}. +$$ + +If $f(a)$ and $g(a)$ are both zero + +$$ +\lim_{x \to a} \frac{f'(a)(x - a) + \frac{f''(a)}{2}(x-a)^2 + O((x-a)^3)}{g'(a)(x-a) + \frac{g''(a)}{2}(x-a)^2 + O((x-a)^3)}, +$$ + +enzovoort. \ No newline at end of file diff --git a/docs/wiskunde/calculus/transcendental-functions/exponential-and-logarithmic-functions.md b/docs/wiskunde/calculus/transcendental-functions/exponential-and-logarithmic-functions.md new file mode 100755 index 0000000..8248fb6 --- /dev/null +++ b/docs/wiskunde/calculus/transcendental-functions/exponential-and-logarithmic-functions.md @@ -0,0 +1,50 @@ +# Exponential and logarithmic functions + +## The natural logarithm + +The natural logarithm is defined as having its derivative equal to $\frac{1}{x}$. For $x > 0$, then + +$$ +\frac{d}{dx} \ln x = \frac{1}{x}. +$$ + +### Standard limit + +$$ +\lim_{h \to 0} \frac{\ln (1+h)}{h} = 1 +$$ + +## The exponential function + +The exponential function is defined as the inverse of the natural logarithm + +$$ +\ln e^x = x. +$$ + +Furthermore $e$ may be defined by, + +$$ +\begin{array}{ll} +\lim_{n \to \infty} (1 + \frac{1}{n})^n = e, \\ +\lim_{n \to \infty} (1 + \frac{x}{n})^n = e^x. +\end{array} +$$ + +### Derivative of exponential function + +The derivative of $y = e^x$ may be calculated by [implicit differentation](../differentation.md#implicit-differentation): + +$$ +\begin{array}{ll} +y = e^x &\implies x = \ln y, \\ + &\implies 1 = \frac{1}{y} \frac{dy}{dx}, \\ + &\implies \frac{dy}{dx} = y = e^x. +\end{array} +$$ + +### Standard limit + +$$ +\lim_{h \to 0} \frac{e^h - 1}{h} = 1 +$$ \ No newline at end of file diff --git a/docs/wiskunde/calculus/transcendental-functions/inverse-functions.md b/docs/wiskunde/calculus/transcendental-functions/inverse-functions.md new file mode 100755 index 0000000..95ff60a --- /dev/null +++ b/docs/wiskunde/calculus/transcendental-functions/inverse-functions.md @@ -0,0 +1,69 @@ +# Inverse functions + +## Injectivity + +A function $f$ is called injective if for all $x_1,x_2 \in \mathrm{Dom}(f), \space x_1 \neq x_2$ implies that $f(x_1) \neq f(x_2).$ Meaning that for every $y \in \mathrm{Rang}(f)$ there is precisely one $x \in \mathrm{Dom}(f)$ such that $y = f(x)$. Meaning, every $x$ has an unique $y$. + +## Inverse function + +If $f$ is injective, then it has an inverse function $f^{-1}$. The value of $f^{-1}(x)$ is the unique number $y$ in the domain of $f$ for which $f(y) = x$. Thus, + +$$ +y = f^{-1}(x) \iff x = f(y) +$$ + +Suppose $f$ is a continuous function, $f$ is injective if $f$ is strictly increasing or decreasing. That is, $f' \leq 0 \vee f' \geq 0$. + +### Derivative of inverse function + +When $f$ is differentiable and injective $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$. + +**Proof:** + +$$f(y) = x \implies f'(y) \frac{dy}{dx} = 1$$ + +$$\frac{dy}{dx} = \frac{1}{f'(y)} = \frac{1}{f'(f^{-1}(x))}$$ + +Without knowing the inverse function a value of the inverse derivative may be determined. + +## The arcsine function + +Always $\arcsin$ not $\sin^{-1}$ that is wrong since $\sin$ is not injective. + +For $x \in [-\frac{\pi}{2},\frac{\pi}{2}] \space \arcsin(\sin x) = x$ + +For $x \in [-1,1] \space \sin(\arcsin x) = x$ + +The arccosine function is similar. + +## Example question + +Prove that $\forall x \geq 0$: $\arctan(x + 1) - \arctan(x) < \frac{1}{1 + x^2}$. + +For $x = 0$: $\frac{\pi}{4} < 1$. + +For $x > 0$: Consider the function $f(t) = \arctan(t)$ on the interval $[x, x+1]$. Apply the [Mean-value theorem](../differentation.md/#mean-value-theorem) of $f$ at the interval $[x,x+1]$, + +$$\frac{f(x+1) - f(x)}{(x+1) - 1} = f'(c).