d s'y}{ r math{atik5t * r d5c braiqe5t ---------------- n< e ;5l8~c vm #cj.aa.bjac * basy n |d$bare v7sj f d l%c + 67 braiqez3l3 h71s&&bc vm braiqe5tkomitee r d5sp>c l`nd7 >bskdl ,/kxi~j math{atik5t d s'y}{ r math{atik5t * r d5c braiqe5t 0e s'y}{atik 756t * 5w)z- u braiqe- 5t. ie /v7!7(, ,q!}>e -vlf`lt>u z privatc, n-kx7zieqc zwe$c } 7w8n5t. d tityblatt } ;,}t3l . 'cop'yr>hts. redaktj: petra :drid&, z8r# vivian :drid&, basy g8nth7 kappy, m)burg 'yvonne saml+d, l3pz> sz: braiqe- u 5w)zd$: vivian :drid& #, ,1la& #bjae =>c= braiqe5tkomitee r d5sp>c l`nd7 >bskdl >isbn r 5w)z5t-|gabe r >sbs: #igh.c.jcc.jdifd.b '$www.bskdl.org d s'y}{ r math{atik5t * r d5c braiqe5t ---------------- * zw3 braiqeb`ndc 7}7 b+d dk :::: 0% regyw7k ue dk g~z8g>7 f*+zieq7 ,zwcduc fgcd7 }iftuc 7mt: &?g u moni'que di{-58l*-}iftu h+s-eggcb7g7-}iftu hir5m+n-}iftu fr0dr# u am:ie me'y7-b1m+n-}iftu m>ros-kulturprozct dr. jean }0g7-}iftu f 0 f<e ,/}8tzu dkc wr: br>it( ;tz, m)burg r67 h7rm+n, h+nov7 ,tgl0d7 r ,/kxi~j math{atik5t . >bskdl ::::::::::::::::::::::::::::: petra :drid&, z8r# >sbs 5w3z7i5e bibliothek f ble, s=- u l%e;h*d7( vivian :drid&, basy s=;h*d7tchilfe basy'- >sbh -b+d r blc- u s=;h*d7tcp`dagogik'- >vbs m)l0s bo4sl7, z8r# >sbs 5w3z7i5e bibliothek f ble, s=- u l%e;h*d7( r#)d h27 &n. haqm+n, hagc )bs;r34 1diotaktile medic r $fern$universit`t * hagc ,?,icd7 . >bskdl g8nth7 kappy, m)burg d5e blc}udic+}:t e.v., m)burg'- 'blista g8nth7 koos, m)burg 'c)l-}r=l-5ule r d5c blc}udic+}:t e.v., m)burg'- 'blista 7n}-d0tr# l?cz, h+nov7 d57 -6 r blc u s=;h*d7tc * }udium u ;ruf e.v.'- >dvbs t*a l?>, d8rc >lvr-louis-braiqe-5ule d8rc, mediczctrum 'yvonne saml+d, l3pz> d5e zctr:b8473 f ble z l3pz> =>dzb= 7# 5mid, w0n b/d%-blc7z0hus*}itut'- >bbi blc- u s=;h*d7tcv7b+d 9}7r34 *h:tsv7z34x =========== 7}7 b+d ::::::: ,?w?t ......................... #a ~wi$lu ...................... #a kompakth v7sus kon('xt/abh`n- g>k ..................... #b n27uc ....................... #c zm &,| 0% regyw7ks ............ #g ,1b1 ........................ #g $la$te>x .................... #i #a gdlegcde tkc zr ,8tgu v ma- th{atik ................... #aa #a.a we4sy zw te'xt- u ma- th{atik5t ............... #aa #a.a.a la'yout ............ #ab #a.a.b +- u abk8nd>usz34c f math{atik5t ........... #af #a.a.c +- u abk8nd>usz34c f te'xt5t ............... #ai #a.a.d doppyleerz34ctk .... #bj #a.a.e h*,3e zm 6sz r 5t- we4sytkc .............. #bc #a.b trcnc u zsh:tc math{ati- *h:tsv7z34x >i 57 |!d$e ................ #bf #a.c +m7kuc zr braiqe5t,8tgu #bh #b ziff7n u zlc ............... #ca #b.a )abi5e ziff7n u zlc .... #ca #b.a.a zlc * ,})d5b,3e .... #cb #b.a.b zlc * &sckt7 5b,3e #ce #b.a.c dezim:br84e ........ #cg #b.a.d p7iodi5e dezim:br84e #ci #b.a.e gl0d7u lg7 zlc ..... #dj #b.a.f ?dnuszlc, dezim:- kla~ifikat?c, datc u uhrztc ................ #db #b.b r9mi5e zlc ............. #dd #c bu4}abc u szz34c ........... #dg #c.a ,?;m7ku zr kcnz34nu v bu4}abc ................. #dg #c.b g~- u kl65bu lat6i57 bu4}abc ................. #dh #c.c gr04i5e bu4}abc ........ #ea #c.d ;e t'ypografi5e |z34nuc #ef #c.e bu4}abc`e s'ymbole .... #fa #c.f kzw?ts'ymbole .......... #fd #c.g szz34c ................. #ff #c.h te'xt * r math{atik5t .. #fg #d 6hc ........................ #fi #d.a kcnz34nu v 6hcs'ymbolc #fi #d.b qzct, qmiqe ............ #gj *h:tsv7z34x >ii #d.c w*ky- u t{p7aturma'~e .. #ga #d.d 6hcs'ymbole 1s bu4}abc #gb #d.e -!g~7us- u -kl67uspr`fi- 'xe ..................... #ge #d.f !whuss'ymbole .......... #gg #e op7!ns- u ry!nsz34c ........ #ha #f klax7n u sckrte }r#e ....... #ia #f.a aq&m6% z klax7n ........ #ic #f.b 6fe klax71s!d$e .... #ih #f.e sckrte }r#e ............ #ajc #f.f te'xtklax7n * r math{a- tik ..................... #ajf #g pf3le ...................... #aji #g.a modul)e pf3le .......... #aji #g.b def*0r( pf3le .......... #aac #g.c ;5tu v pf3lc ........... #aae #h 6fiii zw3t7 b+d ::::::::: #aj qjektivtk ................. #aci #aj.a 6fiv gcde klax7n ............. #ahb +h`n& ......................... #ahg >a#a 5te re4cv7,rc 8 ,=7e z3lc #ahg >a#a.a additj ............... #aij >a#a.b subtraktj ............ #aia >a#a.c multiplik!n .......... #aic >a#a.d divisj ............... #aid >a#a.e l*e)e additj ......... #aif >a#a.f d l9sc v g4uc ........ #aih >a#b !7uc * r math{atik5t ..... #bja >a#b.a &!7( s'ymbole ........ #bja >a#b.b n2e s'ymbole ......... #bjb >a#b.c zlc .................. #bjd >a#b.d xponctc u *diz% ...... #bjd >a#b.e br84e ................ #bje >a#b.f bu4}abc .............. #bje >a#b.g klax7n u sckrte }r#e #bjf >a#b.h 6hc .................. #bjg >a#b.i pf3le ................ #bjh >a#b.aj qjektivtk ........... #bjh >a#b.aa we4sy zw te'xt- u math{atik5t ............. #bji >a#b.ab son}>% .............. #bji >a#c glo~ar ................... #baa >a#d math{ati5e z34c, &?dnet n< r #f-pte-braiqe-ta;qe ..... #bai *h:tsv7z34x >v >a#e :pha;ti5% svi ,?w?t ===== d ,?l0gcde regyw7k } d 7&bx 67 g~c ,8)bu r d5c braiqemath{atik5t s3t r ~}=u r (*t7n!n:c math{atik5t f ble). 0e ue * e #aibj7 jrc v -tret7n 6>7 l`nd7 |&)bet u s?g( df, ~ 0 wc y{c( '- s'ymbole 2 | d),yustkc'- 6e wt- g=cde *t7n!n:e 6hk ,1w0sc. spurc 07 6hk sd 6 knapp% jh sp`t7 x n4 7kcnbar. | wn 0 4*%i5e braiqe5t 7w)tus&m`'~ gz ,7s } 3 0 d5e, ,wc kcn7 r d5c braiqemath{atik5t bm ;- tr ,v,6 wt7ctwi$yt'- u t ihc 0 jew3l>c math{atik5tc. - zu& 07 ~wi$luc tratc + 0 ,ye *t7- n!n:7 &m6~kc ,znhcd ,v,6 /abh`ng>e, ,?w?t: ~wi$lu #a 3gc!}>e math{atik5tc. - d5c spr1m ue ab e #aiej7 jrc 6e ,8)bu ,?&nx u * 6{ n2c regyw7k fe}- &h:tc =#aiee', #; ,1la& #aihf'=. - l1fe r zt ~,}c j4 region:e v)i+tc. p ~wi$ytc c 0 not!nc * r >brd u * 9}7- r34, * r >ddr u * r 5w3z |,6. um 0 math{atik5t f 0c spr1m 0d z -6hc u ,p,t 0 |t15b)k math{ati57 lit7atur n wt7 6,z5r`nkc, ue #bjjf vm braiqe5t- komitee r d5sp>c l`nd7 6e ,/kxi~j &- bildet. kompakth v7sus kon('xt/abh`ng>k ::::::::::::::::::::::::::::::: traditjeq ,hc zlr34e braiqez34c * r math{atik5t ,7e w7( o ;d2tuc 3 * r te'xt5t. 6e umd2tu r #fd mc braiqe- z34c 7l1bt 6e sr kompak('- u dh ,8- s#te'- d),yu r math{ati5c not!n. d0s t { kompromi~, ~ 0 braiqez34c 7} d+n 6d2t> sd, wn s klar r math{atik- o te'xt5t ,z&?dnet ,wc ,kc. * -50dcc spc f+d dg 6 p)ad>mcwe4sy }att. 4 0 d),yu math{ati57 s'ymbole ,?w?t: kompakth #b 4 !lg7e komb*!nc v braiqez34c sd s ,pwl * aq&m6c 3 | * math{ati5c kon- ('xtc 6d2t>. db g=t j4 0 kompakth r 0dgabe -l?c. 0 ,?l0gcde math{atik5t ;h`lt 0 trcnu * te'xt- u math{atik5t ,zg/}c r !kze u ,8s#tk r d),yu b. a7d*gs ktc +n`h7uc + 0 te'xt5t 7r34t ,wc, zm ,b,0 * r kcnz34nu r g~- u kl65bu. n27uc ::::: (d s'y}{ r math{atik5t * r d5c braiqe5t) gl0d7t c * zw3 t3le. r ,?l0gcde 7}e t3l ;5bt 0 regyn zr 0dgabe math{ati57 sc kapity bzw. ab5nit( 6. zlr34e ,b,0e -d2tc 0 um- ,eu r regyn. 6 glo~ar kl`rt spezifi- 5e ;griffkc r braiqe5t. h*,3e z 5t- c re4cv7,rc ,wc ,/,#cdc e ,zg+g zr prakti5c )b t r braiqe5t 7l47n. ,*+t ,d( 0 ,z!sze d),yu r ,b,0e * $la$te>x 9. 6 w#t>% +l0gc } %, 0 ,?w?t: n27uc #c ,#>k r 3gcc *t7pret!n r braiqe5t,b- ,0e ,8pr8fc z ,kc. df }=t s=cdc 0 5w)z5td),yu zr -f8gu. t r $la$te>x- d),yu wd e ta}l%cdc ebcf 6e mk zr kontroqe +&botc. 6e gdlegcde n27u ;trifft 0 mk r kx/ik!n zw e -50dcc l%7gruppc. % ue d1 &<(t, ~ 0 medial ,/50dc |gabc 2 braiqe- u 5w)z5t p)aqy -wcdet ,wc ,kc. 0 ,b,0e sd nux70rt u r te'xt ue p ,1&b1t, ~ 0 &}:tu * braiqe- u 5w)z5t - wc g4 }. e zw3tc t3l bildet 6 ryiefb+d, * { ,pwl 0 taktilc 5w)z5ts'ymbole 3 | 0 braiqect,~uc s`mt7 math{ati57 z34c 1s { regyw7k ,1&fht sd. dm wd 0 kx/ik!n zw blc, s=;h*d7tc u s=cdc ,*0rtc 7l47t. 1 r websi( . braiqe5tkomitees r d5sp>c l`nd7 =>bskdl= ,kc wt7e ,b,0e +&s=c u 7!gzt ,wc ='$www.bskdl." org'.=. 0 *h:tc n27uc r 5t sd - +h+g (>a#b !7uc * r math{atik5t) zs&fa~t. 0% regyw7k } n 3 l=rw7k konzip0rt. f ble u s=cde ,/,#cde, ,8tgcde ,p,2 ,?w?t: n27uc #d l%cde r braiqe5t ,s 0e hdr34u 0 gd- !sze r braiqe5t =s0he (s'y}{ r d5c blc5t)= spezieq 1 { &b0t r math{atik 7!gzc. % b1t ao 1 { gdregyw7k 1 u ,et d~ kcntx ,?1s. basy, j+uar #bjae - namc r ,/kxi~j math{atik5t . braiqe5tkomitees r d5sp>c l`nd7 d redaktjsteam ,?w?t: n27uc #e ,?w?t: n27uc #f zm &,| 0% regyw7ks ================== ,1b1 :::: 0% regyw7k wd * braiqe- u 5w)z5t h71s&&bc. 6e f6e gl0d7u t dezim:- kla~ifik!n d0nt r ?ict0ru - w7k u 7- l47t 0 kx/ik!n b r )b t e -50dcc me- di:c |ga;f?mc. 0 nux70ru r ,b,0e sp0gyt 0e gl0d7u wid7. 6fhcd ,wc gdlegcde tkc u h*,3e zr 0dgabe v math{atik * r braiqe5t - kapity #a 7l|t7t. 0 kapity #b bs #aa fhc 0 6zync y{c( r not!n u d7c &,| 6. ab5~cd foku~0rc 0 kapity #ab bs #ae 1 |&w`hl( &b0( r math{atik. * +h`ngc ,wc +reguc zm )bc t 5tc re4cv7,rc &&bc, 0 !7uc u n27uc 07 ,8)bu r math{atik5t ,1&li}et u braiqespezifi5e f<1s!d$e * 6{ glo~ar 7kl`rt. f 6e 5neqe su4e }=c 6e li}e a7 ;- hdytc braiqez34c u 6 sx-5b,3e 6e |nme bildc 0 ,b,0e - (+h+g >a#a 5te re4cv7,rc 8 ,=7e z3lc), 0 nr * braiqe5t 756c. hr }=t nb um,eus- u &}:tusmkc 0 prakti5e )b t r braiqe- 5t - ,?d7gd. * 6{ zw3tc t3l ,sc ryiefd),yuc blc l%cdc math{ati5e 5w)z5ts'ymbole 7,r- bar mc l`nd7 h`lt 1 97 webs3( ='$www.bskdl.org'.= 6e ,/s3( zr braiqemath{atik5t f d h7,/ladc v dokumctc u d saxyn v ,b- ,0c ;r3t. ,*0r( ,wc 6&ladc, zr 7wt7u r ,b,0saxlu ,b,ztgc. zm &,|: ,1b1 #h $la$te>x :::::::: 3 mk . -g4s r braiqe5t t 67 zw3tc d),yu ue f ta}l%cde, 0 n 1 0 visueqe d),yu z$gr3fc ,kc, 6e $la$te>x-5b,3e &w`hlt. % k n +&nx ,wc, ~ ae ta}l%cdc $la$te>x kcnc. u dcn4 ht c &z3gt, ~ 5 t w>c $la$te>x-kcntxsc zm ,b,0 fe}&,yt ,wc k, ob 6 z34c n4 * 6{ x- ponctc ~h:tc } o n. u &rade s4e /&- w~hc gilt % m} z m*im0rc, wn 0 ,b- ,0e }ud0rt ,wc. r 6d2t>k h:b7 ;}=t 0 $la$te>x-d)- ,yu 1s ?>*al-$la$te>x u n 1s 67 r -- 6fx-'compil7 d)- 1s `stheti5 6w+dfr3e 5w)z5t 7,yc ,k(. um d l%c r $la$te>x-|!d$e z 7l47n, ue f 0e #h-pte-braiqe &w`hlt. zm 5neqc n<5gc -wcdet7 $la$te>x- 5l8~yw9rt7 wd 6e ,1li}u yektroni5 1 zm &,|: $la$te>x #i '$www.bskdl.org +&botc. 0% regyw7k ,p,2 0 ,1li}u 3gnc c n f d 7l7nc v $la$te>x. zm &,|: $la$te>x #aj #a gdlegcde tkc zr ,8tgu v math{atik ========================== #a.a we4sy zw te'xt- u math{atik5t :::::::::::::::::::::: %!, +k8nd>usz34c f 6e pa~a& * ma- th{atik5t %'. abk8nd>usz34c f 6e pa~a& * ma- th{atik5t %'. +k8nd>usz34c f 6e pa~a& * te'xt5t %'. abk8nd>usz34c f 6e pa~a& * te'xt5t braiqez34c u d7c komb*!nc &bc zm t3l ,/50de s'ymbole * r te'xt- u r math{atik5t 0d. 0 kcnz34nu r ,8g`n& zw e bdc 5tc } dh v g~7 ;d2tu. dr3 -50dce tkc }=c hr,f zr -f8gu: '- la'youttk '- +- u abk8nd>ustk '- doppyleerz34ctk #a-#a.a #aa 0 wahl r tk } kon('xtabh`ng>. #a.a.a la'yout :::::::::::::: * dokumctc t g~{ math{ati5{ +t3l re4nc l%cde t math{atik5t. zr kcn- z34nu . we4sys v r te'xt- * 0 ma- th{atik5t &n8gt % dh oft 5, z3lc t math{ati5{ *h:t t hilfe . la'youts vm 8>c fl0'~('xt ab,zhebc. 6e sr h|f> &nz( &}:tusmk zr kcn- z34nu v math{atik5tz3lc sd 6- u |- r$uc. db ,wc z3lc um 6e +zl v f?mc ;z8g r ,?1sg=cdc te'xtum&bu 6&r$t. r we4sy z$ zr te'xt5t 7fgt 4 e ;g*n 67 n2c z3le - te'xtla'yout. ,wc f 0 math{ati5e pa~a& ,=7e z3lc ;n9t>t, } fgcd% z ;b#jb'= o ta;qcsp:tc =s0he ,b,0e #a.a.a >b#jc u #a.a.a >b#jd'=. #a.a-#a.a.a #ab ,b,0 #a.a.a >b#ja der umre4nungsfaktor von der 3nh3t elektronvolt 'e>v in joule >j i}: #ae>v =#a>v.e' =#a>v.#a,f.#aj|-,*>c' =#a,f.#aj|-,*>j Der Umrechnungsfaktor von der Einheit Elektronvolt eV in Joule J ist: \[1 \text{eV} =1 \text{V} \cdot \text{e} =1 \text{V} \cdot 1,6 \cdot 10^{-19} \text{C} =1,6 \cdot 10^{-19} \text{J}\] ,b,0 #a.a.a >b#jb =+m.: r +f+g r math{atik5t wd ,z- !sz t 6{ 'm * r l*kc r+dsp:( &kcn- z34net.= der umre4nungsfaktor von der 3nh3t elektronvolt 'e>v in joule >j i}: m #ae>v =#a>v.e' =#a>v.#a,f.#aj|-,*>c' =#a,f.#aj|-,*>j #a.a.a #ac Der Umrechnungsfaktor von der Einheit Elektronvolt eV in Joule J ist: \[1 \text{eV} =1 \text{V} \cdot \text{e} =1 \text{V} \cdot 1,6 \cdot 10^{-19} \text{C} =1,6 \cdot 10^{-19} \text{J}\] ,b,0 #a.a.a >b#jc =+m.: * $la$te>x wd 1 0 ta;q)i5e d),yu nr rudimct`r h*&w0sc.= satz formel ::::::::::::::::: :::::::::: d0 multiplikation a.b =b.a i} kommutativ. d0 multiplikation 2a.b`.c' i} assoziativ. =a.2b.c` d0 #a verh`lt si4 a.#a =a bez8gli4 der multiplikation n2tral. \[\text{Satz} & \text{Formel} \\ \text{Die Multiplikation ist kommutativ.} & a \cdot b =b \cdot a #a.a.a #ad \\ \text{Die Multiplikation ist assoziativ.} & (a \cdot b) \cdot c =a \cdot (b \cdot c) \\ \text{Die 1 verh„lt sich bezglich der Multiplikation neutral.} & a \cdot 1 '$=A"\'.'= ,b,0 #a.a.a >b#jd =+m.: * $la$te>x wd 1 0 ta;q)i5e d),yu nr rudimct`r h*&w0sc.= spezielle v0recke: bez34nung umfang fl`4eninhalt ::::::::: :::::::::: ::::::::::::: v0reck a +b +c +d ;d1,8#b<" 2h1, +h1;` trapez a +b +c +d m.h1a dra4en- #b'a +#b'b #a;.d1,d1; v0reck paralle- #b'a +#b'b a.h1a logramm rhombus #d'a ..... #a;.d1,d1; quadrat #d'a ..... a|; #a.a.a #ae \[\text{Spezielle Vierecke} \\ \text{Bezeichnung} & \text{Umfang} & \text{Fl„cheninhalt} \\ \text{Viereck} & a +b +c +d & \frac{d_{1}}{2}(h_{1} +h_{2}) \\ \text{Trapez} & a +b +c +d & m \cdot h_{a} \\ \text{Drachenviereck} & 2a +2b & \frac{1}{2}d_{1}d_{2} \\ \text{Parallelogramm} & 2a +2b & a \cdot h_{a} \\ \text{Rhombus} & 4a & \frac{1}{2}d_{1}d_{2} \\ \text{Quadrat} & 4a & a^{2}\] #a.a.b +- u abk8nd>usz34c f math{atik5t ::::::::::::::::::::::::::: %!, +k8nd>usz34c f 6e pa~a& * ma- th{atik5t %'. abk8nd>usz34c f 6e pa~a& * ma- th{atik5t #a.a.a-#a.a.b #af am 6d2t>}c ,wc 0 ,8g`n& v r te'xt- zr math{atik5t u z$ t +- u abk8n- d>usz34c m)k0rt. d +k8nd>usz34c }=t /mtbar ? { 7}c z34c r math{atik5t. 1'~7 am z3lc+f+g g=t ( 8b7,3e 6 leerz34c ,?an. % }=t j4 /mtbar h*t7 67 9ffncdc te'xtklax7 o 6{ ,7c s'ymbol, 1 d | te'xt oe 6 leerz34c fgc ,k(. d abk8nd>usz34c }=t /mtbar h*t7 { ltc z34c r math{atik5tpa~a&. d1 fgt 6 leer- o szz34c. t { +k8nd>usz34c 6&l3(( math{atik- 5tpa~agc sd zw*gcd t { abk8nd>usz34c ab,z5~c. s ,dc nr v kzc, t r doppy- leerz34ctk ab&grcztc te'xtpa~agc ,/- bro4c ,wc. szz34c am cde 67 math{ati5c pa~a& &h9rc * r regy n zm math{ati5c |d$ s}. r ,8s#tk h:b7 ,wc s n< { abk8n- d>usz34c &,5c, wo s e ,?