$$ + +Let $\arctan(c) = y$ then, $c = \tan y$, + +$$ +\begin{array}{ll} +\frac{dy}{dc} (c = \tan y) &\implies 1 = \sec^2 (y) \frac{dy}{dc} = (\tan^2 y + 1) \frac{dy}{dc} \\ + &\implies 1 = (c^2 + 1) \frac{dy}{dc} \\ + &\implies \frac{dy}{dc} = \frac{1}{c^2 + 1}. +\end{array} +$$ + +Obtaining, + +$$\arctan(x+1) - \arctan(x) = f'(c) = \frac{1}{c^2 + 1}.$$ + +For some $c \in (x,x+1)$, since $c > x$ + +$$\frac{1}{1 + c^2} < \frac{1}{1 + x^2},$$ + +thereby + +$$\arctan(x+1) - \arctan(x) < \frac{1}{1 + x^2}.$$ \ No newline at end of file diff --git a/docs/wiskunde/start.md b/docs/wiskunde/start.md new file mode 100755 index 0000000..a5abee3 --- /dev/null +++ b/docs/wiskunde/start.md @@ -0,0 +1,3 @@ +# Wiskunde + +Welkom op de wiskunde pagina. \ No newline at end of file diff --git a/mkdocs.yml b/mkdocs.yml new file mode 100755 index 0000000..76c56bf --- /dev/null +++ b/mkdocs.yml @@ -0,0 +1,76 @@ +site_name: Mijn aantekeningen +theme: + name: material + features: + - navigation.instant + - navigation.instant.prefetch + - navigation.tabs + - toc.follow + palette: + - scheme: slate + primary: black + accent: teal + +nav: + - 'Home': index.md + - 'Wiskunde': + - 'Start': wiskunde/start.md + - 'Calculus': + - 'Sets and numbers': wiskunde/calculus/sets-and-numbers.md + - 'Limits': wiskunde/calculus/limits.md + - 'Continuity': wiskunde/calculus/continuity.md + - 'Differentation': wiskunde/calculus/differentation.md + - 'Transcendental functions': + - 'Inverse functions': wiskunde/calculus/transcendental-functions/inverse-functions.md + - 'Exponential and logarithmic functions': wiskunde/calculus/transcendental-functions/exponential-and-logarithmic-functions.md + - 'Extremes values': wiskunde/calculus/extremes-values.md + - 'Concavity and inflections': wiskunde/calculus/concavity-and-inflections.md + - 'Taylor polynomials': wiskunde/calculus/taylor-polynomials.md + - 'Integration': wiskunde/calculus/integration.md + - 'Integration techniques': wiskunde/calculus/integration-techniques.md + - 'Improper integrals': wiskunde/calculus/improper-integrals.md + - 'First-order differential equations': wiskunde/calculus/first-order-differential-equations.md + + - 'Natuurkunde': + - 'Start': natuurkunde/start.md + + - 'Scheikunde': + - 'Start': scheikunde/start.md + - 'Basis': + - 'Materialen en stoffen': scheikunde/basis/materialen-en-stoffen.md + - 'Scheidingsmethoden': scheikunde/basis/scheidingsmethoden.md + - 'Reacties': scheikunde/basis/reacties.md + - 'Chemisch rekenen': scheikunde/basis/chemisch-rekenen.md + - 'Energie omzetting': scheikunde/basis/energie-omzetting.md + - 'Bindingstypen': scheikunde/basis/bindingstypen.md + - 'Evenwichten': scheikunde/basis/evenwichten.md + - 'Organische chemie': + - 'Organische verbindingen': scheikunde/organische-chemie/organische-verbindingen.md + - 'Lewis theorie': scheikunde/organische-chemie/lewis-theorie.md + - 'VSEPR theorie': scheikunde/organische-chemie/vsepr-theorie.md + - 'Isomerie': scheikunde/organische-chemie/isomerie.md + - 'Cis-trans isomerie': scheikunde/organische-chemie/cis-trans-isomerie.md + - 'Spiegelbeeld isomerie': scheikunde/organische-chemie/spiegelbeeld-isomerie.md + - 'Reacties': scheikunde/organische-chemie/reacties.md + - 'Reactie kinetiek': scheikunde/reactie-kinetiek.md + - 'Zuren en basen': scheikunde/zuren-basen.md + - 'Redox chemie': scheikunde/redox-chemie.md + - 'Analyse methoden': scheikunde/analyse-methoden.md + + - 'Filosofie': + - 'Start': filosofie/start.md + + - 'Programmeertalen': + - 'Start': programmeertalen/start.md + +markdown_extensions: + - pymdownx.arithmatex: + generic: true + +extra: + generator: false + +extra_javascript: + - javascripts/mathjax.js + - https://polyfill.io/v3/polyfill.min.js?features=es6 + - https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js \ No newline at end of file