+&,ytc pt #f n ;n9t>c =s0he (#c.g szz34c)=. ,b,0 #a.a.b >b#ja =+m.: ,85t 1s 6{ l=rbu4.= #a.a.b #ag #ad d0 zerlegung von !,ax|; +bx +c'. in linearfaktoren ==================== o * kz5t: #ad 0 z7legu v !,ax|;' +bx +c'. * l*e)fakt?c ====================== 14 Die Zerlegung von $ax^{2} +bx +c$ in Linearfaktoren ,b,0 #a.a.b >b#jb alle kennen d0 formel !,e =mc|;'., aber nur wenige ver}ehen s0. Alle kennen die Formel $e =mc^{2}$, aber nur wenige verstehen sie. ,b,0 #a.a.b >b#jc d0 newton5e me4anik i} b3 ge5win- digk3ten im ber34 der li4tge5windig- k3t =!,c =#c.#aj|(m8s'.= ni4t mehr g8ltig. Die newtonsche Mechanik ist bei Geschwindigkeiten im Bereich der #a.a.b #ah Lichtgeschwindigkeit ($c =3 \cdot 10^{8} \frac{\text{m}}{\text{s}}$) nicht mehr gltig. #a.a.c +- u abk8nd>usz34c f te'xt5t ::::::::::::::::::::::::::: %'. +k8nd>usz34c f 6e pa~a& * te'xt5t %'. abk8nd>usz34c f 6e pa~a& * te'xt5t te'xt5t *n7h:b 67 math{atik5tpa~a& k ebcf t +- u abk8nd>usz34c &kcnz34- net ,wc. d +k8nd>usz34c }=t /mtbar ? { 7}c te'xtz34c h*t7 6{ + r grcz,ye ,?kxcdc leerz34c. d abk8nd>usz34c fgt /mtbar 1 d lte te'xtz34c. * 67 math{atikpa~a& }=t r te'xt6- 5ub n?m:7,3e - sybc !kzusgrad =kz-, ,q- o basis5t= 2 r 8>e ,8tgce fl0'~- ('xt. 0e +- u abk8nd>usz34c ,dc | *n7h:b 67 math{atik5tpa~a& -wcdet ,wc, 0 i7s3ts t +- u abk8nd>usz34c ab&grczt }. #a.a.b-#a.a.c #ai dg ,dc * 6{ t +k8nd>usz34c &kcn- z34netc te'xt65ub k6e math{ati5c 6- 58be ~h:tc 9. ,b,0 #a.a.c >b#ja >i1, =#aj.>i1>b '.=ri4twert='.' =#abja \[I_{1} =10 \cdot I_{B} \quad \text{(Richtwert)} =120 \text{\mu A}\] #a.a.d doppyleerz34ctk :::::::::::::::::::::: ,8aq d?t, wo 6 we4sy zw te'xt- u math{atik5t 7w)(t ,wc k, ,dc sr kze 658be r jew3ls ,7c 5t t r doppyleer- z34ctk &kcnz34net ,wc. ? { 7}c z34c - ,7c 5ts'y}{ }=t 6 doppyleerz34c. r we4sy z$ zm ,?h7>c 5ts'y}{ wd 7n2t t 6{ doppyleerz34c +&z3gt. d doppyleerz34c m~ zw z34c }=c, d)f ao n am +f+g o am cde 67 z3le zm 6sz kxc. d cde 6% t 6{ doppyleerz34c 6&l3(- tc 65ubs m~ ebcf 4 6 doppyleerz34c &kcnz34net ,wc. nr wn 6 65ub * ma- #a.a.c-#a.a.d #bj th{atik5t am cde 6% absz% }=t, k 1 0% abk8nd>cde doppyleerz34c -z#(t ,wc, da 6 n27 absz e 4 0 doppyleer- z34ctk ;wktc 5twe4sy oeh* ,1hebt. * r regy &h9rc szz34c am 5lu~ 67 math{ati5c pa~a& n zm math{ati5c |- d$. s ,dc dcn4 ? { + r grcz,ye }=cdc doppyleerz34c'- wo n9t> t pt #f'- &- ,5c ,wc =s0he (#c.g szz34c)=. fgt 6 math{atik1sd$ direkt 1 6 fhcd% *t7- ptjsz34c =+fhusz34c, 9ffncde klax7=, k 0 doppyleerz34ctk n +&wcdet ,wc. ,b,0 #a.a.d >b#ja d0 gl34ung x|; =#af i} na4 'x 1fzul9sen. Die Gleichung $x^{2} =16$ ist nach $x$ aufzul”sen. ,b,0 #a.a.d >b#jb basis 'a und exponent 'n 3ner potenz sind f8r a *=n i. allg. ni4t vert15bar: a|n *=n|a =b3sp0l f8r 3ne 1snahme: !,#b|/ =#d|;'.=. Basis $a$ und Exponent $n$ einer #a.a.d #ba Potenz sind fr $a \neq n$ i. allg. nicht vertauschbar: $a^{n} \neq n^{a}$ (Beispiel fr eine Ausnahme: $2^{4} =4^{2}$). ,b,0 #a.a.d >b#jc damit i} a|n f8r alle ganzzah- ligen exponenten =!,n &e>g'.= defi- n0rt, allerdings f8r n 9=#j mit der 3n5r`nkung a *=#j =denn f8r a =#j w8rden d0 definitionen f8r a|-n und !,a|)'.'- wegen !,a|) =a|n"-n'.'- 1f divisionen dur4 null f8hren=. Damit ist $a^{n}$ fr alle ganzzahligen Exponenten ($n \in G$) definiert, allerdings fr $n \leq 0$ mit der Einschr„nkung $a \neq 0$ (denn fr $a =0$ wrden die Definitionen fr $a^{-n}$ und $a^{0}$ - wegen $a^{0} =a^{n -n}$ - auf Divisionen durch Null fhren). #a.a.d #bb ,b,0 #a.a.d >b#jd unter 3a 2a o=#j` ver}ehen wir ... Unter $\sqrt{a} \; (a \geq 0)$ verstehen wir ... ,b,0 #a.a.d >b#je alle kennen ja d0 for- mel e =mc|;', aber nur wenige ver}ehen s0. Alle kennen ja die Formel $e =mc^{2}$, aber nur wenige verstehen sie. #a.a.e h*,3e zm 6sz r 5twe4sytkc :::::::::::::::::::::::::::::::: f l%cde m~ x klar 7kcnbar 9, ob s &rade 0 te'xt- o math{atik5t l%c. f 0 wahl r jew3ls &3gnetc tk gytc fgc- de ,8leguc u pr*zipic: '- 0 +- u abk8nd>usz34c m)k0rc e 5t- we4sy 6d2t>. '- 6e t { +k8nd>usz34c 6&l3(( ma- th{atik5tpa~a& m~ t { abk8nd>us- z34c ;cdet ,wc. #a.a.d-#a.a.e #bc '- la'youttkc grczc yeg+t u klar e gytus;r34 r jew3l>c 5t ab. '- 0 doppyleerz34ctk 3gnet c |!d$ nr f sr kze 658be'- m} oe z3lc- umbr84e. '- wn 6e math{ati5e pa~a& t 6{ +- fhusz34c o 67 te'xtklax7 ;g*nt, d)f s n t r doppyleerz34ctk +- &k8nd>t ,wc. '- * r regy &h9rc szz34c am 5lu~ 67 math{ati5c pa~a& n zr pa~a& s}. s sd dh /mtbar rts vm abk8nd>usz3- 4c %'. z ,ec. wn 0 abk8nd>u 4 doppyleerz34c 7fgt, ,wc s j4 ? 0c =&&bccf t pt #f'= &,5c, dm s n a6 }=c. '- 6 kz7 math{ati57 |d$ am cde 6% te'xtabsz% k t r doppyleerz34ctk 6&l3(t ,wc. d abszcde kcnz34net g4zt> | d cde . 65ub%. ab5~cde szz34c ,wc /mtbar n< { math{ati5c |d$ &,5c u &&bccf t 6{ ,?+&,ytc pt #f -s=c. '- 8b7,3e ,wc te'xt658be * math{a- ti5c pa~agc - sybc !kzusgrad 2 r uml0gcde te'xt &,5c. '- 6zyne o w>e w9rt7 * math{ati5c #a.a.e #bd pa~agc =zm ,b,0 (u), (dh), (% gilt)= ,kc * basis5t =t kcnz34nu r g~5bu= &,5c ,wc, oe 0 math{a- tik5t z -,lc =s0he (#c.h te'xt * r math{atik5t)=. % } zw { ,?t3l- hfc -z#t 1 e 5twe4sy u 6{ evctueq }9rcdc }ilbru4, ? a{ * kz5t('xtc, ab,zw`gc. ,?s#t } b uml1tbu4}abc u '~ &botc, 0 * r math{atik5t 3 ,7e z34c, ? a{ 3 bru4}r# u 5~cde klax7, &l%c ,wc ,kc. szz34c ,mc &&bccf t ,?+&,yt{ pt #f -s=c ,wc. ,b,0 #a.a.e >b#ja f2-x` =-f2x` f8r alle x &e>d o f2-x` =-f2x` f ae x &e>d o f2-x` =-f2x` '.f ae'. x &e>d \[f(-x) =-f(x) \; \text{fr alle} \; x \in D\] #a.a.e #be #a.b trcnc u zsh:tc math{ati57 |!d$e :::::::::::::::::::::::::::::: %' z3lctrcnz34c + r ,ye 6% leerz3- 4cs %" z3lctrcnz34c zw zw3 /mtbar ;n<- b)tc z34c %" zsh:(pt * r 5w)z5t }=t * r regy jd7 ma- th{ati5e (sz) a6 1 67 z3le. 6c trcn}r# zr kcnz34nu 6% math{ati5c z3lcumbru4s gb % * r 5w)z5t n. * r braiqe5t nixt 6 math{ati57 (sz) oftms = 3 nr 6e z3le 6. dh wd 6 z34c ;n9t>t, d 1 0 f?t,eu * r n}c z3le ,1m7k~ mb#aj u #ad.b >b#jf'=. '- * situ!nc, * ec 6 leerz34c obl>a- t?i5 }, e |d$ a |,6r3'~c 8e, k %" =pt #d'= +,ye . leerz34cs &,et ,wc. ? a{ b qjektivc u br8- 4c, a | * matrizc wd 0e tk -wc- det. %" =pt #d'= * 0c f/ktjc } n t { pt #d z -we4syn, r fe}7 ;,}t3l 6>7 s'ymbole'- zm ,b,0 %"e =2ro= o %") =grad-z34c='- } o ? 6{ bu4}abc 3 akzctz34c }=t. #a.b #bg #a.c +m7kuc zr braiqe5t,8tgu :::::::::::::::::::::::::::: %'<= 9ffncde u 5~cde klax7 f +m7kuc zr braiqe5t,8tgu wn 3gcs f d * braiqe5t um,z,ecde w7k r z34c;,} 7wt7t ue o t'ypografi- 5e ;hc d)&,yt o 7kl`rt ,wc ,mc, } % nw, +m7kuc zr braiqe5tt5c 0dgabe r ,?la& +,zbgc. ebcf ,s( 1 0 ,1l9su v ta;qc o 0 v7b:is0ru bzw. d wg,lc v abbilduc h*&w0sc ,wc. +m7kuc, ye 0 gze ,8tgu . w7k% ;- tfc, ,wc * 6{ 3gcc ab5nitt o kapity t ~,~cd7 ,85t am +f+g . w7k% bzw. jd% b+d% . braiqebu4% zs&fa~t. hr ,wc 0 6&fhtc z34c * 67 li}e ,1&fht. gilt 0 +m7ku nr f 6zyne pa~agc - w7k, wd 0e * e klax7n f +m7kuc zr braiqe5t,8tgu + r jew3l>c ,ye 6- &f8gt. ,p,t ,wc s n 3 te'xt r 5w)z- 5t,?la& &l%c. 0 +m7kuc zr braiqe5t,8tgu'- ,pwl * 6{ 3gcc ab5nitt 3 | * klax7n'- ,wc * { !kzusgrad 2 r 8>e te'xt &,5c. #a.c #bh s0he ,b,0e #f.c >b#ja u #ad.c >b#jg. #a.c #bi #a.c #cj #b ziff7n u zlc =============== #b.a )abi5e ziff7n u zlc :::::::::::::::::::::::: 0 math{atik5t -f8gt 8 zw3 d),yus- f?mc f 0 )abi5c ziff7n: '- ,})d5b,3e '- &sck( 5b,3e * r ,})d5b,3e ,wc ziff7n t dcsybc braiqez34c 2 0 lat6i5c bu4}abc 'a bs 'j &bildet. v 0c ,/53dc s c 4 d ,?+- ,yc . zlz34cs. 0 ziff7n r &scktc 5b,3e ;}=c 1s braiqez34c, * ec 0 pte 6e r3he t0f7 3 * r ,})d5b,3e &,et sd. | 0e braiqez34c sd t ,=7c ;d2tuc ;legt u ,yc nr * ;}ixtc kon('xtc ziff7n dar. #b-#b.a #ca #b.a.a zlc * ,})d5b,3e :::::::::::::::::::::: %# zlz34c %a ziff7 6s %b ziff7 zw3 %c ziff7 dr3 %d ziff7 v0r %e ziff7 f8nf %f ziff7 se4s %g ziff7 s0bc %h ziff7 a4t %i ziff7 n2n %j ziff7 nuq * r math{atik5t ;}=t 6e )abi5e zl gd!sz'- 2 * r te'xt5t |'- 1s { zl- z34c u 67 o ,=7c ziff7n. d0s wd 3 ,})d5b,3e ;z34net. n< { zlz34c %# ,yc 0 braiqez34c r bu4}abc 'a bs 'j 0 ziff7n #a bs #i u #j dar, u zwar gd!sz bs zm n}c leerz34c, z3lccde o }r# bzw. ,7c sz- z34c =wo,b d dezim:kxa u r dezim:pt n n 3 szz34c gytc=. 0 wku . zlz34cs 7}re$t c 8: '- 0 ziff7n =* r ,})d- o &scktc 5b- ,3e= #b.a.a #cb '- d dezim:trcnz34c %, '- d gl0d7usz34c %. '- 0 9ffncde klax7 %2 b r 0dgabe v p7iodi5c dezim:br84c =s0he (#b.a.d p7iodi5e dezim:br84e)= '- 0 +k8nd>u f 6e ;e t'ypografi5e |- z34nu %! '- e apo}roph %' bzw. e }r# %- /mtbar h*t7 { zlz34c ,1&hobc wd 0 wku . zlz34cs 4 jd% ,7e z34c ,p,2 '- 6 leerz34c '- d z3lccde'- 1'~7 b z3lctrcnu t pt #d %" jde )t v }r# =zm ,b,0 b*de- o 5r`g}r#= hebt 0 wku . zlz34cs 1, p ~ zlc n< 0{ }s 6 n2% zlz34c ;n9t>c. 6e |nme bildc }r#e - +5lu~ + zlz34c, 0 * pr3s+gabc +,ye 67 nuq ? { dezim:- z34c }=c. 6e zl } nr d+n am z3lccde z trcnc, wn d0s /v7m3d }, zm ,b,0, wn 0 !lge r zl 0 &sam( z3lcbr3( ,85r3(t. 6 apo}roph, r 0 ,ye v fhcdc ziff7n 7,et, }=t - +5lu~ + d zlz34c ? r 7}c #b.a.a #cc ziff7 =s0he ,b,0 #b.a.a >b#jd'=. h*,3: e ziff7n * ,})d5b,3e g=t gd!sz 6 zlz34c ,?1s. % k j4 s*n,q 9, z.b. * 5tc re4cv7,rc, d zlz34c v e ziff7n tw wg,zr$c o !gz d1 z -z#tc, um 0 ,8s#tk z whc =s0he (+h+g >a#a 5te re4cv7,rc 8 ,=7e z3lc)=. ,b,0 #b.a.a >b#ja #c \[3\] ,b,0 #b.a.a >b#jb #bde \[245\] ,b,0 #b.a.a >b#jc #ac':#bg-#ad':#ac $uhr \[13:27-14:13 \; \text{Uhr}\] ,b,0 #b.a.a >b#jd #'je \['05\] #b.a.a #cd #b.a.b zlc * &sckt7 5b,3e ::::::::::::::::::::::::: %, ziff7 6s %; ziff7 zw3 %: ziff7 dr3 %/ ziff7 v0r %? ziff7 f8nf %+ ziff7 se4s %= ziff7 s0bc %( ziff7 a4t %* ziff7 n2n %) ziff7 nuq - +5lu~ + 6>e z34c r math{atik5t ,kc gze zlc oe zlz34c * &sckt7 5b,3e &,5c ,wc. dd wd r |d$ um 6 z34c !kz7. ,z,{ k 0 f/ktj r zl * 6{ kom- paktc math{ati5c |d$ l47 &d2(t ,wc. 0 &sck( 5b,3e wd f ncn7 v 6fb#ja #a: =#d,; \[\frac{1}{3} =\frac{4}{12}\] ,b,0 #b.a.b >b#jb ;x|; 8 x|:< =x|-, \[\frac{x^{2}}{x^{3}} =x^{-1}\] ,b,0 #b.a.b >b#jc e|#b,cjbe ??#aj \[e^{2.3025} \approx 10\] #b.a.b #cf #b.a.c dezim:br84e :::::::::::::::::: %, dezim:trcnz34c =kxa= * |nme!fqc =s0he n, ob * r ,?la& 6 kxa o 6 pt }=t. r pt #c %. wd 3 dezim:trcnz34c -m0dc, da 7 * r braiqe5t 3 gl0d7us- z34c f lge zlc ;legt }. 6e |nme bildc gyd;!tge * 5w3z7 fr+kc u rappc. hr k 0 * r 5w3z 8be 5b,3e t dezim:pt * r braiqe5t ,b;h:- tc ,wc =s0he | (#b.a.e gl0d7u lg7 zlc)=. ,wc * ,7c kon('xtc dezim:pte * r ,?la& -wcdet u } d0s v ;d2tu, k * 67 braiqe5tt5c +m7ku d1 h*&w0sc ,wc. }r#e, 0 * gyd;!tgc +,ye 67 o zw37 nuqc }=c, ,wc 4 d z34c pte #c,f %- d)&,yt. #b.a.c #cg ,b,0 #b.a.c >b#ja #b,cd \[2,34\] ,b,0 #b.a.c >b#jb #de,ih \[45.98\] ,b,0 #b.a.c >b#jc #-,ej \[-,50\] ,b,0 #b.a.c >b#jd #ajj,- \[100,-\] ,b,0 #b.a.c >b#je #ajj,-- \[100,--\] ,b,0 #b.a.c >b#jf =+m.: d),yu v 5w3z7 fr+kc * r 5w3z.= $fr.#c.ej #b.a.c #ch \[\text{Fr.} \; 3.50\] #b.a.d p7iodi5e dezim:br84e ::::::::::::::::::::::::::: %#...2...` p7iodi57 dezim:bru4 b p7iodi5c dezim:br84c wd * r 5w)z5t 0 c 0dholcde ziff7nfge ,8}r#c. * r braiqe5t wd 0e ziff7nfge * r/de klax7n %2...` oe zlz34c &- ,et. ,b,0 #b.a.d >b#ja #j,2c` \[0,\overline{3}\] ,b,0 #b.a.d >b#jb #a,2bhegad` \[1,\overline{285714}\] ,b,0 #b.a.d >b#jc #c,db2h` \[3,42\overline{8}\] #b.a.c-#b.a.d #ci #b.a.e gl0d7u lg7 zlc ::::::::::::::::::::: %. gl0d7usz34c 0 gl0d7u lg7 zlc * gruppc v #c ziff7n 7fgt 4 d gl0d7usz34c pt #c %. /&<(t . * r ,?la& -wcdetc z34cs =pt, leerz34c, apo}roph, kxa=. b gyd;!tgc * 5w3z7 fr+kc u rappc wd * r regy * 5w)z5t r pt u ~,~cd * braiqe5t pt #c %. ,pwl 3 dezim:- trcn- 3 | 3 gl0d7usz34c -wcdet =s0he | (#b.a.c dezim:br84e)=. d0s fht |'~7} sytc z d2tus5>kc, da n< { ltc pt n dr3, sn nr zw3 ziff7n fgc u dh 3 rappc z 7kcnc sd. 0 * r 5w)z5t -br3(( gl0d7u lg7 zlc 4 leerz34c wd n ,8nx, da d leerz34c 0 wku . zlz34cs ,1hebt u 0 n2e zif- f7ngruppe 0dum t 6{ zlz34c &kcnz34- net ,wc ,m(. eo } 0 gl0d7u t apo}rophc =5w3z u l04tc}6= f 0 braiqe5t /&3gnet. r apo}roph wd t ,{sybc braiqez34c 2 d +k8nd>usz34c f kl6bu4}abc d)&,yt, d 0 wku . zlz34cs ,1hebt. #b.a.e #dj ,b,0 #b.a.e >b#ja #bc.ced \[23.354\] ,b,0 #b.a.e >b#jb #a.cbd.cdb \[1\;324\;342\] ,b,0 #b.a.e >b#jc #ai.jda.ejj \[19'041'500\] ,b,0 #b.a.e >b#jd #bf.jid.cah,gea.fbh \[26\;094\;318,751\;628\] ,b,0 #b.a.e >b#je #a.jjj \[1.000\] ,b,0 #b.a.e >b#jf #ajjj \[1000\] #b.a.e #da #b.a.f ?dnuszlc, dezim:kla~ifikat?c, datc u uhrztc ::::::::::::::::::::::::::::::: 0 -b*duc 1s zlc u *t7ptjsz34c ,wc gd!sz 2 * r ,?la& &,5c. db } z ;<- tc: '- 6 pt - +5lu~ + o zw zlc } k6 de- zim:pt u wd dh t pt #c %. d)&- ,yt. n< { gl0d7uspt ~!fqt d zlz3- 4c. '- * uhrzt+gabc ,wc doppypte 3 *t7- ptjsz34c 4 pt #f %' +&k8nd>t. '- * datums+gabc sd b*de}r#e n t pt #f %' +,zk8nd>c. 0 nb#ja #c.e.aa \[3.5.11\] ,b,0 #b.a.f >b#jb #g.aj.d.c \[7.10.4.3\] #b.a.f #db ,b,0 #b.a.f >b#jc #bd.ab.bjaj \[24.12.2010\] ,b,0 #b.a.f >b#jd #bjaj-#ab-#bd \[2010-12-24\] ,b,0 #b.a.f >b#je #ab.de $uhr \[12.45 \; \text{Uhr}\] ,b,0 #b.a.f >b#jf #ab':#de $uhr \[12:45 \; \text{Uhr}\] wo 0 te'xt5t f s4e zlc&f8& 6e -- !kzcde 5b,3e b0(t, d)f 0e ebcf * r math{atik5t -wcdet ,wc. ,b,0 #b.a.f >b#jg =+m.: * ,1gabc r gd5ulmath{atik, * ec 6 7&bx 6,ztgc }, wd oft 6 zlz34c n< { g4hsz34c &,et. dd wd / ,7{ d g4hsz34c ss v 6{ 'g ,/53dbar.= #b.a.f #dc #, #b +#c =# #; #a +#d =# \[1. \quad 2 +3 = \\ 2. \quad 1 +4 =\] ,b,0 #b.a.f >b#jh #:e,, \[3.5.11\] ,b,0 #b.a.f >b#ji #;/ab#bjaj \[24.12.2010\] #b.b r9mi5e zlc ::::::::::::::: %i r9mi5e ziff7 6s %v r9mi5e ziff7 f8nf %x r9mi5e ziff7 z=n %l r9mi5e ziff7 f8nfz> %c r9mi5e ziff7 h/d7t %d r9mi5e ziff7 f8nfh/d7t %m r9mi5e ziff7 t1scd #b.a.f-#b.b #dd r9mi5e zlc ,wc 3 bu4}abc'=fgc= &- ,5c. 3 g~bu4}abc sd s ,{n< }s t %> +,zk8nd>c. 3 kl6bu4}abc &,5c ;,dc s * r math{atik5t h|f> k67 +k8nd>u =s0he (#c.b g~- u kl65bu lat6i57 bu4}abc)=. ,8}r#ce r9mi5e zlc, 0 ggt f ,=fb#ja >i', >iv', >mdcccxxv \[\text{I, IV, MDCCCXXV}\] ,b,0 #b.b >b#jb i', xvi \[\text{i, xvi}\] ,b,0 #b.b >b#jc >x: =#aj.jjj \[\text{\overline{X}} =10\;000\] #b.b #de ,b,0 #b.b >b#jd $:>iv =#djjj \[\text{\overline{IV}} =4000\] #b.b #df #c bu4}abc u szz34c =================== #c.a ,?;m7ku zr kcnz34nu v bu4}abc :::::::::::::::::::::::::: +k8nd>usz34c %' kl6bu4}abc %> 6 o ,=7e g~bu4}abc %$ 6 g~bu4}abe, &fgt v 6{ o ,=7c kl6bu4}abc %< gr04i5e bu4}abc %! #, ;e t'ypografi5e |z34nu % #; ;e t'ypografi5e |z34nu oe kcnz34nu sd ae bu4}abc * r ma- th{atik5t lat6i5e kl6bu4}abc * ic mod7nc 5tf?mc. lat6i5e g~bu4}abc u fr{de bzw. t'ypografi5 spezieqe bu4}abc ,mc ~,~cd &kcnz34net ,wc. #c-#c.a #dg #c.b g~- u kl65bu lat6i57 bu4}abc ::::::::::::::::::::::::: +k8nd>usz34c %' kl6bu4}abc %> 6 o ,=7e g~bu4}abc %$ 6 g~bu4}abe, &fgt v 6{ o ,=7c kl6bu4}abc * r math{atik5t } 0 g~- u kl65bu v bu4}abc sr ~53dcd f d7c ;d2tu. % m~ dh /;d*gt 1 6d2t>k &<(t ,wc. gd!sz } f lat6i5e kl6bu4}abc * r math{atik5t k6e +k8nd>u 7f?d7. j4 ,mc s d?t t pt #f %' &kcnz34net ,wc, wo s 1 gd r ,?h7g=cdc z34c 3 tw ,7% &d2(t ,wc ,kc, zm ,b,0: '- bu4}abc 'a bs 'j /mtbar n< * ,})dziff7n &,5cc zlc: s 8c 3 wt7e ziff7n &l%c ,wc. '- /mtbar n< { bru4cdez34c: d bru4- cdez34c } idcti5 t { +k8nd>usz34c f gr04i5e bu4}abc. '- /mtbar n< t ptc #d,e %> +&k8n- d>tc g~bu4}abc: s ,ktc 3 wt7e g~- #c.b #dh bu4}abc *t7pret0rt ,wc. ae n +&k8nd>tc bu4}abc sd 3 lat6i- 5e kl6bu4}abc z l%c. - ,*e r 6d2t>k wd {pfohlc, kl6- bu4}abc l0b7 6m z oft t pt #f %' 3 6m z w> +,zk8nd>c. 6 g~bu4}abe wd nr d+n 4 0 pte #d,f %$ +&k8nd>t, wn ( m*de}cs 6 kl6bu4}abe fgt. son} sd 6zyne g~- bu4}abc x t e ptc #d,e %> +,zk8n- d>c. 0 wku r +k8nd>usz34c f g~- bzw. kl6bu4}abc wd ,1&hobc 4: '- d n}e leerz34c '- d z3lccde'- 1'~7 bm z3lctrcnz34c pt #d %" '- d n}e 1'~7:pha;ti5e braiqe5tz34c jeg7 )t ,b,0 #c.b >b#ja #aj'a', #ajk', #en', #c>b \[10a, 10k, 5n, 3B\] #c.b #di ,b,0 #c.b >b#jb !{>abcd', >efgh', >ijkl!} \[\{ABCD, EFGH, IJKL\}\] ,b,0 #c.b >b#jc >abc'd \[ABCd\] ,b,0 #c.b >b#jd $ab$cd \[AbCd\] ,b,0 #c.b >b#je l9sungsmenge >l1>a L”sungsmenge $L_{A}$ ,b,0 #c.b >b#jf ;#a +n 8 #e<'x \[\frac{1 +n}{5}x\] #c.b #ej #c.c gr04i5e bu4}abc :::::::::::::::::::: +k8nd>usz34c %< gr04i5e bu4}abc 0 gr04i5c bu4}abc * r math{atik5t %a :pha %b ;ta %g gaxa %d dyta %e epsilon %z zeta %j eta %h theta %i iota %k kappa %l lambda %m m'y %n n'y %x 'xi %o omikron %p pi %r rho %s s>ma %t t1 #c.c #ea %u 'ypsilon %f phi %c 4i %y psi %w omega %v digaxa %q koppa 0 bu4}abc . gr04i5c :pha;ts ,wc t dcsybc braiqe5tz34c 2 0 . lat6i5c :- pha;ts &,5c. dh ,mc gr04i5e bu4}abc |!d$ 3 s4e &kcnz34net ,wc. gr04i5e kl6bu4}abc ,wc 4 d z3- 4c %< +&k8nd>t. b gr04i5c g~bu4}a- bc wd ,z!sz 0 +k8nd>u f g~5bu %> bzw. %$ zw 0{ z34c u { 7}c bu4}abc &,et. 0 +k8nd>u f gr04i5e bu4}abc gilt f 6e +k8nd>u r g~5bu 4 pte #d,e %> bzw. pte #d,f %$ u f ae bu4}abc bs: '- zm n}c leerz34c o z3lccde'- 1'~7 bm z3lctrcnz34c %" bzw. '- zm n}c 1'~7:pha;ti5c braiqe5tz34c jeg7 )t. #c.c #eb n< 67 fge v gr04i5c bu4}abc l3tc dh 0 +k8nd>uc f g~- bzw. kl6bu4}a- bc %> o %$ bzw. %' 6c we4sy z lat6i5c bu4}abc 6. fqs *n7h:b 67 fge v gr04i5c bu4}abc 6e +k8nd>u f g~- bzw. kl6bu4}abc nw wd, m~ 0 +k8nd>u f gr04i5e bu4}abc %< 0dholt ,wc. h*,3: d fr8h7e +k8nd>usz34c f gr04i5e g~bu4}abc % wd n = +&wcdet. ,b,0 #c.c >b#ja $winkel b#jb d0 summe der winkel b#jc b#jd v =;#bt< \[v =\frac{2\pi r}{T}\] ,b,0 #c.c >b#je >a1>o =b#jf e|"+2b#jg =+m.: f 0 5b,3e . g~bu4}abcs dyta 3 diff7czz34c s0he (#c.e bu4}abc`e s'ymbole).= >p =;&d>e 8 &d't<' =;&d>w 8 &d't< \[P =\frac{\Delta E}{\Delta t} =\frac{\Delta W}{\Delta t}\] #c.c #ed ,b,0 #c.c >b#jh b#ji >f1>w =#a;c1usz34c %! #, ;e t'ypografi5e |z34nu % #; ;e t'ypografi5e |z34nu ,3c * math{ati5c ,?lagc ;e d$f?mc 1 -50dce math{ati5e ;d2tuc h*, ,mc 0 ,/50de | * r braiqe5t kcnt &musz34c pt #e %! u pte #d,e,f % zr fr3c -- f8gu. um ye )t v |z34nu % c - 6zyfq hdyt, m~ * 67 braiqe5tt5c +m7ku 7l|- t7t ,wc =s0he (#a.c +m7kuc zr braiqe5t,8tgu)=. da d z34c pte #d,e,f % | 6hc +k8nd>t, } f bu4}abc d z34c pt #e %! z {pf=lc =s0he (#d.a kcnz34nu v 6hcs'ymbo- lc)=. d +k8nd>usz34c f 0 ;e t'ypografi5e |z34nu }=t /mtbar ? { 7}c ;tfcdc bu4}abc bzw. ? r evctueqc +k8nd>u f gr04i5e bu4}abc u!,o g~-!,kl65bu. 0 +k8nd>u gilt f 0 evctueq /mtbar d1 fgcdc z34c %< =gr04i5=, %> o %$ bzw. %' =g~- u kl65bu='- bzw. komb*!nc dv'- u f ae bu4}abc bs: '- zm n}c leerz34c '- zm z3lccde'- 1'~7 bm z3lctrcnz3- 4c %" '- zm n}c 1'~7:pha;ti5c braiqe5tz34c jeg7 )t #c.d #eg 0 +k8nd>usz34c ,dc | ? ,7c z34c r braiqe5t }=c, ec 4 6e t'ypografi5e abhebu ,7e ;d2tuc ,zkxc. * 0{ fq gilt 0 +k8nd>u: '- ? 6{ zlz34c f 0 gze zl '- direkt ? 67 ziff7 nr f 0e 6e zif- f7 '- ? ac ,7c s'ymbolc led> f d d1 fgcde s'ymbol 0 +k8nd>u 4 pt #e %! d)f n b q- jektivc -wcdet ,wc, da pt #e 0 -}`r- ku 6% qjektivs 6l3(t =s0he (#aj.b -- }`rk( qjektive)=. . wt7c } s d?t n 7l1bt, wo s 3 t3l 6% s'ymbols &l%c ,wc ,k(. zm ,b,0 bildet 6 ,?+&,yt7 pt #e * komb*!n t e$>c klax7n %{ bzw. %} n ea fett &d$(, sn &5w3f( klax7n %!{ bzw. %!} =s0he (#f.b 6fu 4 % d)f dg d?t n -wc- det ,wc, wo s t { 7}c t3l 67 ,/c zs- fa~cdc m)k0ru -we4syt ,wc ,k( =s0he (#h.b zsfa~cde m)k0ruc)=. 0 +k8nd>u 4 pte #d,e,f % } di- rekt ? 67 6zync ziff7 67 zl n ,z!l>, um -we4sluc t r +k8nd>u 67 6h ,?,z- #c.d #eh b2gc. wd led> 6 t3l 6% |d$s h7,?&hobc, dm * 67 7l|t7u d1 6&g+gc ,wc k, {pf0hlt c 0 tk r h?izont:c zsfa~uc =s0he (#ae.b h?izont:e zsfa~uc u l0- gcde klax7n)=. f h7,?,zhebcde klax7- paare ,kc 0 spezieqc braiqe5tklax7n -wcdu f*dc =s0he (#f.c spezieqe braiqe5tklax7n)=. ,b,0 #c.d >b#ja w0 l1ten der vektor !v:, und d0 }recke !>ab'? Wie lauten der Vektor $\vec{\mathbf{v}}$ und die Strecke $\mathbf{AB}$? ,b,0 #c.d >b#jb !#dbbf o #dbbf \[\mathbf{4226}\] #c.d #ei ,b,0 #c.d >b#jc #a!bc!de', #ce!f!f \[1\mathbf{2}3\mathbf{4}5, \; 35\mathbf{66}\] ,b,0 #c.d >b#jd $:,!>ab \[\vec{\mathbf{AB}}\] ,b,0 #c.d >b#je $$:,>f1>g$5 \[\vec{\mathbf{F}_{\mathbf{G}}}\] ,b,0 #c.d >b#jf :!>ab', $:!>a1,!>b1,$5 \[\underline{\mathbf{AB}}, \; \underline{\mathbf{A}_{1} \mathbf{B}_{1}}\] #c.d #fj #c.e bu4}abc`e s'ymbole :::::::::::::::::::::::: %&d g~% dyta 3 diff7czz34c %&s suxcz34c %&p qduktz34c %&e } y{ct v %"d r/d% 'd =f p)tieqe abl3tu= %"h 'h-'qu7, reduz0r( pl+$5e kon- }+( %"p w37}ra'~5% 'p %$$n mc& r nc zlc %$$z mc& r gzc zlc %$$q mc& r r!n:c zlc %$$r mc& r reeqc zlc %$$c mc& r komple'xc zlc %$$h mc& r 'quat7nionc %$$p qjektive &rade f vle math{ati5e s'ymbole, d7c f?- mc * r 5w)z5t 1 6zyne bu4}abc z$g=c, 0 a n t 0c bu4}abc idcti5 sd, gb % 3gce braiqe5ts'ymbole. p ,hc 0 s'ymbole f suxe u qdukt * r 5w)z5t 0 f?m r gr04i5c g~bu4}abc s>ma u pi. s ,wc * r braiqe5t ,1gd #c.e #fa i7 !g~e j4 n 2 0e bu4}abc ;hdyt, sn jew3ls t { f s fe}&legtc s'ym- bol %&s f suxe u %&p f qdukt &- ,5c =s0he (#aa.a f/ktjc)=. 0 reduz0r( pl+$5e kon}+( =| 3 ('h- 'qu7) ;k+nt= wd * r 5w)z5t 4 6 ,4- &}r#c% kl6% 'h, * r braiqe5t 4 0 fe}e z34cfge %"h ab&bildet. +:og wd b p)tieqc abl3tuc d * r 5w)z5t &- 5w/gce kl6e 'd * r braiqe5t t %"d 0d&&bc. | 0 t doppy}r#c &z34netc g~bu4}abc f 0 ,})dmcgc ,wc * r braiqe5t 4 3- gce, jew3ls 1s dr3 braiqez34c ;}=cde s'ymbole 0d&&bc: %$$n f ne zlc, %$$z f gze zlc, %$$q f r!n:e zlc usw. u gytc 3 1'~7:pha;ti- 5e s'ymbole =s0he ,b,0 #ab >b#ji'=. b ;d)f ,kc wt7e s'ymbole n< 0{ mu}7 &bildet ,wc. 0 n259pfu m~ * e ,?;m7- kuc o e +m7kuc zr braiqe5t,8tgu 7l|- t7t ,wc =s0he (#a.c +m7kuc zr braiqe5t,8tgu)=. dg } r gr04i5e kl6bu4}abe pi n v e ,7c gr04i5c kl6bu4}abc z ,/53dc u } ~,~cd e regyn f 0 d),yu gr04i57 bu4}abc z ;hdyn =s0he (#c.c gr04i5e #c.e #fb bu4}abc)=. ,b,0 #c.e >b#ja "h =h8#b

b#jb $$n =!{#a',#b',#c',#d',#e'," ...!}'; $$n1) =!{#j',#a'," #b',#c',#d',#e',...!} \[\mathbb{N} =\{1,2,3,4,5,...\}; \; \mathbb{N}_{0} =\{0,1,2,3,4,5,...\}\] ,b,0 #c.e >b#jc t2;"d|;usz34c %$ bzw. %> /mtbar n< { 5l8~yz34c z ,ec. d kzw?t m~ v d1fgcdc )gumctc usw. 4 6 leer- z34c o 6 +k8nd>usz34c =zlz34c, kl6- 5bz34c o.`.= ab&grczt ,wc. % l0gt - 7me~c . ,8tgcdc o 5bcdc, kzw9rt7, f 0 3gce, t { z34c %6 ;- #c.f #fd g*ncde s'ymbole xi}0rc, eo z 5bc. d 5l8~yz34c %7 f*det | f 0 6l3tu -50dc7 &ometri57 s'ymbole -wcdu =s0- he (#ad.a &ometri5e s'ymbole)=. * 0c s'ymbolc fgt 1 d 5l8~yz34c j4 n0 6 bu4}abe. ,b,0 #c.f >b#ja #aj 7mod #h =#b \[10 \; \text{mod} \; 8 =2\] ,b,0 #c.f >b#jb #a 7>or #a =#a aber' #a 7>xor #a =#j \[1 \; \text{OR} \; 1 =1 \; \text{aber} \; 1 \; \text{XOR} \; 1 =0\] ,b,0 #c.f >b#jc =+m.: f 0 8be d),yu s0he ,b,0 #ad.b >b#ja'= 7sin#cj") =#j,e \[\sin 30^{\circ} =0,5\] #c.f #fe ,b,0 #c.f >b#jd 6l1,)'x =6l'x o 7log1,)'x =7lg'x \[\log_{10}x =\lg x\] #c.g szz34c ::::::::::: %' +k8nd>usz34c f szz34c kxc * r math{atik5t szz34c ?, m~ ihc r pt #f %' ,?+&,yt ,wc, um s v ,7c s'ymbolc z ,/53dc. r szpt %. ,p,2 r &dkc}r# %'- sd hr,v |&nx, da k6e -we4slus&,r ;}=t. | klax7n r te'xt5t wd r pt #f ,?+- &,yt. r/de te'xtklax7n %= 756c ,p- ,t jew3ls t 6{ pt #f. obwl b e$>c te'xtklax7n %'= 6 pt #f ;r3ts ;,}- t3l . s'ymbols }, ,mc s t 6{ wt7c pt #f +&k8nd>t ,wc =s0he (#f.f te'xt- klax7n * r math{atik)=. #c.f-#c.g #ff #c.h te'xt * r math{atik5t :::::::::::::::::::::::::: f te'xt658be * math{ati5c pa~agc wd * r regy * 0 te'xt5t &we4syt. dc- n4 ,dc 6zyne w9rt7 u kze phrasc &,5c ,wc, oe 0 math{atik5t z -,lc. d+n m~ basis5t -wcdet ,wc. * r math{atik5t wd 0 g~5bu gd!sz &kcnz34net, p | b w9rt7n. dg wd b 6{ 5twe4sy zr te'xt- 5t 0 g~5bu 2 - 8>c te'xt &hd,ht. n?m:7,3e g=t { te'xt 6 leerz34c ,?1s. d0s s?gt * r regy df, ~ 7 n zm ,b,0 t v)iablc -we4syt ,wc k. 0 d5c bu4}abc '`, '9, '8 u '~ ,wc t braiqez34c &,5c, 0 ebcf math{ati5e s'ymbole o t3le dv abbildc. ,pf7n % c um kl6bu4}abc hdyt, ,mc s * kriti- 5c situ!nc t pt #f %' -s=c ,wc. ,plge 0 math{atik5t n -,lc wd, ;- d)f % bm z3lcumbru4 ? o n< 6{ w?t 6% z3lctrcnz34cs =s0he (#a.b trcnc u zsh:tc math{ati57 |!d$e)=. f szz34c gytc 0 * r math{atik5t 8bc regyn. s sd dh * e m3}c !fqc t ,?+g=cd{ pt #f %' z kcnz34nc =s0he #c.h #fg (#c.g szz34c)=. ,b,0 #c.h >b#ja a o=#j', n',z &e$$n und' n *=#a \[a \geq 0, \; n,z \in \mathbb{N} \; \text{und} \; n \neq 1\] ,b,0 #c.h >b#jb &s>f =#j ==o v =konstant \[\sum F =0 \Rightarrow v =\text{konstant}\] ,b,0 #c.h >b#jc f2x` =;$kraft 8 beschleunigte' $masse< =;>f 8 m< \[f(x) =\frac{\text{Kraft}} {\text{beschleunigte Masse}} =\frac{F}{m}\] #c.h #fh #d 6hc ====== #d.a kcnz34nu v 6hcs'ymbolc ::::::::::::::::::::::::::: % kcnz34c f 6hcs'ymbole 6hcs'ymbolc wd d 6hckcnz34c % /mtbar ,?+&,yt. % } /7heb, ob s e4( ma'~6hc 2 met7 o hilfs6hc 2 qzct d)- ,yc. bildc ,=7e 6hc 6c 6hckomple'x, ;- d)f % nr 6% kcnz34cs, ,plge r kom- ple'x n 4 leerz34c o w7( ,/bro4c wd. ;z0ht c 6e 6h direkt 1 6c w7t, wd s t { ,?+&,ytc kcnz34c /mtbar + 0c +&5lo~c. leerz34c * 67 5w)z5t,?la& ,wc hr ign?0rt. 0 kcnz34nu v 6hc 7,et 0 t'ypo- grafi5c mt, 0 r 5w)z5t zr ,/53du r v)iablc v 6hc- u f/ktjss'ymbolc zr -f8gu }=c. um 67 mc -we4slus&,r -- 50dc7 s'ymbolt'ypc ,?,zb2gc, ;d0nt c 0 5w)z5t'- vm l%cdc oft nr /;wu~t #d-#d.a #fi wh&nx'- d$t57 f6hc. t'ypi5e visueqe kcnz34nusm7kme sd &rade &,e( bu4}abc f 6hc u kursive f v)iablc'- o | 6 leerz34c =q o h:b= ? 6hcs'ymbolc, a n ? v)iablc. ,b,0 #d.a >b#ja #j,cl =#cjjml \[0,3 \text{l} =300 \text{ml}\] ,b,0 #d.a >b#jb #a>h =#a>v.s8>a \[1 \text{H} =1 \frac{\text{V} \cdot \text{s}}{\text{A}}\] #d.b qzct, qmiqe :::::::::::::::: %#j) qzct %#j)) qmiqe ,b,0 #d.b >b#ja #bj#j) von #eakm \[20\% \; \text{von} \; 51 #d.a-#d.b #gj \text{km}\] ,b,0 #d.b >b#jb #g#j)) \[7 \permil\] #d.c w*ky- u t{p7aturma'~e :::::::::::::::::::::::::: %") grad =kr*gy= %"* m*u( =}r#= %"** sek/de =doppy}r#= %rad radi+t =rad= %rad|; 'quadratradi+t ,b,0 #d.c >b#ja #abj") \[120^{\circ}\] ,b,0 #d.c >b#jb 6a#ij") =b#jc #bj")#e"*#aj"** o #bj") #e"* #aj"** \[20^{\circ} \; 5' \; 10''\] #d.d 6hcs'ymbole 1s bu4}abc ::::::::::::::::::::::::::: %" akzctz34c |&w`hl( 6hcs'ymbole 1s bu4}abc %m ...... met7 %cm ..... zctimet7 %mm ..... miqimet7 %v ..... volt %>mv .... megavolt %m>a .... miqiamp7e %w ... mikrowatt %<>w .... ohm %k<>w ... kiloohm %$hz .... h7tz %k$hz ... kiloh7tz %e>v .... yektroncvolt #d.c-#d.d #gb %$me>v o %>m'e>v megayektroncvolt %s ...... sek/de %sec .... sek/de %min .... m*u( %>"a .... "ang}r9m 0 bu4}abc, 0 d 6hcs'ymbol bildc, ,wc - +5lu~ + d kcnz34c f 6hc n< e 8bc regyn &,5c =s0he (#c bu4}abc u szz34c)=. } 6 bu4}abe t akzct ;,}t3l 6% 6hc- s'ymbols, wd 07 4 pt #d %" u e gd- bu4}abc d)&,yt. ,b,0 #d.d >b#ja #aatm =#aja,cbek$pa \[1 \text{atm} =101,325 \text{kPa}\] ,b,0 #d.d >b#jb #a$hz =#as|-, \[1 \text{Hz} =1 \text{s}^{-1}\] ,b,0 #d.d >b#jc #a>"a =#ajjpm =#aj|-,)m #d.d #gc \[1 \AA =100 \text{pm} =10^{-10} \text{m}\] ,b,0 #d.d >b#jd #a<>w =#aj|*;cm8s< o #a<>w =#aj|*cm8s \[1 \Omega =10^{9} \frac{\text{cm}}{\text{s}}\] ,b,0 #d.d >b#je #a>a =#a;>c8s< =#a>c.s|-, o #a>a =#a>c8s =#a>c.s|-, \[1 \text{A} =1 \frac{\text{C}}{\text{s}} =1 \text{C} \cdot \text{s}^{-1}\] ,b,0 #d.d >b#jf $kapazit`t1min =;#aes 8 #b<>w<' =#g,e;s8<>w< =#g,e>f o $kapazit`t1min =#aes8#b<>w' =#g,es8<>w =#g,e>f #d.d #gd \[\text{Kapazit„t}_{\text{min}} =\frac{15 \text{s}}{2 \Omega} =7,5 \frac{\text{s}}{\Omega} =7,5 \text{F}\] #d.e -!g~7us- u -kl67uspr`fi'xe ::::::::::::::::::::::::::::::: -!g~7us- u -kl67uspr`fi'xe ,wc -- wcdet, um 6 ,=f<% bzw. 6c bru4t3l 67 gd6h z bildc. am ;k+n(}c sd d0jc>c . *t7n!n:c 6hcs'y}{s =>si=, zm ,b,0 ('k) =(kilo-), d t1scdfb#ja #aj|-:m =#amm' =ein $tausendstel $meter \[10^{-3} \text{m} =1 \text{mm} =\text{ein Tausendstel Meter}\] #d.d-#d.e #ge ,b,0 #d.e >b#jb $ein $millionstel $meter' =#a.#aj|-+m =#at$byte =#a.jjj>g$byte' =#a.jjj.jjj>m$byte' =#aj|*k$byte =#aj|,;$byte Speicherkapazit„t von x Terabyte! Aber was, bitte sehr, ist ein Terabyte? \[1 \text{TByte} =1\;000 #d.e #gf \text{GByte} =1\;000\;000 \text{MByte} =10^{9} \text{kByte} =10^{12} \text{Byte}\] #d.f !whuss'ymbole :::::::::::::::::: |&w`hl( !whuss'ymbole %"e 2ro =2rozone= %>eur 2ro =2rozone= %ct 2ro-'cct =2rozone= %$fr. fr+kc =5w3z= %>chf fr+kc =5w3z= %"s doqar =? a{ >usa= %>usd doqar =>usa= %"c 'cct =? a{ >usa= %"l pf/d =? a{ g~brit+nic= %>gbp pf/d =g~brit+nic= %>tl pf/d!,lira =t8rk3= %>trl pf/d!,lira =t8rk3= %dkr krone =d`nem)k= %>dkk krone =d`nem)k= %$kc krone =t5e4i5e rk= %>czk krone =t5e4i5e rk= %i>r rupie =*dic= #d.e-#d.f #gg %>inr rupie =*dic= %"s>a doqar =1}r:ic= %>aud doqar =1}r:ic= %>nz"s doqar =n2seel+d= %>nzd doqar =n2seel+d= %"y 'yc =japan= %>jpy 'yc =japan= %"y 'yuan =4*a= %>cny 'yuan =4*a= !whuss'ymbolc wd 2 ,7c 6hcs'ymbolc d 6hckcnz34c % ,?+&,yt. }=t 6e t pt ab&5lo~ce !whus6h ? { w7t, wd k6 leerz34c zw e bdc &,et. ,b,0 #d.f >b#ja #g"e \[7 \euro\] ,b,0 #d.f >b#jb #e"e #bgct \[5 \euro \; 27 \text{ct}\] ,b,0 #d.f >b#jc "e#e.fch,ej #d.f #gh \[\euro 5.638,50\] ,b,0 #d.f >b#jd >eur#hc$mio. \[\text{EUR} \; 83 \text{Mio.}\] ,b,0 #d.f >b#je =+m.: s0he 7kl`ru zm pt * 5w3z7 gyd;!tgc * (#b.a.c dezim:br84e).= $fr.#ba.ej \[\text{Fr.} \; 21.50\] ,b,0 #d.f >b#jf $preis =#cj"e8m|; \[\text{Preis} =30 \euro /\text{m}^{2}\] #d.f #gi #d.f #hj #e op7!ns- u ry!nsz34c ====================== f s'ymbole, d7c namc t 6{ }7n &- kcnz34net sd, ,wc - +5lu~ + 0 li}c ;e regyn ;,5c. >a h|f> &,|( z34c %+ plus %- m*us %. mal =pt='* %: &t3lt 4, -h`lt c z =doppypt= %= g4 %o, !g~7 3 %9. kl67 3 >b op7!nsz34c %+ plus %- m*us %+- plus!,m*us %-+ m*us!,plus %. mal =pt='* %( mal =kr2z= %!( mal =}7n= %) -kn8pft t =kuq7, -kettusz34c, kr3sop7ator= #e>a-#e>b #ha %: &t3lt 4, -h`lt c z =doppypt= %8 bru4}r#'* =leerz34cregyn s0he (#i br84e)= %6 fakult`t'* >c ry!nsz34c %= g4 %*= /g4 %== idcti5 g4, kongruct =zlc- the?0= %*== n idcti5 g4, *kongruct =zlc- the?0= %:= def*itjs&m`'~ g4 =doppypt g4- hsz34c= %=: def*itjs&m`'~ g4 =g4hsz34c doppypt= %:=: -t15bar =doppypt g4hsz34c doppypt= %? `, `'quiv:ct, qp?tjal %*? n `, n `'quiv:ct, n qp?tjal %?? /&!r g4 %o, !g~7 3 %*o, n !g~7 3 %o= !g~7 o g4 %9. kl67 3 %*9. n kl67 3 %9= kl67 o g4 #e>b-#e>c #hb %oo, g~ g %99. kl6 g %o,9. !g~7 o kl67 3 %9.o, kl67 o !g~7 3 %o=9. !g~7, g4 o kl67 %9=o, kl67, g4 o !g~7 %0= ~spr#t %0?? ~spr#t /&!r >d t3lt =zlcthe?0= %"l t3lt %*"l t3lt n >e mcgcl=re =s0he (#ab mcgcl=re)= %3. -6>t t %0. &5nittc t %1. -m*d7t um, oe %|. s'yxetri5e diff7cz %3: vy =-b+dsthe?0= %0: et =-b+dsthe?0= %&e } y{ct v %*&e } n y{ct v %&* ht zm y{ct %2. } ~h:tc *, } t3lmc& v %2= } ~h:tc * o g4 %`, ~h`lt, } ob7mc& v %`= ~h`lt o } g4 #e>c-#e>e #hc >f logik =s0he (#ac logik)= %0, u %3, o %:* n >g &ometr0 =s0he (#ad.a &ometri5e s'ymbole)= %?= kongruct =&ometr0= %*?= *kongruct =&ometr0= %:0 qjektiv z %=0 p7spektiv z %#. sckrt 1 %"% p)aqy z =d zw3( ,qz34c } t3l . s'ymbols.= %"%= p)aqy u g4 =d zw3( ,qz34c } t3l . s'ymbols.= >h pf3le =s0he (#g pf3le)= %::o pf3l n< rts %:, pf3l n< rts %9:: pf3l n< l*ks %!: pf3l n< l*ks %9::o doppypf3l t 6f<{ 5aft %!:, doppypf3l t 6f<{ 5aft %==o implik!nspf3l =pf3l n< rts t doppyt{ 5aft= %4=, implik!nspf3l =pf3l n< rts t #e>e-#e>h #hd doppyt{ 5aft= %9==o `'quiv:czpf3l =doppypf3l t doppyt{ 5aft= %4!=, `'quiv:czpf3l =doppypf3l t doppyt{ 5aft= %>:, ,z?dnuspf3l %4!; pf3l n< obc %4;, pf3l n< /tc ? ,bnahe ac op7!ns- u ry!nsz34c } 6 leerz34c z ,ec, n< ihc dg n. da vle op7!ns- u ry!nsz34c k6c pt r ob7c ptr3he =pte #a u #d'= ~h:tc, 7- l47t r +5lu~ + d /mtbar d1fgcde z34c d 7kcnc r v7tik:c positj r pte t { f*g7. d leerz34c ? 6{ op7!ns- bzw. ry!nsz34c ~!fqt nr n< z34c r braiqe- 5t, 1 0 oeh* k6 leerz34c fgc d)f. d0s sd ? a{ 0 op7!ns- u ry!nsz34c, 9ffncde klax7n, xponctc u *diz% ,p,2 d wurzyz34c. d leerz34c ? { m:pt wd oft wg&,lc, um 0 zs&h9r>k bd7 t3l1s!d$e z -d2t- c. dm r pt n 3 gl0d7uspt &l%c wd, m~ 6e d1 fgcde zl t zlz34c -s=c ,wc. #e>h-#e #he f e bru4}r# u d fakult`tz34c gytc 0 obc 7l|t7tc aq&m6c leerz34cregyn f op7!ns- u ry!nsz34c n. 0 0dgabe v br84c wd - kapity (#i br84e) |fh ;hdyt. d fakult`tz34c %6 fgt /mtbar 1 e t7m. 6 leerz34c n< { fakult`tz34c 5~t 6e -we4slu t 6{ r vlc s'ymbole, 0 t { 5l8~yz34c %6 ;g*nc, 1s. fqs c + 07 ,ye k6 leerz34c 7gb, m~ f 6- d2t>k &s?gt ,wc. zm ,b,0 k ? 6e 9ff- ncde klax7 6 m:pt =ggf. t 67 braiqe- 5tt5c +m7ku= o a r zsh:(pt %" 6- &f8gt ,wc, um 0 6d2t>k z &!whl3}c =s0he ,b,0 #e >b#aj'=. r sckrte bzw. 5r`& }r# 4 6 5w)z5t- s'ymbol, r 0 ;d2tu . s'ymbols ne- g0rt, wd * r braiqe5t 4 6 ,?+&,y- t% %* 0d&&bc. 6zyne ry!nss'ymbole ,kc * r 5w)z5t -50dce f?mc ,hc. r ,/e }r# bm s'ym- bol f (!g~7 o g4) k zm ,b,0 waagrt o 5r`g d)&,yt 9. d braiqe5ts'ymbol }=t jew3ls f ae g`ng>c v)i+tc . 5w)z5t- s'ymbols. #e #hf h*,3e: f m)k0ruc + s'ymbolc, 0 2 op7!ns- bzw. ry!nsz34c |s=c, s0he (#h 6fb#ja #h +#g =#g +#h \[8 +7 =7 +8\] ,b,0 #e >b#jb x -#e =#b \[x -5 =2\] ,b,0 #e >b#jc #fc.#e =#cae o #fc .#e =#cae \[63 \cdot 5 =315\] ,b,0 #e >b#jd a.b =b.a \[a \cdot b =b \cdot a\] #e #hg ,b,0 #e >b#je #cd (#e =#agj oder' #cd !(#e =#agj \[34 \times 5 =170 \; \text{oder} \; 34 *5 =170\] ,b,0 #e >b#jf #h.2-#g` =-#ef o #h .2-#g` =-#ef \[8 \cdot (-7) =-56\] ,b,0 #e >b#jg #aj :#d =#b,e \[10 :4 =2,5\] ,b,0 #e >b#jh a :b =c \[a :b =c\] ,b,0 #e >b#ji n6 =#a.#b.#c.#d...." .2n -#a`.n #e #hh o n6 =#a .#b .#c .#d ....' .2n -#a` .n \[n! =1 \cdot 2 \cdot 3 \cdot 4 \cdot ... \cdot (n -1) \cdot n\] ,b,0 #e >b#aj 2a +#a`6 =a6.2a +#a` o 2a +#a`6 =a6"2a +#a` \[(a +1)! =a! (a +1)\] ,b,0 #e >b#aa f )g2x` \[f \circ g(x)\] #e #hi #e #ij #f klax7n u sckrte }r#e ======================= fgcde s'ymbole ,wc 1 r (*ncs3() di- rekt + d ;ne 9ffncde klax7 %} e$>e 5~cde klax7 %!{ &5w3f( 9ffncde klax7 %!} &5w3f( 5~cde klax7 %${ spitze 9ffncde klax7 %$} spitze 5~cde klax7 %'{ }umpfw*kl>e 9ffncde klax7 %'} }umpfw*kl>e 5~cde klax7 %!>{ g1'~5e 9ffncde klax7 =ob7e grcze= %!>} g1'~5e 5~cde klax7 =ob7e grc- ze= %!<{ g1'~5e 9ffncde klax7 =,/e grc- ze= %!<} g1'~5e 5~cde klax7 =,/e grcze= %#2 r/de spezieqe 9ffncde braiqe- 5tklax7 %#` r/de spezieqe 5~cde braiqe5t- klax7 #f #ia %#{ e$>e spezieqe 9ffncde braiqe- 5tklax7 %#} e$>e spezieqe 5~cde braiqe5t- klax7 %#!{ &5w3f( spezieqe 9ffncde braiqe5tklax7 %#!} &5w3f( spezieqe 5~cde braiqe- 5tklax7 %$!{ z3lczsfa~usklax7: ,=7e z3lc zsfa~cde g~e l*ke &5w3f( klax7 %"l sckrt7 }r# %"% sckrt7 doppy}r# =d zw3( ,qz34c } t3l . s'ymbols.= %'<= braiqe5tt5e +m7kusklax7n =9ff- ncd u 5~cd= %<8 ;g*n 67 n2c z3le f l0gcde zsfa~cde klax7n s0he (#ae.b h?izont:e zsfa~uc u l0gcde klax7n). #f #ib #f.a aq&m6% z klax7n :::::::::::::::::::: * r math{atik5t } 0 &n1e 0dgabe . ,/50ds zw 9ffncdc u 5~cdc klax7n /7- !l. d%h:b sd 0 klax7n r braiqe5c te'xt5t f math{ati5e |!d$e /&3gnet. * r math{atik5t sd dh 3gce klax7f?mc 7f?d7. #f.b 6f. * r 5w)z5t ,wc |'~7e klax7n ggt tw !g~7 &d$t 3 0 - *n7c. * e m3}c !fqc m~ 0 braiqe5t 0c math{ati5 /- ;d2tcdc ,/50d n 0d&bc. j4 k % s*n,q 9, e ,/50d * 0 braiqe5t z ,8nhc, ea 1s !gdc r kl)h o w3l 0 ,8tgu r$5l8~e 1 0 5b,3e - ?>*al 7mc ,s. * 0c !fqc #f.a-#f.b #ic ,kc spezieqe braiqe5tklax7n -wcdet ,wc =s0he (#f.c spezieqe braiqe5t- klax7n)=. ,b,0 #f.b >b#ja f2x` =2x +#b`2x -#b` \[f(x) =(x +2)(x -2)\] ,b,0 #f.b >b#jb !{$apfel', $birne',' $orange!} \[\{\text{Apfel}, \; \text{Birne}, \; \text{Orange}\}\] ,b,0 #f.b >b#jc }-#%'; -#aj=} 3.{#aj:'; #%{ \[\left] -\infty; \frac{-10}{7} \right] \cup \left[ \frac{10}{3}; \infty \right[\] ,b,0 #f.b >b#jd 2#a +;#b8t<`|; .{;#a8t<' -2;t8#b< -#a`|-,}|-; \[\left(1 +\frac{2}{t}\right)^{2} #f.b #id \cdot \left[\frac{1}{t} -\left(\frac{t}{2} -1\right)^{-1}\right]^{-2}\] ,b,0 #f.b >b#je =+m.: f 0 d),yu t h7,?hebu r !g~7c 1'~cklax7n s0he ,b,0 #f.c >b#ja.= 6l 22x +#g`|;` =#j \[\lg \left( (x +7)^{2} \right) =0\] #f.c spezieqe braiqe5tklax7n :::::::::::::::::::::::::::: %#2 r/de spezieqe 9ffncde braiqe- 5tklax7 %#` r/de spezieqe 5~cde braiqe5t- klax7 %#{ e$>e spezieqe 9ffncde braiqe- 5tklax7 %#} e$>e spezieqe 5~cde braiqe5t- klax7 %#!{ &5w3f( spezieqe 9ffncde braiqe5tklax7 #f.b-#f.c #ie %#!} &5w3f( spezieqe 5~cde braiqe- 5tklax7 0 spezieqc braiqe5tklax7n ,kc ,/- 50d -wcdu f*dc, zm ,b,0, '- um &}:tustkc r 5w)z5t zr trcnu math{ati57 |!d$e 0d,z&bc, 0 c * r braiqe5t n o nr 5w7 re:is0rc ,lc '- um ;s h7,?&hobce klax7n d),z,yc '- um 0 ;e h7,?hebu 6zyn7 |d$st3le 0d,z&bc '- um 6zyne t3le kompliz0rt7 |!d$e ss gl0d7n z ,kc. * r 5w)z5t ,wc ;d*guc f 0 g8lt>k 6% ,?1sg=cdc |d$s oft r|m ab&,et u am cde d7sybc z3le &,5c. 0 kzc braiqe5tz3lc ,lc 6e s4e tk sytc z. d 65~c r ;d*guc * spezieqc braiqe5t- klax7n s?gt f 0 n9t>e abtrcnu u ,3t g4zt> d1 h*, ~ 0 klax7n syb7 * r 5w)z5t,?la& n 756c. sd * r 5w)z5t klax7paare ;s'- ea 4 f)be o fettd$'- h7,?&hobc, p b0tc 0 spezieqc braiqe5tklax7n 6e yeg+t7e d),yu 3 r 6sz 6% +k8nd>usz34cs f ;e t'ypografi5e |z34nuc =s0he (#c.d ;e #f.c #if t'ypografi5e |z34nuc)=. * 0{ fq m~ 0 f?m r klax7n * 67 +m7ku zr braiqe,8- tgu fe}&h:tc ,wc =s0he (#a.c +m7kuc zr braiqe5t,8tgu)=. 0 spezieqc braiqe5tklax7n ,kc | -- wcdet ,wc, um ;s h7,?&hobce t3l1s- !d$e z kcnz34nc. | * 0{ fq m~ 0 f?m r klax7n * 67 +m7ku zr braiqe,8tgu fe}&h:tc ,wc =s0he (#a.c +m7kuc zr braiqe5t,8tgu)=. * r math{ati5c not!n r 5w)z5t b0(t 0 r|me -t3lu r s'ymbole subtile m- kc, d vh 6zyn7 s'ymbole ,z,6 kl),z- ,yc. b r ,8tgu komple'x7 |!d$e * 0 braiqe5t k % dh v ,?t3l 9, 0 zsh`n& 07 s'ymbole 4 6 ,z!sz% klax7paar d2t z mn:is0rc, ~ + 0c ,yc * r 5w)z5t k6e klax7n xi}0rc. ,b,0 #f.c >b#ja =+m.: * 0{ ,b,0 sd 0 |'~7c klax7n rot.= '<=d0 speziellen braille5rift- klammern %#2...#` }ehen f8r rote klammern.'<= 6l #22x +#g`|;#` =#j #f.c #ig \[\lg \textcolor{red}{\left(} (x+7)^{2} \textcolor{red}{\right)} =0\] s0he | ,b,0 #f.d >b#jf. #f.d ,=z3l>e klax71s!d$e :::::::::::::::::::::::: ,wc * r 5w)z5t ,=7e z3lc t math{a- ti5c |!d$c 4 6e g~e &5w3f( klax7 zs- &fa~t, wd * r braiqe5t d s'ym- bol %$!{ ? { 7}c |d$ &,5c. 0 z3lc- we4sy r 5w)z5t ,wc t %<8 &kcnz34- net, /abh`ng> dv, ob * r braiqe5t 6e n2e z3le ;gonnc wd =s0he ,b,0e #f.d >b#je u #f.d >b#jf'=. f klax7paare, 0 c 8 ,=7e z3lc 7}re$c, gb % * r braiqe5t zw3 d)- ,yusmkc. 8b7,3e wd 0 9ffncde klax7 z ;g*n r 7}c z3le u 0 5~cde klax7 - +5lu~ + 0 lte z3le &,et. 0 z3lcwe4sy r 5w)z- 5t ,wc | hr t %<8 &kcnz34net. 6e #f.c-#f.d #ih n2e ph'ysik:i5e z3le } n 7f?d7. ? u n< 0{ z34c ,kc je nd, 2 % r ,8s#tk d0nt, leerz34c &,et ,wc, j4 d)f d 7}e r bdc braiqez34c n 3 z34c f bru4cde mi~d2(t ,wc ,kc. m+4m } % zwe$m`'~>, 0 5w)z5td),yu 6% math{ati5c |d$s 8 ,=7e z3lc * 0 braiqe5t z ,8nhc, ,b,0s,3e zr -+51u b r 6fhu . matri'x;griffs. * 07 d)- ,yusf?m 756c 0 klax7s'ymbole ,/,6 1 jd7 z3le. 0 6zync t7me *n7h:b r klax7n ,wc p |&,#et, ~ 0 sp:tc gut 7kcnbar sd. leerz34c *n7h:b v t7mc sd t pt #d %" |,zf8qc, um 0 ,z&h9- r>k r y{c( z -d2tc. g~e leerr|me sd z -m3dc =s0he ,b,0 #f.d >b#jd'=. h*,3: 0e d),yusf?m fht n zr sybc ,8bli$- b)k 2 * r 5w)z5t. s } ,z,{ sr ,1w`n- d> z 5bc u 3gnet c w>7 f 0 rout*e)b. ,b,0 #f.d >b#ja 2n<8k` 2#e<8#c` \[\left( \begin{array}{c} n \\ k #f.d #ii \end{array} \right) \left( \begin{array}{c} 5 \\ 3 \end{array} \right)\] oder \[\binom{n}{k} \binom{5}{3}\] ,b,0 #f.d >b#jb 2#a #b #c #d <8 #b #c #d #a` | 6d2t> u ,8s#t } 0 vm ,})d abw34c- de, !kz7e 5b,3e: 2#a#b#c#d <8 #b#c#d#a` \[\left( \begin{array}{cccc} 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \end{array} \right)\] ,b,0 #f.d >b#jc 2a1,, a1,; ... a1#an <8' a1;, a1;; ... a1#bn <8' ... <8' a1n#a a1n#b ... a1nn` \[\left( \begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1n} #f.d #ajj \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & & & \vdots \\ a_{n1} & a_{n2} & \dots & a_{nn} \end{array} \right)\] ,b,0 #f.d >b#jd 2#j x"+#a` 2x"-#a #j ` o 2#j x"+#a <8 x"-#a #j` \[\left( \begin{array}{cc} 0 & x +1 \\ x -1 & 0 \end{array} \right)\] ,b,0 #f.d >b#je f2x` =$!{#a f8r rationale x' <8 #j f8r irrationale x o f2x` =$!{#a '.f8r rationale'. x' <8 #j '.f8r irrationale'. x 6e mk * r kz5t: f2x` =$!{ #a f r!n:e x' <8 #j f irr!n:e x \[f(x) =\left\{ 1 \; \text{fr rationale} \; x \\ #f.d #aja 0 \; \text{fr irrationale} \; x \right.\] ,b,0 #f.d >b#jf =+m.: s0he | (#f.c spezieqe braiqe5tklax7n).= a#a.f d l9sc v g4uc)=. tretc 0 }r#e paar,3e 1, zm ,b,0 3 ;tgs}r#e o 3 det7m*+tc}r#e b matri- zc, ,wc s 2 klax7n ;hdyt. #f.d-#f.e #ajc ,b,0 #f.e >b#ja >p2#b"l#e` \[P(2 | 5)\] ,b,0 #f.e >b#jb "l-#c"l =#c \[|-3| =3\] ,b,0 #f.e >b#jc b#jd y ="l"l#cx"l -#c"l \[y =||3x| -3|\] ,b,0 #f.e >b#je 6cb#jf =+m.: s0he | (>a#a.f d l9sc v g4uc).= #f.e #ajd x =#cx -#d "l-x \[x =3x -4 \quad | -x\] ,b,0 #f.e >b#jg !{k &e$$z "l p 9=k 9=q!}' =:{p...q} \[\{k \in \mathbb{Z} | p \leq k \leq q\} =:[p ... q]\] ,b,0 #f.e >b#jh "la1, b1, <8 a1; b1;"l \[\left| \begin{array}{cc} a_{1} & b_{1} \\ a_{2} & b_{2} \end{array} \right|\] ,b,0 #f.e >b#ji "%a.x"% ="la"l."%x"% \[\|a \cdot x\| =|a| \cdot \|x\|\] #f.e #aje #f.f te'xtklax7n * r math{atik :::::::::::::::::::::::::::::: | * math{ati5c pa~agc tretc klax7n 1, 0 eh7 6e te'xt- 3 6e math{ati5e f/ktj 7f8qc. z 0{ zwe$ ,dc 0 klax7n r te'xt5t * 0 math{atik5t (imp?t0rt) ,wc. s ,wc d+n 4 6c ,?1sg=cdc pt #f %' v math{ati5c s'ymbolc =2 { g4hs- z34c= ,/50dc =s0he (#c.g szz34c)=. n ae klax7n * 67 math{ati5c um&bu ,hc 0 3gcte f/ktj math{ati57 klax7n. ,b,0e hr,f sd f?mynux7n, g4us;d*guc u 6hc h*t7 g4uc. hr } % r 5bcdc bzw. ,8tgcdc pn ,8,lc, te'xt- o math{ati5e klax7n z w`hlc. ,b,0 #f.f >b#ja >i1, =>i1; =>i '=$gesamtstrom'= \[I_{1} =I_{2} =I \quad \text{(Gesamtstrom)}\] ,b,0 #f.f >b#jb =+m.: 0 f?mynux70ru, 0 * r 5w)z5t m3} am rtc s3tcr+d }=t, wd * braiqe- #f.f #ajf 5t ;,?zugt l*ks pz0rt. * r zw3tc r fgcdc v)i+tc }=t s * te'xt5t; % wd 7} d+< 4 la'youttk * 0 math{atik5t &we4syt. s0he (#a.a.a la'yout).= '=#a.a'= s =v1).t +#a;at|; o =#a.a'= s =v1).t +#a;at|; o 2#a.a` s =v1).t +#a;at|; \[s =v_{0} \cdot t +\frac{1}{2} at^{2} \quad (1.1)\] ,b,0 #f.f >b#jc x|; +px +q =#j '=p *=#j'= o x|; +px +q =#j 2p *=#j` \[x^{2} +px +q =0 \quad (p \neq 0)\] #f.f #ajg #f.f #ajh #g pf3le ======== 0 braiqemath{atik5t -f8gt 8 zw3 d),yus)tc f pf3le. '- * r modul)c 0dgabe ,wc pf3le 1s y{ctc f ,#u =h?izontal, v7tikal o diagonal=, 5aft- u spitzcf?m zs&- ,et. '- f 6>e h?izont:e pf3le }=c ,z!sz def*0r( d),yuc zr -f8gu. #g.a modul)e pf3le :::::::::::::::::: pf3lmodule %4 5l8~yz34c f pf3ld),yuc %; 6f<7 v7tik:7 5aft %: 6f<7 h?izont:7 5aft %? 6f<7 diagon:7 5aft =l*ks obc!, rts /tc= %* 6f<7 diagon:7 5aft =l*ks /tc!, rts obc= %= doppyt7 h?izont:7 5aft %;; &}r#yt7 6f<7 v7tik:7 5aft #g-#g.a #aji %:: &}r#yt7 6f<7 h?izont:7 5aft %?? &}r#yt7 6f<7 diagon:7 5aft =l*ks obc!,rts /tc= %** &}r#yt7 6f<7 diagon:7 5aft =l*ks /tc!,rts obc= %== &}r#yt7 doppyt7 h?izont:7 5aft %! 6f kl67 'qu7}r# 6% ,z?dnuspf3ls h?izont:e pf3le, b ec ,zm3} 1 d 5l8~yz34c -z#(t wd %:, pf3l n< rts t 6f<{ 5aft u 6f<7 spitze %!: pf3l n< l*ks t 6f<{ 5aft u 6- f<7 spitze %!:, pf3l n< l*ks u rts t 6f<{ 5aft u 6f:, ,z?dnuspf3l t r modul)c d),yu ,kc pf3le t -50- dcc 5aft- u spitzcf?mc * a4t -50dce ,#uc 0d&&bc ,wc. #g.a #aaj 6 pf3ls'ymbol wd je n< ,?hdcs6 r y{c('- ` 2 * 6{ b1ka}cs'y}{'- 2 fgt zs&,et: '- 5l8~yz34c f pf3le '- }r# 4 e 5aft bzw. 'qu7}r# bm ,z- ?dnuspf3l '- spitze am l*kc, b v7tik:c pf3lc am ,/c cde . 5afts =fqs ,?hdc= '- 5aft '- spitze am rtc, b v7tik:c pf3lc am ob7c cde . 5afts =fqs ,?hdc= dr3- u ,=f:, =,z?dnuspf3l= ,wc m3}cs oe 5l8~yz34c &,5c. sd wt7e pf3lf?mc'- zm ,b,0 6e &bo- gce spitze'- d),z,yc, k 6% r fgcdc z34c zw { 5l8~y- u { d1 fgcdc z34c 6&5obc ,wc: %/ %+ %( %) %< u %% =d - ltc fq z -wcdcde z34c } d zw3( ,qz34c=. 6 s4% z34c ,k( | 3 6 wt7% 5aft- o spitzcf?mz34c d0nc. 0 ;d2tu . -wcdetc z34cs } * 67 braiqe- 5tt5c +m7ku z 7l|t7n =s0he (#a.c +- m7kuc zr braiqe5t,8tgu)=. #g.a #aaa 0 modular 0d&&bcc pf3le ,hc -50dce f/ktjc u d7c ;zug z ;nb#ja x >:,6,t'x \[x \mapsto \arctan x\] #g.a #aab ,b,0 #g.a >b#jb f:': x >:,3x \[\overline{f}: x \mapsto \sqrt{x}\] ,b,0 #g.a >b#jc #l1x":,p f2x` =#l1x"4*,p f2x`' =#l1x"4?,p f2x` \[\lim_{x \rightarrow p} f(x) =\lim_{x \nearrow p} f(x) =\lim_{x \searrow p} f(x)\] ,b,0 #g.a >b#jd f1; )f1,': $!{>d1, :,$$r' <8 x >:,f1,2x` \[f_{2} \circ f_{1}: \left\{ D_{1} \to \mathbb{R} \\ x \mapsto f_{1}(x)\right.\] #g.b def*0r( pf3le :::::::::::::::::: %::o pf3l n< rts t 6f<{ 5aft u 6- f<7 spitze #g.a-#g.b #aac %9:: pf3l n< l*ks t 6f<{ 5aft u 6- f<7 spitze %9::o pf3l n< l*ks u rts t 6f<{ 5aft u 6fb#ja #a -#a8x =#j 9==o #a8x =#a' 9==o x =#a \[1 -\frac{1}{x} =0 \Leftrightarrow \frac{1}{x} =1 \Leftrightarrow x =1\] ,b,0 #g.b >b#jb g #.h ==o a:,1g )a:,1h =#j \[g \perp h \Rightarrow \vec{a}_{g} \circ \vec{a}_{h} =0\] #g.c ;5tu v pf3lc ::::::::::::::::: * r braiqemath{atik5t ,wc pf3l- ;5tuc'- /&<(t i% r|mc ;zugs zm pf3l * r ,?la&'- - +5lu~ + e pf3l }s * klax7n &,5c. 1 e pf3l fgt pt #d %" u 0 ;5tu wd * spezieqc r/dc braiqe5t- klax7n %#2...#` 6&5lo~c. 6 we4sy zr te'xt5t m~ &kcnz34net ,wc. 1 0 5~cde klax7 fgt 6 leerz34c o 6 szz3- #g.b-#g.c #aae 4c. :t7n!v ,dc je n< *h:t r/de math{a- ti5e klax7n %2 u %` o a r/de te'xtklax7n %=...= -wcdet ,wc. ? 0 9ffncde klax7 wd b math{ati5c klax7n pt #d %" u b te'xtklax7n pt #f %' &,et. d 5ts'y}{ *n7h:b r klax7n ~- spr#t r &w`hltc klax7)t. }=t 0 ;5tu * r ,?la& s} * klax7n u sd 0e v *h:t7 ;d2tu, ,wc s ,8nx u 4 0 obc ;,5cc klax7n 7!gzt. ,b,0 #g.c >b#ja #e ::o"#2+#b#` ... ::o"#2.#c#`' ... ::o"#2-#d#` ... o #e ::o"2+#b` ... ::o"2.#c` ...' ::o"2-#d` ... \[5 \stackrel{+2}{\longrightarrow} ... \stackrel{\cdot 3}{\longrightarrow} ... \stackrel{-4}{\longrightarrow} ...\] #g.c #aaf ,b,0 #g.c >b#jb #c,hf .#d,g ::o"#2'.gerundet'.#`' #d .#d,e o #c,hf .#d,g ::o'=gerundet=' #d .#d,e \[3.86 \cdot 4.7 \stackrel{\text{gerundet}} {\longrightarrow} 4 \cdot 4.5\] #g.c #aag #g.c #aah #h 6fusz34c %> +k8nd>usz34c f 6fusz34c f 6fusz34c f zsfa~cde ob7e m)k0ruc % +k8nd>usz34c f zsfa~cde ,/e m)- k0ruc %$ -}`rkusz34c f zsfa~cde m)k0ruc %! zw3t% -}`rkusz34c f zsfa~cde m)k0ruc b -5usz34c f zsfa~cde m)- k0ruc %$5 abk8nd>usz34c f -}`rk( zsfa~cde m)k0ruc %!5 zw3t% abk8nd>usz34c f -}`rk( zsfa~cde m)k0ruc m)k0ruc, 0 * r 5w)z5t rts obc o rts /tc am s'ymbol }=c %* }r# =5r`g o &rade= %/ }7n %( kr2z =5r`g= #h #aai %+ plusz34c %- m*usz34c %4 hakc =-s#7usmath{atik='* m)k0ruc, 0 * r 5w)z5t 8 o / { s'ymbol }=c %: waagrt7 }r# %? 5l+gcl*ie =tilde= %; pt %) kr3s, kuq7 %0 d<'* %= g4hsz34c %2 bogc %:, pf3l n< rts %!: pf3l n< l*ks %4! k3l t spitze rts'* %4, k3l t spitze l*ks'* '* 1 0 z34c f e -s#7usmath{ati5c ha- kc, d d< u 0 k3le m~ jew3ls 6 leer- o szz34c fgc, da s son} t ,7c z34c -we4syt ,wc ,kc. m)k0ruc sd ,z!sze, 0 * r 5w)z5t ob7h:b, ,/h:b o rts v 6{ s'ymbol &- ,5c ,wc, um d~ ;d2tu z !7n. #h #abj ,b,0e hr,f sd: '- pf3le 8 bu4}abc, 0 s 3 vekt?c kcnz34nc. '- }r#e 8 bu4}abc, 0 s 3 }re$c kcn- z34nc. '- }r#e n< bu4}abc, 0 s 3 &ometri5e abbilduc kcnz34nc. '- }r#e n< f/ktjss'ymbolc, 0 diff7c- ti:abl3tuc m)k0rc. t0f- o ho4&,y( bu4}abc u zlc + 6{ s'ymbol !zlc dg n z e m)k0ruc. s ,wc 3 *diz% ;hdyt =s0he (#aj.c *diz% u xponctc)=. eo w> sd s'ymbole f 6hc 2 grad o w*kym*u( m)k0ruc. r * r 5w)z5t 8be }r# 8 c 0dholcdc ziff7n u ziff7nfgc * p7iodi5c dezi- m:br84c wd * r braiqe5t n 4 6e m)- k0ru 0d&&bc =s0he (#b.a.d p7iodi5e dezim:br84e)=. % wd braiqe5tt5 zw 6fusz34c ,wc direkt ?, 0 ab- k8nd>usz34c /mtbar h*t7 { z34c bzw. #h #aba r z34cfge &,5c. h*,3: 0 fr8h7c s'ymbole f k3l t spitze rts %"o u spitze l*ks %"9 uc 4 0 * r li}e ,1&fhtc s'ymbole 7,et. #h.a 6fusz34c l3(t 0 m)k0ru 6 u gb +, ob s * r 5w)z5t obc bzw. obc rts o /tc bzw. /tc rts }=t. d +k8n- d>usz34c wd b ob7c m)k0ruc 8b7,3e wg&,lc. } 6 s'ymbol ,pwl t m)k0ruc 3 | t *diz% bzw. xponctc -s=c, p ,wc 0 m)- k0ruc * r regy ? lt7c &,5c =s0he (#aj.c *diz% u xponctc)=. ,wc + 6{ hps'ymbol ,=7e m)k0ruc d7sybc )t 4 6e 6&klax7( zl 7,et, wd 0e 3 *de'x &,5c. #h-#h.a #abb ,b,0 #h.a >b#ja f*', f**', f***', f|2#d` \[f', f'', f''', f^{(4)}\] ,b,0 #h.a >b#jb y** =f2x', y', y*` \[y'' =f(x,y,y')\] ,b,0 #h.a >b#jc >a?x? +b? =#j \[\tilde{A}\tilde{x} +\tilde{b} =0\] ,b,0 #h.a >b#jd f:': x >:,3x \[\overline{f}: x \mapsto \sqrt{x}\] ,b,0 #h.a >b#je c/1k \[{\overset{*}{c}}_{k}\] ,b,0 #h.a >b#jf a*1n \[a'_{n}\] #h.a #abc ,b,0 #h.a >b#jg =+m.: 0 }r#-m)k0ru }=t n< { hp- s'ymbol t ,/{ *de'x u 7fa~t ,p,t bde.= y1,>* \[{y_{1}}'\] ,b,0 #h.a >b#jh y;1n"+#a \[\dot{y}_{n +1}\] ,b,0 #h.a >b#ji man4mal wird mit $$q<+ d0 menge der positiven rationalen zahlen und mit $$q>+1) d0- selbe menge mit der zahl #j bez34net. o man4mal wird mit !,$$q<+'. d0 menge der positiven rationalen zahlen und mit !,$$q>+1)'. d0- selbe menge mit der zahl #j bez34net. Manchmal wird mit $\mathbb{Q}_{+}$ die Menge der positiven rationalen #h.a #abd Zahlen und mit $\mathbb{Q}^{+}_{0}$ dieselbe Menge mit der Zahl 0 bezeichnet. #h.b zsfa~cde m)k0ruc ::::::::::::::::::::: zsfa~cde m)k0ruc ;z0hc c 1 ,=7e s'ymbole u ,wc * r braiqe5t ? 0c &- ,et. braiqe5tt5 sd s qjektive =s0he (#aj qjektivtk)=. 0 m)k0ru wd }s 4 6% r bdc +k8n- d>usz34c f zsfa~cde ob7e %$ bzw. ,/e % m)k0ruc 6&l3(t. fqs *n7h:b . m)k0rtc ;r34s wt7e zsfa~cde m)- k0ruc o qjektive ,?kxc, m~ d +k8n- d>usz34c um d z34c %$ bzw. %! z ;g*n 7wt7t u am 5lu~ zw*gcd d ~,~cde 5lu~z34c &,et ,wc. =s0he hr,z 0 7l|- t7u zr -}`rku v qjektivc, (#aj.b -- }`rk( qjektive).= 0 m)k0ru gilt, bs 0 wku ,1&hobc wd 4: '- 6 5lu~z34c '- 6 leerz34c #h.a-#h.b #abe '- d z3lccde'- 1'~7 bm z3lctrcnz3- 4c %" o '- 6e wt7e m)k0ru bzw. 6 ,7% qjektiv ,b,0 #h.b >b#ja >u =$2>ab +$:>ab \[U =\frown{AB} +\overline{AB}\] ,b,0 #h.b >b#jb :>ab =:>cd \[\underline{AB} =\underline{CD}\] ,b,0 #h.b >b#jc $$:>a 3.>b$5 =>a: 0.>b: o $:>a"3.>b =>a: 0.>b: \[\overline{A \cup B} =\overline{A} \cap \overline{B}\] ,b,0 #h.b >b#jd $$:v|;$5 ='.mittlere quadrat.' ge5windigk3t'. \[\overline{v^{2}} =\text{mittlere quadrat. Geschwindigkeit}\] #h.b #abf ,b,0 #h.b >b#je $$:,>p1)>p$5 \[\vec{P_{0}P}\] ,b,0 #h.b >b#jf $:>a*>b* "%$:>ab \[\overline{A'B'} \parallel \overline{AB}\] ,b,0 #h.b >b#jg $$:>t1,; 0.>t1,($5 \[\overline{T_{12} \cap T_{18}}\] ,b,0 #h.b >b#jh $$:2>a 0.>b` 3.2>a 0.>c`' 3.2>b 0.>c`$5 \[\overline{(A \cap B) \cup (A \cap C)\cup (B \cap C)}\] #h.b #abg #h.b #abh #i br84e ======== %8 bru4}r# %; bru4+f+g %< bru4cde %%< cde s`mt7 br84e =d zw3( ,qz34c } t3l . s'ymbols.= #i.a zlcbr84e u &mi5( zlc ::::::::::::::::::::::::: zlcbr84e, b ec ,pwl !zl7 3 | ncn7 1s positivc gzc zlc ;}=c, ,wc 2 fgt d)&,yt: r !zl7 wd * r ,})d5b,3e &,5c u r ncn7 * r &scktc 5b,3e oe 3gc% zlz34c u oe leerz34c +&f8gt. h*,3: sd * math{ati5c |!d$c zlcbr84e t br84c * 6f<7 o |fh7 5b,3e komb*0rt, p k | f zlcbr84e 0 ~,~cde 5b,3e &- w`hlt ,wc, um ae br84e 6h z &}:tc. #i-#i.a #abi bde ;,}t3le 67 &mi5tc zl, 0 gze zl u r zlcbru4, ,wc t 6{ zlz34c -s=c u oe leerz34c +,6 &,5c. ,b,0 #i.a >b#ja #a; \[\frac{1}{2}\] ,b,0 #i.a >b#jb #bg:+ \[\frac{27}{36}\] ,b,0 #i.a >b#jc #c/ -#a: =#i,; -#d,; =#e,; \[\frac{3}{4} -\frac{1}{3} =\frac{9}{12} -\frac{4}{12} =\frac{5}{12}\] ,b,0 #i.a >b#jd ;#aj8-#b< =-#e \[\frac{10}{-2} =-5\] #i.a #acj ,b,0 #i.a >b#je #aj; +#aj8-#b =#j o ;#aj8#b< +;#aj8-#b< =#j \[\frac{10}{2} +\frac{10}{-2} =0\] ,b,0 #i.a >b#jf #d#a/ \[4\frac{1}{4}\] ,b,0 #i.a >b#jg #a#c/ +#b#a: =#ba,; +#bh,;' =#di,; =#d#a,; o #a#c/ +#b#a:' =#ba,; +#bh,;' =#di,;' =#d#a,; \[1\frac{3}{4} +2\frac{1}{3} =\frac{21}{12} +\frac{28}{12} =\frac{49}{12} =4\frac{1}{12}\] #i.a #aca #i.b 6fu r |fhc bru45b,3e -z#(t ,wc =s0he (#i.c |fhe bru45b,3e)=. d s'ymbol f e bru4}r# %8 fgt /mtbar 1 e !zl7. ebcf oe leerz34c 5~t r ncn7 direkt +. 0e -6f<( 5b,3e } | d?t ,z!l>, wo leerz34c - !zl7 o ncn7 4 e zsh:(- pt %" 7,et ,wc. s 3gnet c n f kom- ple'xe |!d$e, 0 4 0 ,/!d$u v leerr|- mc /,8s#t ,wc. ,b,0 #i.b >b#ja a8b \[\frac{a}{b}\] ,b,0 #i.b >b#jb a8#b \[\frac{a}{2}\] #i.b #acb ,b,0 #i.b >b#jc #a8x|: \[\frac{1}{x^{3}}\] ,b,0 #i.b >b#jd #ex8#cx"-#b \[\frac{5x}{3x -2}\] f ,b,0e r 6fb#jb u #d.d >b#je. #i.c |fhe bru45b,3e :::::::::::::::::::: zlc- u ,7e 6fu f gr04i5e bu4}abc -we4syt ,wc ,kc. dh wd 6 lat6i57 kl6bu4}abe - +5lu~ + d bru4cdez34c t #i.c #acd pt #f %' +&k8nd>t u zw { bru4cde- z34c u 0 +k8nd>u f g~bu4}abc r zs- h:(pt %" 6&f8gt =s0he (#a.b trcnc u zsh:tc math{ati57 |!d$e) ,p,2 ,b- ,0e #c.b >b#jf u #aa.c >b#jd'=. h*,3: ~,g r fr8h7c pra'xis ,mc ae zlc /- mtbar n< { bru4}r# * r ,})d5b,3e &- ,5c ,wc =s0he (#b.a.a zlc * ,})d5b- ,3e)=. ,b,0 #i.c >b#ja ;-a8b< o ;-a 8 b< \[\frac{-a}{b}\] ,b,0 #i.c >b#jb >a =;hb8#b<m|; \[A =\frac{hb}{2}\text{m}^{2}\] ,b,0 #i.c >b#jc ;#ex 8 #cx -#b< \[\frac{5x}{3x -2}\] #i.c #ace ,b,0 #i.c >b#jd ;#dx|; +#cx -#a 8 2x +#b`" 2x -#a`|;< =a8x"+#b +b8x"-#a' +c82x"-#a`|; o ;#dx|; +#cx -#a 8 2x +#b`" 2x -#a`|;< =;a 8 x +#b<' +;b 8 x -#a< +;c 8 2x -#a`|;< \[\frac{4x^2 +3x -1}{(x +2)(x -1)^2} =\frac{a}{x +2} +\frac{b}{x -1} +\frac{c}{(x -1)^2}\] ,b,0 #i.c >b#je ;x 8 x|; +y|;<' .2;#bx 8 x +y< -;x -y 8 x<` \[\frac{x}{x^2 +y^2} \cdot \left(\frac{2x}{x +y} -\frac{x -y}{x}\right)\] ,b,0 #i.c >b#jf x";x +#a 8 x|; -#a< \[x\frac{x +1}{x^{2} -1}\] #i.c #acf ,b,0 #i.c >b#jg ;a|; +b|; 8 #c2a -b`<'x \[\frac{a^{2} +b^{2}}{3(a -b)}x\] s0he | ,b,0e #ad.b >b#je u #aa.c >b#jd. #i.d ,=fb#ja ;;a8x< 8 ;b8x|:<< \[\frac{a /x}{b /x^{3}}\] oder \[\frac{\frac{a}{x}} {\frac{b}{x^{3}}}\] ,b,0 #i.d >b#jb ;#c/ -#a: 8 #a; +#a: -#a/< \[\frac{\frac{3}{4} -\frac{1}{3}} {\frac{1}{2} +\frac{1}{3} -\frac{1}{4}}\] ,b,0 #i.d >b#jc ;u -;#a8u< 8 u -;u 8 u +;#a8u%< \[\frac{u -\frac{1}{u}}{u -\frac{u}{u +\frac{1}{u}}}\] :::::::::::: cde . 7}c b+d% #i.d #ach d s'y}{ r math{atik5t * r d5c braiqe5t ---------------- * zw3 braiqeb`ndc zw3t7 b+d *h:tsv7z34x =========== 7}7 b+d ::::::: ,?w?t ......................... #a ~wi$lu ...................... #a kompakth v7sus kon('xt/abh`n- g>k ..................... #b n27uc ....................... #c zm &,| 0% regyw7ks ............ #g ,1b1 ........................ #g $la$te>x .................... #i #a gdlegcde tkc zr ,8tgu v ma- th{atik ................... #aa #a.a we4sy zw te'xt- u ma- th{atik5t ............... #aa #a.a.a la'yout ............ #ab #a.a.b +- u abk8nd>usz34c f math{atik5t ........... #af #a.a.c +- u abk8nd>usz34c f te'xt5t ............... #ai #a.a.d doppyleerz34ctk .... #bj #a.a.e h*,3e zm 6sz r 5t- we4sytkc .............. #bc #a.b trcnc u zsh:tc math{ati- *h:tsv7z34x >i 57 |!d$e ................ #bf #a.c +m7kuc zr braiqe5t,8tgu #bh #b ziff7n u zlc ............... #ca #b.a )abi5e ziff7n u zlc .... #ca #b.a.a zlc * ,})d5b,3e .... #cb #b.a.b zlc * &sckt7 5b,3e #ce #b.a.c dezim:br84e ........ #cg #b.a.d p7iodi5e dezim:br84e #ci #b.a.e gl0d7u lg7 zlc ..... #dj #b.a.f ?dnuszlc, dezim:- kla~ifikat?c, datc u uhrztc ................ #db #b.b r9mi5e zlc ............. #dd #c bu4}abc u szz34c ........... #dg #c.a ,?;m7ku zr kcnz34nu v bu4}abc ................. #dg #c.b g~- u kl65bu lat6i57 bu4}abc ................. #dh #c.c gr04i5e bu4}abc ........ #ea #c.d ;e t'ypografi5e |z34nuc #ef #c.e bu4}abc`e s'ymbole .... #fa #c.f kzw?ts'ymbole .......... #fd #c.g szz34c ................. #ff #c.h te'xt * r math{atik5t .. #fg #d 6hc ........................ #fi #d.a kcnz34nu v 6hcs'ymbolc #fi #d.b qzct, qmiqe ............ #gj *h:tsv7z34x >ii #d.c w*ky- u t{p7aturma'~e .. #ga #d.d 6hcs'ymbole 1s bu4}abc #gb #d.e -!g~7us- u -kl67uspr`fi- 'xe ..................... #ge #d.f !whuss'ymbole .......... #gg #e op7!ns- u ry!nsz34c ........ #ha #f klax7n u sckrte }r#e ....... #ia #f.a aq&m6% z klax7n ........ #ic #f.b 6fe klax71s!d$e .... #ih #f.e sckrte }r#e ............ #ajc #f.f te'xtklax7n * r math{a- tik ..................... #ajf #g pf3le ...................... #aji #g.a modul)e pf3le .......... #aji #g.b def*0r( pf3le .......... #aac #g.c ;5tu v pf3lc ........... #aae #h 6fiii zw3t7 b+d ::::::::: #aj qjektivtk ................. #aci #aj.a 6fiv gcde klax7n ............. #ahb +h`n& ......................... #ahg >a#a 5te re4cv7,rc 8 ,=7e z3lc #ahg >a#a.a additj ............... #aij >a#a.b subtraktj ............ #aia >a#a.c multiplik!n .......... #aic >a#a.d divisj ............... #aid >a#a.e l*e)e additj ......... #aif >a#a.f d l9sc v g4uc ........ #aih >a#b !7uc * r math{atik5t ..... #bja >a#b.a &!7( s'ymbole ........ #bja >a#b.b n2e s'ymbole ......... #bjb >a#b.c zlc .................. #bjd >a#b.d xponctc u *diz% ...... #bjd >a#b.e br84e ................ #bje >a#b.f bu4}abc .............. #bje >a#b.g klax7n u sckrte }r#e #bjf >a#b.h 6hc .................. #bjg >a#b.i pf3le ................ #bjh >a#b.aj qjektivtk ........... #bjh >a#b.aa we4sy zw te'xt- u math{atik5t ............. #bji >a#b.ab son}>% .............. #bji >a#c glo~ar ................... #baa >a#d math{ati5e z34c, &?dnet n< r #f-pte-braiqe-ta;qe ..... #bai *h:tsv7z34x >v >a#e :pha;ti5% svi #aj qjektivtk ============= %3 ...... wurzy %| ...... ob77 *de'x =h*tc= o x- ponct %1 ...... ,/7 *de'x =h*tc= %| o %#| ..... ,?d77 ob77 *de'x %1 o %#1 ..... ,?d77 ,/7 *de'x %$ ...... +k8nd>usz34c f zsfa~cde ob7e m)k0ruc % ...... +k8nd>usz34c f zsfa~cde ,/e m)k0ruc %5 ...... 5lu~z34c f 6fusz34c o bs s 4 6 ,7% z34c o 6e ,7e +k8nd>u ,1&hobc wd. % wd braiqe5tt5 zw 6f sd. #aj #adj #aj.a 6f, s?gt d wg,lc f !kz7e'- u dh !g~tc- t3ls ,8s#t7e'- |!d$e. * zw3fys!fqc } % j4 x ss, d 5lu~z34c z ,ec. um 1 0 -}`rku 6% qjektivs -z#tc z ,kc, ,mc leerz34c 4 e zsh:(pt %" 7,et ,wc. 0e tk } ? a{ b kzc |!d$c t #aj.a #ada op7!nsz34c !nz, ea b 6{ plusz34c * 6{ xponctc. ,b,0 #aj.a >b#ja x|; +x|n \[x^{2} +x^{n}\] ,b,0 #aj.a >b#jb a|;b|:c|/ \[a^{2}b^{3}c^{4}\] ,b,0 #aj.a >b#jc a|x5b|y5c|z \[a^{x}b^{y}c^{z}\] ,b,0 #aj.a >b#jd a|-#c .a|"+#e =a|-#c"+#e =a|#b o a|-: .a|"+#e =a|-#c"+#e =a|; \[a^{-3} \cdot a^{+5} =a^{-3 +5} =a^{2}\] #aj.a #adb ,b,0 #aj.a >b#je ;x|n 8 n6< \[\frac{x^{n}}{n!}\] ,b,0 #aj.a >b#jf ;x|#bn"+#a 8 2#bn +#a`6< \[\frac{x^{2n +1}}{(2n +1)!}\] #aj.b -}`rk( qjektive ::::::::::::::::::::: ~h`lt 6 qjektiv wt7e qjektive, leerz34c o bru4}r#e, m~ % (-}`rkt) ,wc. d0s 7fgt 4 d ,ec . qjektiv- v7}`rkusz34cs %$ ? d z34c f d ;- tfcde qjektiv. r gytus;r34 . qjek- tivs wd zw*gcd 4 6 -}`rkt% 5lu~z3- 4c %$5 ;cdet. -}`rk( qjektive ,kc | -5 ,z&h9r>e 5lu~z34c hebt 0 wku . -}`rktc qjek- #aj.a-#aj.b #adc tivz34cs 1. wn ae qjektive + d7sybc ,ye ab&5lo~c ,wc ,sc, k d saxy5lu~- z34c %%5 0 fge v 5lu~z34c 7,ec =d zw3( ,qz34c } t3l . z34cs=. ,b,0 #aj.b >b#ja $3x|;$5 =x \[\sqrt{x^{2}} =x\] ,b,0 #aj.b >b#jb n$|;|:$5 =n|( \[n^{2^{3}} =n^{8}\] ,b,0 #aj.b >b#jc =+m.: 0 7}e wurzy 5~t e xponctc 'n t 6, 0 zw3( dg n.= x$|n8#b$5 =$3x|n$5 =3x|n \[x^{frac{n}{2}} =\sqrt{x^{n}} =\sqrt{x}^{n}\] ,b,0 #aj.b >b#jd |/$3b.|:!3b|;.3b%5 \[\sqrt[4]{b \cdot \sqrt[3]{b^{2} \cdot \sqrt{b}}}\] #aj.b #add ,b,0 #aj.b >b#je $zerfallsgesetz': >n =>n1)' .2#a;`$|";#a 8 >t1#a;<$5 \[\text{Zerfallsgesetz:} N =N_{0} \cdot \left( \frac{1}{2} \right)^{\frac{1}{T_{1/2}}}\] #aj.c *diz% u xponctc ::::::::::::::::::::: } * r 5w)z5t 0 ho4- o t0f,yu 6% o ,=77 s'ymbole v math{ati57 ;d2tu, wd 0e * r braiqemath{atik5t 3 (ob77) bzw. (,/7 *de'x) &kcnz34net. xponctc sd t'ypografi5 n v ,7c ob7c *diz% z ,/53dc u ,wc dh =- ,/50d zr fr8h7c pra'xis= | * r braiqe5t t ,{sybc braiqez34c 6&l3(t. #aj.c.a h*t7e *diz% u xponctc ::::::::::::::::::::::::::::: *diz% rts vm hps'ymbol ,wc | * r braiqe5t /mtbar rts v 0{ s'ymbol &- ,5c. s ,wc t { z34c f 6c ob7c %| bzw. ,/c %1 *de'x 6&l3(t. fqs 7f?- #aj.b-#aj.c.a #ade d7, wd 0% z34c t 6{ qjektivv7}`r- kusz34c komb*0rt. } 6 s'ymbol t ,=7c h*t7c ,z!szc -- s=c, p ,wc 0e n<,6 ,8tgc. jd7 ,zsz } 6zyn 6,zl3tc. 6 acf ,?hdc7 xponct r$t + 0 lte ,ye. 6fb#ja ;x|n 8 n6< o ;x|n8n6< o x|n8n6 \[\frac{x^{n}}{n!}\] ,b,0 #aj.c.a >b#jb f1n52x`', f1n"+#a52x` \[f_{n}(x), f_{n +1}(x)\] ,b,0 #aj.c.a >b#jc 2>p1#bn"-#a`|r \[\left(P_{2n -1}\right)^{r}\] #aj.c.a #adf ,b,0 #aj.c.a >b#jd e|"+2b#je $zeit1$bob =$zeit1$alice' =;$entfernung 8' $geschwindigkeit< \[\text{Zeit}_{\text{Bob}} =\text{Zeit}_{\text{Alice}} =\frac{\text{Entfernung}} {\text{Geschwindigkeit}}\] ,b,0 #aj.c.a >b#jf a1n|k =2a1n`|k \[a_{n}^{k} =(a_{n})^{k}\] ,b,0 #aj.c.a >b#jg 2x1n|i`|r \[({x_{n}}^{i})^{r}\] #aj.c.a #adg ,b,0 #aj.c.a >b#jh >f$1n1k$52x` \[F_{n_{k}}(x)\] ,b,0 #aj.c.a >b#ji 2x$1n|i$5`|r \[(x_{n^{i}})^{r}\] ,b,0 #aj.c.a >b#aj &s$1#j"9=i"9=m <8 #j"9.j"9.n$5' >p2i', j` o &s1#j"9=i"9=m51#j"9.j"9.n' >p2i', j` \[\sum_{\substack{0 \leq i \leq m \\ 0 b#aa ;e$|";x|;8#b<$5 8 3#bb#ab b#ac >k1.i.j|k.l. =g|kr5g|ls5>k1risj \[K_{.i.j}^{k.l.} =g^{kr} g^{ls} K_{risj}\] ,b,0 #aj.c.a >b#ad 2>p$1b#ae -#a;e$|-r|;$5"l1)|>r' =#a;2#a -e$|->r|;$5` \[\left.-\frac{1}{2} e^{-r^{2}} \right|_{0}^{R} =\frac{1}{2} (1 -e^{-R^2})\] ,b,0 #aj.c.a >b#af e$|b#ag #l1t":,#a ;t|usz34c zr -f8gu. 0 kzf?mc %| u %1 sd 0 ,})df?mc. 0 lgf?mc %#| u %#1 sd nr d?t z -wcdc, wo -we4s- lus&,r t *diz% am ,?+g=cdc s'ymbol ;}=t, zm ,b,0 /mtbar n< 67 v)iablc. n< 6{ leer-, g4hs- o op7!nsz34c bzw. 67 9ffncdc klax7 sd -we4sluc |&5lo~c u 0 kzf?mc sd z -wcdc. #aj.c.a-#aj.c.b #aej ,b,0 #aj.c.b >b#ja |:3#h =#b \[\sqrt[3]{8} =2\] ,b,0 #aj.c.b >b#jb |,)6l#c \[^{10}\log 3\] ,b,0 #aj.c.b >b#jc 1i|j5m \[_{i}^{j}m\] ,b,0 #aj.c.b >b#jd x.|n"+#a3y \[x \cdot \sqrt[n +1]{y}\] ,b,0 #aj.c.b >b#je x#|n"+#a3y \[x \sqrt[n+1]{y}\] ,b,0 #aj.c.b >b#jf >a$|x#|n"+#a3y$5 \[A^{x \sqrt[n +1]{y}}\] #aj.c.b #aea ,b,0 #aj.c.b >b#jg |a56l'x =|a56l'b .|b56l'x \[^{a}\log x =^{a}\log b \cdot ^{b}\log x\] ,b,0 #aj.c.b >b#jh a.$|n8m$5$3y8x$5 \[a \cdot \sqrt[\frac{n}{m}]{\frac{y}{x}}\] #aj.c.c *diz% 1s gzc zlc :::::::::::::::::::::::: ;}=t 6 *de'x led> 1s 67 gzc zl, k 0e oe zlz34c +&5lo~c * r &scktc 5b- ,3e + d *de'xz34c &,5c ,wc. b neg!vc zlc wd d m*usz34c oe zsh:(pt %" zw { *de'xz34c u r zl 6&f8gt. ,s 0 positive 3gc5 r zl 4 6 plusz34c ;- tont ,wc, m~ dg 0 ,})d5b,3e zr +wcdu kxc u d plusz34c 4 e zsh:(pt %" vm *de'xz34c &trcnt ,wc. n< r zl * &sckt7 5b,3e ;d)f % k6% qjektiv5lu~z34cs, da d cde 6% 6fb#ja a|;b|:c \[a^{2}b^{3}c\] ,b,0 #aj.c.c >b#jb a1,,a1;; -a1,;a1;, \[a_{11}a_{22} -a_{12}a_{21}\] ,b,0 #aj.c.c >b#jc f1,2u1,', u1;` \[f_{1}(u_{1}, u_{2})\] ,b,0 #aj.c.c >b#jd x|"+#c5x|-: =x|) =#a \[x^{+3}x^{-3} =x^{0} =1\] ,b,0 #aj.c.c >b#je ;x|: 8 #c6< o ;x|:8#c6< \[\frac{x^{3}}{3!}\] #aj.c.c #aec #aj.d wurzyn u ,z!sze ::::::::::::::::::::: 'quadratwurzyn =oe 'qu:ifiz0rcde zl #b'= ,wc t { s'ymbol f wurzyn =ggf. t 6{ qjektivv7}`rkusz34c kom- b*0rt= 6&l3(t. d qjektiv 5~t ae s'ymbole 6, 0 c * r 5w)z5td),yu / { wurzy}r# ;f*dc. zlc, 0 +z3gc, ~ % c um 6e drit(, v0r( o ,7e wurzy hdyt, ,wc 3 ,?d7e ob7e *diz% am wurzyz34c d)&,yt. ,b,0 #aj.d >b#ja 3#b; =3#a \[\sqrt{\frac{2}{2}} =\sqrt{1}\] ,b,0 #aj.d >b#jb 3a"+b \[\sqrt{a +b}\] ,b,0 #aj.d >b#jc $3a2x -#c`|;$5 \[\sqrt{a(x -3)^{2}}\] #aj.d #aed ,b,0 #aj.d >b#jd $3#b3#b$5 =3#b .|/3#b \[\sqrt{2 \sqrt{2}} =\sqrt{2} \cdot \sqrt[4]{2}\] ,b,0 #aj.d >b#je u =|:$3-q8#b +-!32p8#c`|:' +2q8#b`|;%5 \[u =\sqrt[3]{-\frac{q}{2} \pm \sqrt{\left(\frac{p}{3}\right)^{3} +\left(\frac{q}{2}\right)^{2}}}\] #aj.d #aee #aj.d #aef #aa +:'ysis =========== %#% /cd =d zw3( ,qz34c } t3l . s'ymbols.= %&s suxcz34c %&p qduktz34c %1 ,/e grcze =h*t77 ,/7 *de'x= %| ob7e grcze =h*t77 ob77 *de'x= %) -kn8pft t =kr3s, kuq7= %6l log)ithmus %6(l log)ithmus nt:is %6/l log)ithmus du:is %6,l +tilog)ithmus %6:l 7!gzus- o kompl{ct`rlog)ith- mus %6e xponcti:f/ktj %6n num7us %6{ )gumct %~ *(gral %~~ doppy*(gral %~) uml1f*(gral %~~) h8qc*(gral %~<: ,/% *(gral %~: ob7% *(gral %>~ *(gral ;7 )t #aa #aeg %"l *(gr:}r# %* abl3tus}r# %; abl3tuspt %"d r/d% 'd =f p)tieqe abl3tu= %&d g~% dyta 3 diff7czz34c %#l lim% %#l<: lim% *f7ior %#l: lim% sup7ior #aa.a f/ktjc :::::::::::: %#% /cd =d zw3( ,qz34c } t3l . s'ymbols.= %&s suxcz34c %&p qduktz34c %1 ,/e grcze =,/7 *de'x= %| ob7e grcze =ob77 *de'x= %) -kn8pft t =kr3s, kuq7= bm suxc- u qduktz34c ,wc ,/e u ob7e grczc &m`'~ r ,?la& 3 ,/e u ob7e *diz% 0d&&bc. r ,~,3e fgcd wd * r braiqe5t zn 0 ,/e grcze +&&bc. ? { 1 0 grcz+gabc fgcdc |d$ }=t 8b7,3e 6 leerz34c. #aa-#aa.a #aeh ,b,0 #aa.a >b#ja &s a1b#jb &s1b#jc &s$1k1, =#a$5$|n1,$5 \[\sum_{k_{1} =1}^{n_{1}}\] #aa.b log)ithmus- u xponcti:f/ktjc :::::::::::::::::::: %6l log)ithmus %6(l log)ithmus nt:is %6/l log)ithmus du:is %6,l +tilog)ithmus %6:l 7!gzus- o kompl{ct`rlog)ithmus %6e xponcti:f/ktj %6n num7us %6{ )gumct #aa.a-#aa.b #aei 0 obc ,1&fhtc s'ymbole ,wc g47ma- '~c f ae v)i+tc r ;tfcdc kzw?ts'ym- bole * r 5w)z5t -wcdet. 0 s'ymbole * r braiqe5t }=c ebcf ,pwl f g~- 3 | f kl6&,5ce kzw?ts'ymbole. ;g*nt d )gumct t 6{ +k8nd>usz34c =zm ,b,0 f zlc o g~- bzw. kl6bu4}a- bc=, d)f % + d s'ymbol +5~c. * ,7c !fqc m~ 6 leerz34c zw s'ymbol u )gu- mct }=c. % } gd!sz ,z!l>, 0e s'ymbole t { aq&m6c 6l3tusz34c f kzw?ts'ymbo- le %7 u e * r 5w)z5t 8bc bu4}abc z bildc. zm ,b,0 k f %6l | %7log &,5c ,wc =s0he (#c.f kzw?ts'ymbo- le)=. 0e |!d$e sd !lg7, ,kc j4 d?t zwe$m`'~> 9, wo g~- u kl6&,5ce kz- w?ts'ymbole ,/50dc ,wc, 6e cg7e ,8- 6}ixu t r 5w)z5t &wht ,wc ,s o bm l%cdc 6e -tr1th t -6zyt ,?kxcdc s'ymbolc n ,?1s&,et ,wc k. ,b,0 #aa.b >b#ja |,)6l#c \[^{10}\log 3\] #aa.b #afj ,b,0 #aa.b >b#jb 6(l 2#c -x` \[\ln (3 -x)\] ,b,0 #aa.b >b#jc |a56l'x =|a56l'b .|b56l'x \[^{a}\log x =^{a}\log b \cdot ^{b}\log x\] ,b,0 #aa.b >b#jd 6e* 2z` =6e 2z` \[\exp'(z) =\exp (z)\] ,b,0 #aa.b >b#je 6e'z :=&s1k"=#j|#% ;z|k8k6< \[\exp z :=\sum_{k =0}^{\infty} \frac{z^{k}}{k!}\] #aa.c *(gr:- u diff7cti:re4nu ::::::::::::::::::::::::::::: %~ *(gral %~~ doppy*(gral #aa.b-#aa.c #afa %~) uml1f*(gral %~~) h8qc*(gral %~<: ,/% *(gral %~: ob7% *(gral %>~ *(gral ;7 )t %1 ,/e grcze =,/7 *de'x= %| ob7e grcze =ob77 *de'x= %"l *(gr:}r# %* abl3tus}r# %; abl3tuspt %"d r/d% 'd =f p)tieqe abl3tu= %&d g~% dyta 3 diff7czz34c %#l lim% %#l<: lim% *f7ior %#l: lim% sup7ior ,/e u ob7e grczc ,wc &m`'~ r ,?la& 3 ,/e u ob7e *diz% 0d&&bc. r ,~,3e fgcd gb m zn 0 ,/e grcze +. ? { d1 fgcdc |d$ }=t 8b7,3e 6 leerz34c. ,b,0 #aa.c >b#ja ~1a|b ... \[\int_{a}^{b} ...\] #aa.c #afb ,b,0 #aa.c >b#jb ~$1x1k"-#a$5$|x1k"+#a$5 \[\int_{x_{k -1}}^{x_{k +1}}\] ,b,0 #aa.c >b#jc >a =~1)|: x|;dx' ="l#a:x|:"l1)|: \[A =\int_{0}^{3} x^{2}dx =\left| \frac{1}{3} x^{3} \right|_{0}^{3}\] ,b,0 #aa.c >b#jd e$|b#je #l1n":,#%', #l1h":,#j \[\lim_{n \to \infty}, \; \lim_{h \to 0}\] ,b,0 #aa.c >b#jf e$|#l!1x :,x1)!5 f2x`$5 \[e^{\lim_{x \to x_{0}} f(x)}\] #aa.c #afc #aa.c #afd #ab mcgcl=re ============ %!{ &5w3f( 9ffncde klax7 %!} &5w3f( 5~cde klax7 %$$n mc& r nc zlc %$$z mc& r gzc zlc %$$q mc& r r!n:c zlc %$$r mc& r reeqc zlc %$$c mc& r komple'xc zlc %$$h mc& r 'quat7nionc %$$p qjektive &rade %&o leere mc& %&a :eph %&, f ae %&? % gb %&e } y{ct v'* %*&e } n y{ct v'* %&* ht zm y{ct'* %2. } ~h:tc *, } t3lmc& v'* %2= } ~h:tc * o g4'* %`, ~h`lt, } ob7mc& v'* %`= ~h`lt o } g4'* %3. -6>t t'* %0. &5nittc t'* %1. -m*d7t um, oe'* #ab #afe %|. s'yxetri5e diff7cz'* %( 'c)t%i5% qdukt =m:kr2z='* %"l sckrt7 }r#, p ~ %* }r# 3 m)k0ru f kompl{ct`re mc- gc %|c ho4&,yt% 'c 3 m)k0ru f kom- pl{ct`re mcgc '* ? 0c s'ymbolc } 6 leerz34c z ,ec, n< ihc dg n =s0he (#e op7!ns- u ry!nsz34c)=. 0 s'ymbole f 0 ,})dmcgc =mc& r nc zlc, r gzc zlc usw.= sd fe}e &bilde 1s dr3 braiqez34c. s sd n 3 bu4}abc t +k8nd>u z -}=c. `e s'ymbole, 0 hr n ,1&li}et sd, ,dc +:og &bildet ,wc. ,b,0 #ab >b#ja >a 3.>b \[A \cup B\] ,b,0 #ab >b#jb >l 1.2>m 0.>n`' =2>l 1.>m` 3.2>l 1.>n` \[L \setminus (M \cap N) #ab #aff =(L \setminus M) \cup (L \setminus N)\] ,b,0 #ab >b#jc >a |.>b =>b |.>a \[A \triangle B =B \triangle A\] ,b,0 #ab >b#jd >a 0.>u =!{#c', #e', #g!} \[A \cap U =\{3,5,7\}\] ,b,0 #ab >b#je !{rot', gr8n!} 3.!{gelb',' blau!} =!{rot', gr8n', gelb',' blau!} \[\{\text{rot, grn}\} \cup \{\text{gelb, blau}\} =\{\text{rot, grn, gelb, blau}\}\] ,b,0 #ab >b#jf !{gerade $zahlen!} 0.!{ungerade' $zahlen!} =&o \[\{\text{gerade Zahlen}\} \cap #ab #afg \{\text{ungerade Zahlen}\} =\varnothing\] ,b,0 #ab >b#jg &,x &e>b \[\forall x \in B\] ,b,0 #ab >b#jh 2>a |.>b`|c =>a|c 3.>b \[(A \triangle B)^{c} =A^{c} \cup B\] ,b,0 #ab >b#ji $$n 2=$$z 2=$$q \[\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q}\] #ab #afh #ac logik ========= %0, u %3, o %:* n %==o pf3l n< rts t doppyt{ 5aft u 6f<7 spitze =implik!nspf3l= %9==o doppypf3l t doppyt{ 5aft =`'quiv:czpf3l= 0 s'ymbole f (u), (o) u (n) ,wc n< 6{ leerz34c, +&5lo~c + d fgcde z34c &,5c =s0he (#e op7!ns- u ry!nsz3- 4c)=. 0 implik!ns- u `'quiv:czpf3le }=c zw leerz34c. ,4&}r#ce pf3le ,wc 3 modul)e pf3le d)&,yt =s0he (#g.a mo- dul)e pf3le)=. ,b,0 #ac >b#ja #c teilt x 0,x 9=#aj o * ,q- bzw. kz5t: #c t3lt x 0,x 9=#aj bzw.: #c '.t3lt'. x 0,x 9=#aj #ac #afi \[3 \; \text{teilt} \; x \land x \leq 10\] ,b,0 #ac >b#jb >a ::o >b 9==o :*>a 3,>b \[A \rightarrow B \Leftrightarrow \lnot A \lor B \] #ac #agj #ad &ometr0, trigonometr0 u vekt?c =========================== #ad.a &ometri5e s'ymbole :::::::::::::::::::::::: %7/ dr3e$ %7) kr3s %7= 'quadrat %7% rte$ =d zw3( ,qz34c } t3l . s'ymbols.= %7? rhombus %7+ p)aqyograx %7* ,4me~7 %79 w*ky %7( rt7 w*ky %7:, - uhrz3g7s*n %7!: g e uhrz3g7s*n %$: zsfa~cde m)k0ru f }re$e =waag- rt7 }r# 8 ,=7c bu4}abc= %$2 zsfa~cde m)k0ru f bogc =bogc 8 ,=7c s'ymbolc= %$:, zsfa~cde m)k0ru f vektor =pf3l 8 ,=7c s'ymbolc= #ad-#ad.a #aga %:, m)k0ru f vektor =pf3l 8 6{ s'ymbol= %#. sckrt 1 %"% p)aqy z =d zw3( ,qz34c } t3l . s'ymbols.= 0 t 5l8~yz34c %7 &bildetc s'ym- bole ,kc t o oe ,?+g=cd{ u!,o fgcd{ leerz34c &,5c ,wc. s ,mc j4 klar v /mtbar rts nb ihc }=cdc bu4}abc, zm ,b,0 4 6 +k8nd>usz34c f g~- bzw. kl65bu, &trcnt ,wc. zsfa~cde m)k0ruc 8 ,=7c s'ymbolc }=c /mtbar ? { 7}c 07 s'ymbole u sd qjektive. d cde r m)k0ru &}:(t c n< e aq&m6c regyn f qjektive =s0he (#aj qjektivtk)=. 0 s'ymbole f (sckrt 1) u (p)aqy z) sd ry!nsz34c =s0he (#e op7!ns- u ry!nsz34c)=. ,b,0 #ad.a >b#ja 7=>abcd \[\square ABCD\] #ad.a #agb ,b,0 #ad.a >b#jb 79>pqr =#cj") \[\angle PQR\ =30^{\circ}\] #ad.b w*ky-, h'yp7byf/ktjc u umk=ruc :::::::::::::::::::::::::::: %6a )kus %6s s*us %6c kos*us %6t t+gcs %68 kot+gcs %6- sek+s %62 kosek+s %6,s )kuss*us %6,c )kuskos*us %6,t )kust+gcs %6,8 )kuskot+gcs %6,- )kussek+s %6,2 )kuskosek+s %6(s s*us h'yp7boli'cus %6(c kos*us h'yp7boli'cus %6(t t+gcs h'yp7boli'cus %6(8 kot+gcs h'yp7boli'cus #ad.a-#ad.b #agc %6,(s )eas*us h'yp7boli'cus %6,(c )eakos*us h'yp7boli'cus %6,(t )eat+gcs h'yp7boli'cus %6,(8 )eakot+gcs h'yp7boli'cus 0e s'ymbole gytc f ae v)i+tc r ;- tfcdc kzw?ts'ymbole * 5w)z5t, zm ,b- ,0 f t+gcs * r 5b,3e (tan) o ('tg). ebcf gilt d s'ymbol ,pwl f g~- 3 | f kl6&,5ce kzw9rt7. ;g*nt d )gumct t 6{ +k8nd>usz34c =2 f zlc o g~- bzw. kl6bu4}abc=, d)f % + d s'ymbol +5~c. * ,7c !fqc m~ 6 leerz34c zw s'ymbol u )gumct &,et ,wc. % } ,z!l>, 0e s'ymbole t { aq&m6c 6l3tusz34c f kzw?ts'ymbole %7 u e * r 5w)z5t 8bc bu4}abc 0d,z&bc. p ~}=cde |!d$e sd !lg7, ,kc j4 d?t zwe$m`'~> 9, wo 6e cg7e ,86}ixu t r 5w)z5t 7f?d7 } o 6e -tr1th t -6zyt ,?kxcdc s'ymbolc n ,?1s&,et ,wc k. 0 s'ymbole sd }ruktur0rt ,1&b1t. kot+gcs, sek+s u kosek+s sd math{a- ti5e k=rw7( v t+gcs, kos*us u s*us. d0s wd 4 0 sp0gyu . zw3tc z34cs um 0 h?izont:e a4se d)&,yt. 0 h'yp7byf/k- #ad.b #agd tjc ,wc 4 e 65ub . z34cs %( =6 t0f&,yt% 'h f (h'yp7boli'cus)= &bil- det. b e trigonometri5c f/ktjc ;d2(t 6 6&5obc%, t0f&,yt% 'a %, ()- kus...), b e h'yp7byf/ktjc ()ea...). ,b,0 #ad.b >b#ja 6s#cj") =#j,e \[\sin 30^{\circ} =0,5\] ,b,0 #ad.b >b#jb 6t'x =;6s'x 8 6c'x< o 6t'x =;6s'x86c'x< \[\tan x =\frac{\sin x}{\cos x}\] ,b,0 #ad.b >b#jc 6s ;b#jd 6s 2x +y` =6s'x.6c'y +6c'x.6s'y \[\sin (x +y) #ad.b #age =\sin x \cdot \cos y +\cos x \cdot \sin y\] ,b,0 #ad.b >b#je 6sb#jf 6,t#a =b#jg 6c#bus- #ad.c #agg z34c f ;e t'ypografi5e |z34nuc %! o % kcnt &mb#ja a:,.b:, \[\vec{a} \cdot \vec{b}\] ,b,0 #ad.c >b#jb x:, =2xyz` \[\vec{x} =(x \; y \; z)\] ,b,0 #ad.c >b#jc y:, =2#a<8#b<8#c` \[\vec{y} =\left( \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right)\] ,b,0 #ad.c >b#jd >m:,1d \[\vec{M} _{d}\] #ad.c #agh ,b,0 #ad.c >b#je $:>ab bzw. $:,>ab \[\overline{AB} \; \text{bzw.} \; \vec{AB}\] ,b,0 #ad.c >b#jf 6d >v:, =&n:,.>v:, ="d>v1,8"dx' +"d>v1;8"dy +"d>v1:8"dz \[\text{div} \vec{V} =\vec{\nabla} \cdot \vec{V} =\frac{\partial V_{1}}{\partial x} +\frac{\partial V_{2}}{\partial y} +\frac{\partial V_{3}}{\partial z}\] ,b,0 #ad.c >b#jg '<=im folgenden werden fett ge- druckte bu4}aben mit den punkten #d,e,f % und goti5e mit punkt #e %! gekennz34net.'<= >k|) =!b.6r!t * kz5t: '<=- fgcdc ,wc fett &d$( bu4}abc t e ptc #d,e,f % u goti5e t pt #e %! &kcnz34net.'<= #ad.c #agi >k|) =!b.6r!t \[\mathbf{K}^{0} =\mathfrak{b} \cdot \text{rot} \; \mathfrak{t}\] #ad.c #ahj #ae pzh:t7 u h?izont:e zsfa~uc ============================== #ae.a pzh:t7 :::::::::::: pzh:t7 ,wc ? a{ * l=rw7kc -wcdet. &w9hn }=c s + r ,ye v ziff7n, zlc o ,7c s'ymbolc, 0 % h71s,zf*dc gilt. ,1gd i7 vlf`lt>c -wcdus- u 756usf?mc ,lc s c n ab5~cd def*0rc. % wd d%wgc hr nr ,b,0hf ,1&z3gt, 2 pzh:t7 d)- &,yt ,wc ,kc. da f pzh:t7 wed7 s'ymbole n4 ,?- g=cs,3c ,})dis0rt sd, ,mc ,8tgcde ie &w`hltc d),yuc jew3ls * braiqe5tt5c +m7kuc 7l|t7n. % {pf0hlt c, b r wahl r s'ymbole d1 z a4tc, ~ s braiqe5tt5 2 0 y{c( &hd,ht ,wc ,kc, 0 s d),yc. p ,sc ,b- ,0s,3e pzh:t7 f op7!nss'ymbole wom | 2 op7!nss'ymbole 1 6 leerz34c fgc u +s n}e z34c +5~c. #ae-#ae.a #aha ,b,0 #ae.a >b#ja #g +... =#ae o '<=das vollz34en }eht f8r d0 3nzusetzende zahl.'<= #g +% =#ae \[7 +... =15\] ,b,0 #ae.a >b#jb '<=legende: 7= w3~es quadrat 7= 5warzes quadrat 7/ w3~es dr3eck 7/ 5warzes dr3eck'<= 2a 7=b` 7/2a 7/b` 7=c =#b'b \[(a \square b) \blacktriangle (a \triangle b) \blacksquare c =2b\] #ae.b h?izont:e zsfa~uc u l0gcde klax7n :::::::::::::::::::::::::::::::: %&: ..... +k8nd>usz34c f h?izont:e zsfa~uc %&'=...'= abk8nd>usz34c f h?izont:e #ae.a-#ae.b #ahb zsfa~uc t 7l|t7u 3 te'xt %&2...` abk8nd>usz34c f h?izont:e zsfa~uc t 7l|t7u 3 ma- th{ati57 |d$ * r 5w)z5t 7fgt 0 m)k0ru ,=77 t7me zr 7l|t7u 6% math{ati5c |d$s h|f> 4 l0gcde klax7n o 4 t'ypografi5e h7,?- hebuc =f)be, fettd$ o `%=. d7)t>e zsfa~uc ,kc * r math{atik5t 2 fgt 0d&&bc ,wc: '- /mtbar ? { zs&fa~tc |d$ wd d z3- 4c %&: &,et. '- /mtbar n< { zs&fa~tc |d$ }=t d z34c %& zs t r 7l|t7u =o 6e ;- 5bu= * klax7n. '- je nd, ob 0 7l|t7u 3 te'xt- o 3 math{ati57 |d$ &,5c wd, }=t s * te'xt- o math{ati5c klax7n. '- l0gcde klax7n ,wc n direkt 0d&- &bc. * vlc !fqc } 0 0dgabe t 07 tk zwar m, ,7e'- n ,})dis0r('- tkc ,kc j4 zwe$m`'~>7 9, ea w3l d 650bc r 7l|- t7u 0 ,8s#tk reduz0rt'- u &rade d%- #ae.b #ahc wgc d z0l r zsfa~u * r 5w)z5t -f=lt. ,b,0e f :t7n!vtkc sd: '- 0 sep)a( ,1li}u r m)k0rtc t7me * 67 )t legcde u '- 0 z7gl0d7u 6% kxct0rtc re4c5ritts * ,=7e t3l5rit(. ,b,0 #ae.b >b#ja a.b =&:a +a +... +a&'='b-mal'= \[a \cdot b =\underbrace{a +a +... +a}_{b-\text{mal}}\] ,b,0 #ae.b >b#jb '<=im folgenden 1sdruck sind #j,be und 3#c exakte werte, #j,dcc der n`herungswert.'<=' >a =#j,be.3#c5.s|; ??#j,dccs|; o * kz5t: '<=- fgcdc |d$ sd #j,be u !,3#c'. xak( w7(, #j,dcc r n`h7us- w7t.'<=' >a =#j,be.3#c5.s|; ??#j,dccs|; \[A =\underbrace{0.25}_{exakte} \cdot \underbrace{\sqrt{3}}_{Werte} \cdot s^{2} \approx #ae.b #ahd \underbrace{0.433} _{N„herungswert} s^{2}\] ,b,0 #ae.b >b#jc &s1k"=#a|n &:6c 2b#jd ~1)|#ba#a 5te re4cv7,rc 8 ,=7e z3lc =============================== ac 5tc re4cv7,rc } &m6~, ~ 0 zif- f7n r ;tfcdc zlc * 6 r|m% vh ,z,6 &,yt ,wc. d0s 7l47t d |w`hlc r je- w3ls z m+ipul0rcdc zlct3le. wed7 0 d),yuspra'xc n4 0 ihc ,zgde l0gcdc :g?ithmc sd j4 /iv7seq. | *n7h:b . d5c spr1ms sd s n 6h. * r braiqe5t -fgc 5te re4cv7,rc d0sybc z0le 2 * r 5w)z5t. ,zm3} sp0- gyn ie d),yus,3e u :g?ithmc e 5w)z- 5t&,| wid7. zr iqu}r!n ,k( ,pgar 6 re4cv7,rc r 5w)z5t * braiqe5t ab- &bildet u 7l|t7t ,wc. 0 medi:c ,/50- de r braiqe- u 5w)z5t fhc * r t`gc pra'xis z &r*gc ,/50dc * e jew3l>c -,rc. zm ,b,0: '- t { kugy5b7 k m l4 v 67 z3le zr ,7c spr*gc. um t r braiqe5tm5e 7- n2t 1 6e fr8h7e z3le z we4syn, m~ >a#a #ahg d pap0r z$&dr=t ,wc. ,1gd me4+i57 /&n1>kc r m5e -rut5c n2e z34c ;- zogc 1 5 &,5ce. ,z,{ ,kc 0 z34c r n -,rc &- w`hlt, 0 k6 z$dr=c . pap0rs 7f?- d7n. '- * r 5w)z5t ,kc zlc bzw. ziff7n ,4&}r#c u n2e /mtbar d8 o d),/ &- ,5c ,wc. | d not0rc v ,8tgszlc 1 cg{ r1m } /probl{ati5. s4e -,rc'- 0 | bm !kzc v br84c +wcdu f*dc'- sd * r braiqe5t nr 6&5r`nkt m. '- &w~e braiqe5ts'ymbole sd 7} 7kcn- bar, wn ie positj *n7h:b r braiqef?m 6d2t> }. ;s * ;zug 1 5te re4cv7,rc sd pt #a =ziff7 #a'=, pt #b =dezim:kxa= u pt #c =gl0d7usz34c= 3 potczieqe (}ol- p7}6e) z 7w`hnc. 4 v7tik:e ;wguc r f*g7, 0 * 5tc re4cv7,rc /v7- m3d sd, wd 0 ptr3he *n7h:b r f?m n4ms 5>7 z 7kcnc. | d%wgc wd b 5tc re4cv7,rc * r braiqe5t h|f>7 1 gl0d7usz34c -z#(t 3 * r 5w)z5t. '- 0 d),yu v zlc t zlz34c w34t pr*- >a#a #ahh zipieq v r 5w)z5t ab u ;d*gt 3gce ,8leguc. zlz34c /mtbar ? zlc fhc dz, ~ s oft * ziff7nsp:tc =zm ,b- ,0 * r h/d7t7-sp:(= }=c u d kl)e re4cbild }9rc. d%wgc ,wc s oft t ab,} z e n 9, 6e zt lg * re4nus,1,yuc 6 ,qz34c +,ye . kxas z 5bc. % ,wc ,=7e v)i+tc f 5te re4cv7,rc * r braiqe5t &z3gt, oe ~ ihc ,?5bcd7 4k ,zkxt. 3 ,b,0 6% v r 5w)z5t m)k+t abw34c- dc -,rcs wd ,z,{ 0 l*e)e additj 7l|- t7t. >a#a #ahi >a#a.a additj ::::::::::::: b r additj v gzc zlc sd nr kl6e ,/50de * e -50dcc d),yusf?mc * r braiqe- u 5w)z5t z -z34nc. s ;tfc ? a{ 0 pz0ru r zlz34c, 0 0dholu . plusz34cs u 0 ,/}r34u . 7&bxs%. b !g~7c zlc ,wc gl0d7usz34c =t1sc- d7trcnz34c= t ,?t3l wg&,lc, w3l s l4 t 67 ziff7 #a -we4syt ,wc ,kc. da % * braiqe5t t5 5>7 }, ,wc ,8- tgszlc n p oft not0rt 2 * r 5w)z5t. 6c ,g,} 1 { ti5 z pz0rc o z -50bc k zm ,b,0 d)an 7*n7n, ~ 6 ,8tg n4 z add0rc }. fqs 0 ,8tgszlc dcn4 not0rt ,wc, m~ df 6e z3le fr3 &,lc ,wc. dm ,8!tge v e ,7c zlc z ,/53dc sd, k b 07 z3le d zlz34c wg&,lc ,wc. :t7n!v ,kc * r z3le . suxc}r#% 0 ,8!tge &- ,5c ,wc. ,3c 0 zlc g~e ,/50de * i7 !lge 1, k % !nz 9, 0 leerc ,yc t nuqc ,1,z- f8qc. 0e d0nc { f*g7 zr ?ict0ru zw e ziff7n d%sybc ,ycw7ts d8 u d),/. >a#a.a #aij ,b,0 >a#a.a >b#ja #ade #ade +# be +# be :::: --u- #agj #agj #ade #ade +# be +# be :::: a #agj ::::: ==== =#agj ,b,0 >a#a.a >b#jb # id,geahi # id,geahi # h,i +# h,ijjjj +# a,fce +# a,fcejj :::::::::::: aab # aje,bhfhi ::::::::::: #aje,bhfhi >a#a.b subtraktj :::::::::::::::: b r 5tc subtraktj sd zw3 gd!sz -50dce :g?ithmc z 7kcnc. s ,/53dc c 7} * !fqc, * ec 6e ziff7 * r ,/c zl >a#a.a->a#a.b #aia =subtrahcd= !g~7 } 3 0 ~,~cde ziff7 * r ob7c zl =m*ucd=, 2 b (#de m*us #bg). - d5c spr1m } d 7!gzusv7,rc am wte}c -br3(t. * 0{ -,rc wd 0 ob7e ziff7 4 z=n 7!gzt. - ,b,0 v (#de m*us #bg) wd +,ye v #e t #ae &re4net u 0 ,/e ziff7 * r l*ks dv }=cdc sp:( um #a 7!gzt: 0 #b wd z 67 #c. d ,7e, *t7n!nal ,8w0gcd -wcde( -- ,rc } d ~b8ndyusv7,rc. hr wd n 0 ,/e zl 7!gzt, sn 0 #a =3gct #aj'= f 0 #e r l*ks dv }=cdc ziff7 ~nx. * /s7{ ,b,0 wd 0 #d z 67 #c. r6 braiqe5tt5 ht d 7!gzusv7,rc e ,?t3l, ~ x nr #a dz &!zlt wd, w c l47 m7kc u | not0rc l, 3 0 !7uc bm ~b8ndyusv7,rc. * fgcdc ,b,0c ;z0hc c (,8tgsz3lc) nr 1 d 7!gzusv7,rc. ,b,0 >a#a.b >b#ja #de #de -#bg -#bg ::: a #ah ::: === #ah >a#a.b #aib ,b,0 >a#a.b >b#jb #fcdej #fc.dej -# bbaj -# b.baj :::::: ::::::: #fabdj #fa.bdj ======= ,b,0 >a#a.b >b#jc #ajj #ajj,j #ajj%j -# b,e -# b,e -# b%e ::::::: aaa :::::: #ig,e :::::: #ig,e ===== =# ig,e ===== >a#a.c multiplik!n :::::::::::::::::: b r 5tc multiplik!n sd ? a{ ,/50- de * r positj0ru r t3l7&bxse ,p,2 * r r3hcfge r t3l5rit('- vm !g~tc zm kl6}c ,ycw7t o um&k=rt'- z ;oba#a.b->a#a.c #aic ,b,0 >a#a.c >b#ja #bgc.#ahd #bgc.#ahd ::::::::: ::::::::: # bgc # bgc # bahd +# bahd # ajib +# ajib ::::::::: :::::::: #ejbcb # ejbcb #bgc .#ahd #bgc .#ahd :::::::::: :::::::::: # ajib # ajib # bahd # bahdj # bgc # bgcjj :::::::::: :::::::::: #ejbcb #ejbcb ====== ====== >a#a.d divisj ::::::::::::: b r 5tc divisj v)i0rc - d5c spr1m ? a{ 0 +zl r zw5rit(, 0 not0rt ,wc, ,p,2 r umg+g t { re};tg. * r braiqe5t 75w7t r z3lcwe4sy n< obc d not0rc r t3l- u cd7&bxse. oft >a#a.c->a#a.d #aid ,wc t3l7&bxse zn * 0 z3lc r zw5rit( &,5c u 7} am 5lu~ z 67 zl =cd7&bx= zs&f8gt. d 7&bx wd ~wed7 2 * r 5w)z- 5t obc o a /tc not0rt. oe 6e &w~e 8bu u gd7,ru * r prak- ti5c )b t r braiqe5tm5e k % / um!}c 5> 9, ab,z5`tzc, ob n4 |r34cd 5bpz 1 r blattfl`4e f 0 ,q!}>e divisj t ac zw5rittc zr -f8gu }=t. ,b,0 >a#a.d >b#ja #eeb :#bd =#bc #eeb :#bd =#bc -#dh #b # gb #b ::: # j #c # gb # gb #c :::: # j ,b,0 >a#a.d >b#jb #cea :#ae =#bc >r#f #cj #b ::: # ea # de #c :::: # f re} >a#a.d #aie #cea :#ae #b #cj --- # ea #c # de :::: % # fj #d # fj ::::: # j =#bc,d ====== >a#a.e l*e)e additj ::::::::::::::::::: nb e })k + 0 5w)z5t +&l=ntc re4c- v7,rc uc | + 0 braiqe5tm5e +&pa~( tkc ~wi$yt. % ,s hr 0 l*e)e additj 3 ,b,0 d0nc. d -,rc l c * l47 abw+dlu ebcf f d re4nc am 'comput7 6,ec. ,b,0 >a#a.e >b#ja % ,sc 0 bdc dezim:zlc #i,gc u #c,df add0rt ,wc. zn wd 0 ,1gabe ,1- &,yt: >a#a.d->a#a.e #aif #i,gc +#c,df =# 5ritt #a': 0 +zl r ,yc r !lg}c zl =*klusive dezim:trcnz34c= wd 7mtt u 6s dz &!zlt. d 7gb 0 ma'xim:e !lge . 7&bxs%. r 5bkopf r braiqe5tm5e =hr 4 ae se4s pte d)&,yt= wd p wt n< rts ;wgt, ~ d 7&bx pz ht. - ,b,0 wd dh v0rm 0 leerta}e &!d$t. =bm re4nc am 'comput7 m~ r 'cursor n n< rts ;wgt ,wc.= #i,gc +#c,df =# % 5ritt #b': 0 ziff7n r ltc ,ye ,wc add0rt u d 7&bx h*&,5c. d+< wd zw3m 0 r$ta}e ;t`t>t. =bm re4nc am 'com- put7 wd r 'cursor 6m n< l*ks ;wgt.= #i,gc +#c,df =# %i 5ritt #c': 0 ziff7n r zw3tltc ,ye ,wc add0rt u d 7&bx wd &,5c. - ,b,0 m~ r ,8tg &m7kt o am rtc r+d r z3le not0rt ,wc. r 5bkopf wd l*ks nb 0 &- ,5ce ziff7 . 7&bxs% &,yt. =bm re4nc am 'comput7 wd r 'cursor 6m n< l*ks >a#a.e #aig ;wgt.= #i,gc +#c,df =# %ai 5ritt #d': d kxa wd &,5c u zw3m 0 r$ta}e &!d$t. =bm re4nc am 'comput7 wd r 'cursor 6m n< l*ks ;wgt.= #i,gc +#c,df =# %,ai 5ritt #e': 0 ziff7n l*ks . kxas ,wc add0rt u r ,8tg dz&re4net. 0 zl wd ,1&,5c. #i,gc +#c,df =#ac,ai >a#a.f d l9sc v g4uc :::::::::::::::::::: bm l9sc v g4uc 7&bc c k1m ,7e tkc 3 * r 5w)z5t. dg ~fqc mkc r )b *n7- h:b r z3le'- 2 ,b,0s,3e d ,4}r34c v y{ctc u d 5bc v n2c'- wtg=cd, p ~ 0 l9su = z3lc 6nhc k 3 * r 5w)z5t. 6 sckrt7 }r# %"l =ggt | 3 (op7a- t?c}r#) ;k+nt= k &,5c ,wc, um 6e ,4- >a#a.e->a#a.f #aih ,zfhcde op7!n v r g4u z trcnc. d * r 5w)z5t 8be ,/,6,yc s47 }r#e } * r braiqe5t n d0n; am ;}c ,wc zw3 o dr3 leerz34c, ? u k6e n< { }r# &,lc. ,b,0 >a#a.f >b#ja #cx -#g =#da "l+#g #cx =#dh "l:#c x =#ah ,b,0 >a#a.f >b#jb 2x8#b +#d`|; =#ajj "l3 x8#b +#d =+-#aj "l-#d x8#b =#f oder -#ad "l.#b x =#ab oder -#bh >a#a.f #aii >a#a.f #bjj >a#b !7uc * r math{atik5t ========================= >a#b.a &!7( s'ymbole :::::::::::::::::::: %!{ ... &5w3f( 9ffncde klax7 %!} ... &5w3f( 5~cde klax7 %${ ... spitze 9ffncde klax7 %$} ... spitze 5~cde klax7 %'{ ... }umpfw*kl>e 9ffncde klax7 %'} ... }umpfw*kl>e 5~cde klax7 %#!{ &5w3f( spezieqe 9ffncde braiqe5tklax7 %#!} &5w3f( spezieqe 5~cde braiqe5tklax7 %"l ... sckrt7 }r# =+ ac ,yc= %"% ... sckrt7 doppy}r# =+ ac ,yc= =d zw3( ,qz34c } t3l . s'ymbols.= %4! ... k3l t spitze rts 3 m)k0ru %4, ... k3l t spitze l*ks 3 m)k0ru %") grad =kr*gy= %"* m*u( =}r#= %"** sek/de =doppy}r#= >a#b->a#b.a #bja %rad radi+t =rad= %rad|; 'quadratradi+t %&e ... } y{ct v =mcgcl=re= %| .... ob77 *de'x o xponct =fr8h7 nr ob77 *de'x= %"d ... r/d% 'd =f p)tieqe abl3tu= %4!; pf3l n< obc %4;, pf3l n< /tc %!( ... mal =}7n= >a#b.b n2e s'ymbole ::::::::::::::::::: %!>{ g1'~5e 9ffncde klax7 =ob7e grcze= %!>} g1'~5e 5~cde klax7 =ob7e grcze= %!<{ g1'~5e 9ffncde klax7 =,/e grcze= %!<} g1'~5e 5~cde klax7 =,/e grcze= %$!{ z3lczsfa~usklax7: ,=7e z3lc zsfa~cde g~e l*ke &5w3f( klax7 %7:, - uhrz3g7s*n %7!: g e uhrz3g7s*n >a#b.a->a#b.b #bjb %7? ... rhombus %7+ ... p)aqyograx %> .... kl67 'qu7}r# 6% ,z?dnus- pf3ls %::o pf3l n< rts t 6f<{ 5aft u 6f<7 spitze %9:: pf3l n< l*ks t 6f<{ 5aft u 6f<7 spitze %9::o pf3l n< l*ks u rts t 6f<{ 5aft u 6fa#b.b #bjc >a#b.c zlc :::::::::: 0 ;r3ts praktiz0r( -wcdu v r/dc klax7n f 0 d),yu v p7iodi5c dezim:- br84c ue ,1&nx =s0he (#b.a.d p7iodi- 5e dezim:br84e)=. 1 0 abw34cde pra'xis b e dezim:- u gl0d7usz34c * gyd;!tgc * r 5w3z u l04tc}6 wd 6&g+gc =s0he (#b.a.c de- zim:br84e) u (#b.a.e gl0d7u lg7 zlc)=. r9mi5e zlc ,wc +:og z ,7c bu4}abc- fgc ;hdyt u n t { zlz34c 6&l3(t. >a#b.d xponctc u *diz% :::::::::::::::::::::: xponctc ,wc n2 t ,{sybc z34c %| 2 ob7e *diz% 6&l3(t. n2 d)f 6 m*usz34c oe pt #d =zsh:(- pt= 1 6 *de'xz34c fgc. neg!ve gze zlc ,dc eo n< 6{ *de'x * &sckt7 5b- ,3e &,5c ,wc =s0he (#b.a.b zlc * &- sckt7 5b,3e) u (#aj.c.c *diz% 1s gzc >a#b.c->a#b.d #bjd zlc)=. >a#b.e br84e :::::::::::: n2 ,mc !zl7 u ncn7 ;zogc 1 e ab,} zm bru4}r# g4 &}:(t 9. bru4+f+gs- u '-cdez34c ,mc x paar,3e -wcdet ,wc. 6 1s 67 gzc zl ;}=cd7 ncn7 d)f n< 6{ bru4}r# n * &sckt7 5b,3e }=c. f e ab5lu~ s`mt7 br84e * 6{ ,=fa#b.f bu4}abc :::::::::::::: 0 kcnz34nu v g~- u kl65bu } n2 &- regyt. s } n/ t r te'xt5t kompatib- l7. 0 kcnz34nu f goti5e bu4}abc ue &- }r#c. goti5e bu4}abc ,wc n2 2 ,7e bu4}abc t 67 ;c t'ypografi5c |z34nu ;hdyt. =s0he (#c.d ;e t'ypografi5e |z34nuc).= >a#b.d->a#b.f #bje gr04i5e bu4}abc ,wc n2 6h t { fr8h7c +k8nd>usz34c f gr04i5e kl6- bu4}abc %< +&k8nd>t. g~5bu wd t e aq&m6 8bc +k8nd>usz34c &kcnz34net. d fr8h7e +k8nd>usz34c f gr04i5e g~- bu4}abc ue &}r#c. f eta, theta u 4i gytc 0 bsh7>c :t7n!vz34c. =s0he (#c.c gr04i5e bu4}abc).= 0 fr8h7c d$kcnz34c uc 4 zw3 +k8n- d>usz34c f n n`h7 ;}ix( ;e t'ypo- grafi5e |z34nuc 7,et. 0 ;d2tu } je- w3ls * 67 +m7ku fe},zh:tc. =s0he (#c.d ;e t'ypografi5e |z34nuc).= >a#b.g klax7n u sckrte }r#e ::::::::::::::::::::::::::: um t n27uc * r te'xt5t 6e m} wt- g=cde kompatibil!` z &!whl3}c, uc 0 s'ymbole f &5w3f(, spitze u }umpf- w*kl>e klax7n +&pa~t. n2 }=c ,z!sze zm gdz34c x ? 0{, | b 5~cdc klax7n. =s0he (#f klax7n u sckrte }r#e).= f 6e g~e l*ke &5w3f( klax7, 0 ,=7e z3lc (zsh`lt), ue 6 n2% s'ymbol de- f*0rt =s0he (#f.d ,=z3l>e klax71s- >a#b.f->a#b.g #bjf !d$e)=. l0gcde klax7n ,wc n/,= 3 h?izont:e zsfa~uc ;hdyt u n * i7 )t ,/50dc =s0he (#ae.b h?izont:e zsfa~uc u l0- gcde klax7n)=. d s'ymbol f 6c sckrtc }r# wd n2 gc7eq t %"l d)&,yt, 0 bdc ;tgs}r#e 6% paar% sd ao g4. doppy}r#e =n zw3 6fa#b.h 6hc :::::::::: d fr8h7e 5l8~yz34c f 6hc ue &}r#c. ae 6hc ,wc t { z34c % 6&l3(t. b 6hckomple'xc wd d z34c nr 6m &,et. =s0he (#d.a kcnz34nu v 6hcs'ymbo- lc).= n2 uc | !whus6hc ,1&nx. n2grad, n2w*kym*u( u n2w*kysek/de ,wc n = ,1&fht. >a#b.g->a#b.h #bjg >a#b.i pf3le :::::::::::: 0 (|fhe pf3ls'ymbolik) ue * (mo- dul)e pf3le) um;n+nt, konsi}ct7 &me pf3le ,8nx =s0he (#g.b def*0r( pf3le)=. >a#b.aj qjektivtk ::::::::::::::::: 0 +zl r -}`rkusebcc f qjektive ue v zw3 1 6e reduz0rt. f 0 -}`rku }=c j4 zw3 -50dce +k8nd>uc zr -f8gu, 0 | -5a#b.i->a#b.aj #bjh >a#b.aa we4sy zw te'xt- u math{atik5t ::::::::::::::::::::::::: n2 sd 0 -50dcc methodc zr kcnz34nu . we4sys zw e 5tc xplizit |&fht. 0 doppyleerz34ctk ,p,2 0 kcnz34nu 4 la'yout sd ®yt. =s0he (#a.a we4sy zw te'xt- u math{atik5t).= 0 r$k=r zr te'xt5t f 6 6zyn% o w>e w9rt7 * 67 math{atikpa~a& ue n2 de- f*0rt u m~ n/ n nr +&k8nd>t, sn | ab&k8nd>t ,wc =s0he (#a.a.c +- u ab- k8nd>usz34c f te'xt5t)=. >a#b.ab son}>% :::::::::::::: % wd d)an fe}&h:tc, d fr8h7e (4)- z34c %/ n 0d ,1,znhc. r m:}7n wd n2 3 %!( def*0rt. 0 abtrcnu . )gumcts vm f/ktjss'ym- bol b s*us, log)ithmus u `{ ue &re- gyt =s0he (#c.f kzw?ts'ymbole), (#aa.b log)ithmus- u xponcti:f/ktjc) >a#b.aa->a#b.ab #bji ,p,2 (#ad.b w*ky-, h'yp7byf/ktjc u umk=ruc)=. n2 ,dc ;l0b>e kzw?ts'ymbole xpli- zit t { kzw?t5l8~yz34c %7 &bildet ,wc =s0he (#c.f kzw?ts'ymbole)=. 0 div7sc f/ktjc . pt% #d ,wc 7l|- t7t =s0he (#a.b trcnc u zsh:tc ma- th{ati57 |!d$e) u (#c.e bu4}abc`e s'ymbole)=. 0 * r te'xt5t #bjaa 6&fhtc klax7n f +m7kuc r ,8tgcdc pn f*dc n/ ebcf * r math{atik5t +wcdu =s0he (#a.c +m7- kuc zr braiqe5t,8tgu)=. braiqe5tt5e ;griffkc uc +&pa~t. p } zm ,b,0 v braiqe5t u n pt5t 0 re- de. oe 0 regyn s} z !7n, ue 1 0 ;- griffe (zeqchf) u (,ybar) -z#(t. >a#b.ab #baj >a#c glo~ar =========== - glo~ar ;f*dc c 7kl`ruc z * 0{ bu4 -wcdetc braiqe5tt5c ;griffc. ;griffe, 0 + ,77 ,ye . glo~)s n`h7 7kl`rt ,wc, sd t '* &kcnz34net. abk8nd>usz34c: z34c'* r braiqe5t'*, d n s} 6 s'ymbol'* r 5w)z5t'* 0d- gb, sn +z3gt, ~ r 65ub * te'xt-'* o math{atik5t'* ab&5lo~c u ,p,t r we4sy * d jew3ls ,7e braiqe- 5ts'y}{'* 7fgt }. +- u abk8nd>ustk: tk r braiqe5t'*, 0 n s} 6e ~,~u * r 5w)z5t'* ht u zr ,?,8g=cdc !7u r ;d2tu v z34c'* 4 e we4sy * 6 ,7% braiqe5ts'y}{'* d0nt. +k8nd>usz34c: z34c'* r braiqe5t'*, d n s} 6 s'ymbol'* r 5w)z5t 0dgb, sn +z3gt, 2 d bzw. 0 na#c glo~ar >a->b #baa braiqef?m: r pz, r 4 6 braiqez34c'* 6&nx wd. s ;}=t 1s 6{ sckrt }=cdc fyd, * { 0 pte * zw3 sp:tc +&?dnet sd. s0he | braiqez34c. braiqept: 67 r se4s bzw. a4t pte 6% braiqez34cs'*. braiqe5t: hapti5 wh,znhcd% 5t- s'y}{'*, b { 0 braiqez34c'* 1s se4s o a4t ptc &bildet ,wc. ,/50dc wd zw r #f-pte- u r #h-pte-braiqe- 5t. braiqe5tklax7n, spezieqe: s0he spe- zieqe braiqe5tklax7n. braiqe5t,8tgusklax7n: z34c'* r braiqe5t'*, 0 n s} s'ymbole'* r 5w)z5t'* 0d&bc. s 7mc 7l|t7nde ,z!sze zr braiqe5t,8tgu =s0he (#a.c +m7kuc zr braiqe5t,8tgu)=. braiqe5ts'y}{: s'y}{, * y{ e z34c'* t bli$ 1 e -wcduszwe$ =math{atik, 4{0, fr{dspe usw.= ~,~cd { jew3l>c regyw7k konkre( ;d2tuc ,z&w0sc ,wc. braiqez34c: 6e r #fd =#f-pte-braiqe= bzw. #bef =#h-pte-braiqe= mc kom- b*!nc &,et7 braiqepte'* * 67 braiqef?m'* 65~ . leerz34cs'*. >a#c glo~ar >b #bab doppyleerz34ctk: 6e f?m r kcnz34nu . we4sys zw math{atik-'* u te'xt5t'* bzw. um&k=rt 4 zw3 ,1,6fgcde leer- z34c'* =s0he (#a.a.d doppyleerz3- 4ctk)=. gl0d7usz34c: z34c zr gl0d7u g~7 zlc /abh`ng> v e * r 5w)z5t,?la& &- w`hltc gl0d7usz34c =s0he (#b.a.e gl0d7u lg7 zlc)=. *de'x: 6e zl, v)iable o.`., 0 * r 5w)z5t'* ho4- o t0f&,yt, ? o h*t7 6{ math{ati5c |d$ }=t =s0he (#aj.c *diz% u xponctc)=. kz5t: s0he !kzusgrad. !kzusgrad: 6e r fgcdc dr3 |fhks}ufc f 0 0dgabe v te'xt * r d5c braiqe- 5t'*: '- basis5t: jd7 bu4}abe wd * 6e braiqef?m'* &,5c. '- ,q5t: 0 bu4}abcgruppc 'au, '`u, 'ch, 'ei, 'eu, 'ie, 'sch u 'st ,wc jew3ls t 6{ 3gcc braiqez3- 4c'* &,5c. '- kz5t: % ,wc !kzuc -wcdet, 0 bu4}abcgruppc u gze w9rt7 0d&- bc. >a#c glo~ar >d->k #bac $la$te>x: 6 szs'y}{, t { | math{ati- 5e te'x( 7,yt ,wc ,kc. math{ati5e |!d$e =z.b. br84e= ,wc * l*e)7 5b- ,3e 7,yt. * 0{ regyw7k 7f8qt 0 $la$te>x-5b,3e 0 f/ktj 67 zw3tc d),yusf?m r ,b,0e f braiqel%cde. s d0nc j4 n 3 5b+l3tu f $la$te>x. la'youttk: 6e f?m r kcnz34nu . we4- sys zw math{atik-'* u te'xt5t'* bzw. um&k=rt 4 e &z0ltc 6sz v &}:- tusmtn, z.b. 6- u |r$uc o ta;qc =s0he (#a.a.a la'yout)=. leerz34c: braiqez34c'*, * { k6e pte &,et sd. m)k0ru: 6 ,zsz z 6{ s'ymbol'*, r * r 5w)z5t'* ,8- o ,/h:b bzw. ho4- o t0f&,yt h*t7 0{ }=t. % wd zw 6fa#c glo~ar >l->m #bad regyn, 0 * { ,?l0gcdc w7k fe}&h:tc sd. pa~a&: 6e fge zsh`ngcd7 z34c'* * te'xt-'* o math{atik5t'*. qjektivtk: 6e tk r math{atik5t'* zr 6dimcsj:c 0dgabe v: '- s'ymbolc'*, 0 c 8 ,7e s'ymbo- le'* 7}re$c =zm ,b,0 d wurzyz3- 4c= u '- { grad r ho4- bzw. t0f,yu v *- diz%. % wd zw 6f ab,zbildc, z.b. ;- 5tu + pf3lc =s0he (#g.c ;5tu v pf3lc)=. s'ymbol: hr def*0rt 3 z34c'* r 5w)z- 5t'* o d~ ~,~u * r braiqe5t'*. ,{- ,zfge sd z34c'* r braiqe5t'* | s'ymbole, ,pf7n s 6e 5w)z5tct,~u ,hc, z.b. suxcz34c. dg sd z34c'*, >a#c glo~ar >m->s #bae 0 nr * r braiqe5t'* ,?kxc, z.b. +- k8nd>usz34c'*, k6e s'ymbole. * r braiqe5t'* ,kc s'ymbole 1s ,=7c braiqez34c'* ;}=c. te'xt5t: d gdlegcde s'y}{ r d5c braiqe5t'* =®yt * (d s'y}{ r d5c blc5t)=, t { hps`4 te'xt * -- 50dcc !kzusgradc'*, a | zlc u wt7e z34c'* r 5w)z5t 0d&&bc ,wc. s wd 4 ,z!sze braiqes'y}{e'* =z.b. 0e math{atik5t'*= 7!gzt. ,q5t: s0he !kzusgrad'*. z34c: hr y{ct r 5w)z-'* o braiqe- 5t'*, { 6e 3gce ;d2tu ,z&?dnet }. * r braiqe5t'* k 6 z34c 1s 6{ o ,=7c braiqez34c'* ;}=c. s0he | s'ymbol'*. z3lctrcnz34c: z34c'* r braiqe5t'*, d n s} 6 s'ymbol'* r 5w)z5t'* 0dgb, sn am cde r braiqe5tz3le &,et wd. % ,3t d1 h*, ~ r math{ati5e |d$ n4 n ab&5lo~c } u * r fgcdc z3le f?t- &,et wd =s0he (#a.b trcnc u zsh:tc math{ati57 |!d$e)=. zsh:(pt: z34c'* r braiqe5t'*, d n s} 6 s'ymbol'* r 5w)z5t'* 0dgb, sn 6 leerz34c'* 7,et, um 6c math{ati5c >a#c glo~ar >s->z #baf |d$ *h:t zs,zh:tc. 7 k | z34c zs- h:tc, 0 1s braiqe5tt5c !gdc n di- rekt ,1,6 tfc ,dc. =s0he (#a.b trcnc u zsh:tc math{ati57 |!d$e).= >a#c glo~ar >z #bag >a#c glo~ar >z #bah >a#d math{ati5e z34c, &?dnet n< r #f-pte-braiqe-ta;qe =============================== * 0{ regi}7 ,wc 0 math{atikz34c n< e braiqez34c &?dnet, 1s ec s ;}=c. s ,wc n< fgcd7, 1 louis braiqe z$g=cd7 ,1,yu &r3ht. z ; % %! %$ %< %' 0 r3hc ,ec c 2 fgt zs: '- #, r3he: 0 7}c z=n bu4}abc. d sd d0sybc z34c, 0 f zlc -wcdet ,wc. s ,hc ae m*de}cs 6c pt obc, m*- >a#d z34cli}e #bai de}cs 6c pt * r l*kc h`lf( r braiqef?m u k6c pt /tc. '- #; r3he: 0 z34c r #, r3he t 6{ ,z!szc pt #c. '- #: r3he: 0 z34c r #, r3he t ,z- !szc ptc #c u #f. r bu4}abe 'w wd |&,lc. '- #/ r3he: 0 z34c r #, r3he t 6{ ,z!szc pt #f. am cde 07 r3he ;- f*det c r bu4}abe w. '- #? r3he: 0 z34c r #, r3he um 6e ptr3he t0f7 &,5c. '- #+ r3he: ae 8>c z34c t 6{ pt * r l*kc h`lf( r braiqef?m. '- #= r3he: ae z34c, 0 nr pte * r rtc h`lf( r braiqef?m ,hc. %a ....... #b.a.a'; ziff7 6s '- ..... #c.c'; :pha %b ....... #b.a.a'; ziff7 zw3 '- ..... #c.c'; ;ta %c ....... #b.a.a'; ziff7 dr3 '- ..... #b.b'; r9mi5e ziff7 h/- d7t '- ..... #c.c'; 4i %d ....... #b.a.a'; ziff7 v0r '- ..... #b.b'; r9mi5e ziff7 >a#d z34cli}e %a-%d #bbj f8nfh/d7t '- ..... #c.c'; dyta %e ....... #b.a.a'; ziff7 f8nf '- ..... #c.c'; epsilon %f ....... #b.a.a'; ziff7 se4s '- ..... #c.c'; phi %g ....... #b.a.a'; ziff7 s0bc '- ..... #c.c'; gaxa %h ....... #b.a.a'; ziff7 a4t '- ..... #c.c'; theta %i ....... #b.a.a'; ziff7 n2n '- ..... #b.b'; r9mi5e ziff7 6s '- ..... #c.c'; iota %j ....... #b.a.a'; ziff7 nuq '- ..... #c.c'; eta %k ....... #c.c'; kappa %l ....... #b.b'; r9mi5e ziff7 f8nfz> '- ..... #c.c'; lambda %m ....... #b.b'; r9mi5e ziff7 t1- scd '- ..... #c.c'; m'y %n ....... #c.c'; n'y %o ....... #c.c'; omikron %oo, ..... #e>c; g~ g %o, ...... #e>a, #e>c; !g~7 3 %o,9. .... #e>c; !g~7 o kl67 3 >a#d z34cli}e %d-%o #bba %o= ...... #e>c; !g~7 o g4 %o=9. .... #e>c; !g~7, g4 o kl67 %p ....... #c.c'; pi %q ....... #c.c'; koppa %r ....... #c.c'; rho %s ....... #c.c'; s>ma %t ....... #c.c'; t1 %u ....... #c.c'; 'ypsilon %v ....... #b.b'; r9mi5e ziff7 f8nf '- ..... #c.c'; digaxa %x ....... #b.b'; r9mi5e ziff7 z=n '- ..... #c.c'; 'xi %y ....... #c.c'; psi %z ....... #c.c'; zeta %& ....... #c.c'; 4i %&a ...... #ab'; :eph %&d ...... #c.e', #aa', #aa.c'; g~% dyta 3 diff7czz34c %&e ...... #c.e', #e>e, #ab', >a#b.a'; } y{ct v =mc- gcl=re= %&n ...... #ad.c'; nabla %&o ...... #ab'; leere mc& %&p ...... #c.e', #aa', #aa.a'; q- duktz34c %&s ...... #c.e', #aa', #aa.a'; suxcz34c >a#d z34cli}e %o-%& #bbb %&2...` #ae.b'; abk8nd>usz34c f h?izont:e zsfa~uc t 7- l|t7u 3 math{ati57 |d$ %&, ...... #ab'; f ae %&: ...... #ae.b'; +k8nd>usz34c f h?izont:e zsfa~uc %&? ...... #ab'; % gb %&* ...... #e>e, #ab'; ht zm y{ct %&'=...'= #ae.b'; abk8nd>usz34c f h?izont:e zsfa~uc t 7- l|t7u 3 te'xt %%5 ...... #aj'; 5lu~z34c f s`mte qjektive =d zw3( ,qz3- 4c } t3l . z34cs.= %%< ...... #i'; cde s`mt7 br84e =d zw3( ,qz34c } t3l . s'ymbols.= %{ ....... #f'; e$>e 9ffncde klax7 %~ ....... #aa', #aa.c'; *(gral %~~ ...... #aa', #aa.c'; doppy*(- gral %~~) ..... #aa', #aa.c'; h8qc*(gral %~: ...... #aa', #aa.c'; ob7% *(- gral %~) ...... #aa', #aa.c'; uml1f*(- gral %~<: ..... #aa', #aa.c'; ,/% *(gral >a#d z34cli}e %&-%~ #bbc %} ....... #f'; e$>e 5~cde klax7 %1 ....... #aj'; ,/7 *de'x =h*tc= '- ..... #aa', #aa.a', #aa.c'; ,/e grcze =h*t77 ,/7 *de'x= '- ..... #aj'; ,?d77 ,/7 *de'x %1. ...... #e>e, #ab'; -m*d7t um, oe %2 ....... #f'; r/de 9ffncde klax7 '- ..... #b.a.d'; +f+g r p7iode 6% p7iodi5c dezim:bru- 4% '- ..... #h'; bogc %2= ...... #e>e, #ab'; } ~h:tc * o g4 %2. ...... #e>e, #ab'; } ~h:tc *, } t3lmc& v %3 ....... #aj'; wurzy %3, ...... #e>f, #ac'; o %3: ...... #e>e; vy =-b+dsthe?0= %3. ...... #e>e, #ab'; -6>t t %4 ....... #c.c'; theta '- ..... #g.a'; 5l8~yz34c f pf3l- d),yuc '- ..... #h'; hakc =-s#7usmath{a- tik= %4, ...... #h', >a#b.a'; k3l t >a#d z34cli}e %}-%4 #bbd spitze l*ks 3 m)k0ru %4;, ..... #e>h, >a#b.a'; pf3l n< /tc %4=, ..... #e>h; implik!nspf3l =pf3l n< rts t doppyt{ 5aft= %4! ...... #h', >a#b.a'; k3l t spitze rts 3 m)k0ru %4!; ..... #e>h, >a#b.a'; pf3l n< obc %4!=, .... #e>h; `'quiv:czpf3l =doppypf3l t doppyt{ 5aft= %5 ....... #c.c'; eta '- ..... #h'; abk8nd>usz34c f zs- fa~cde m)k0ruc '- ..... #aj'; 5lu~z34c f 6fb; fakult`t %6a ...... #ad.b'; )kus %6c ...... #ad.b'; kos*us %6d ...... #ad.c'; div =div7gcz= %6e ...... #aa', #aa.b'; xponcti:- f/ktj %6g ...... #ad.c'; grad =gradict= %6l ...... #aa', #aa.b'; log)ithmus %6n ...... #aa', #aa.b'; num7us >a#d z34cli}e %4-%6 #bbe %6r ...... #ad.c'; rot, 'curl =ro- t!n= %6s ...... #ad.b'; s*us %6t ...... #ad.b'; t+gcs %6{ ...... #aa', #aa.b'; )gumct %62 ...... #ad.b'; kosek+s %68 ...... #ad.b'; kot+gcs %6,c ..... #ad.b'; )kuskos*us %6,l ..... #aa', #aa.b'; +tilo- g)ithmus %6,s ..... #ad.b'; )kuss*us %6,t ..... #ad.b'; )kust+gcs %6,2 ..... #ad.b'; )kuskosek+s %6,8 ..... #ad.b'; )kuskot+gcs %6,(c .... #ad.b'; )eakos*us h'yp7- boli'cus %6,(s .... #ad.b'; )eas*us h'yp7bo- li'cus %6,(t .... #ad.b'; )eat+gcs h'yp7- boli'cus %6,(8 .... #ad.b'; )eakot+gcs h'y- p7boli'cus %6,- ..... #ad.b'; )kussek+s %6:l ..... #aa', #aa.b'; 7!gzus- o kompl{ct`rlog)ithmus %6/l ..... #aa', #aa.b'; log)ithmus du:is >a#d z34cli}e %6 #bbf %6(c ..... #ad.b'; kos*us h'yp7bo- li'cus %6(l ..... #aa', #aa.b'; log)ithmus nt:is %6(s ..... #ad.b'; s*us h'yp7boli- 'cus %6(t ..... #ad.b'; t+gcs h'yp7boli- 'cus %6(8 ..... #ad.b'; kot+gcs h'yp7bo- li'cus %6- ...... #ad.b'; sek+s %7 ....... #c.f'; 5l8~yz34c f kz- w?ts'ymbole %7% ...... #ad'; rte$ =d zw3( ,qz3- 4c } t3l . s'ymbols.= %79 ...... #ad'; w*ky %7:, ..... #ad', >a#b.b'; - uhrz3- g7s*n %7/ ...... #ad'; dr3e$ %7? ...... #ad', >a#b.b'; rhombus %7+ ...... #ad', >a#b.b'; p)aqyo- grax %7= ...... #ad'; 'quadrat %7( ...... #ad'; rt7 w*ky %7* ...... #ad'; ,4me~7 %7) ...... #ad'; kr3s %7!: ..... #ad', >a#b.b'; g e uhr- >a#d z34cli}e %6-%7 #bbg z3g7s*n %8 ....... #e>b, #i'; bru4}r# %99. ..... #e>c; kl6 g %9,, ..... #g.b', >a#b.b'; pf3l n< l*ks t &}r#yt{ 5aft u 6f<7 spitze %9,,o .... #g.b', >a#b.b'; pf3l n< l*ks u rts t &}r#yt{ 5aft u 6fh, #g.b', >a#b.b'; pf3l n< l*ks t 6f<{ 5aft u 6f<7 spitze %9::o .... #e>h, #g.b', >a#b.b'; doppypf3l t 6f<{ 5aft %9= ...... #e>c; kl67 o g4 %9=o, .... #e>c; kl67, g4 o !g~7 %9== ..... #g.b', >a#b.b'; pf3l n< l*ks t doppyt{ 5aft u 6f<7 spitze %9==o .... #e>h, #g.b', #ac', >a#b.b'; doppypf3l t doppyt{ 5aft =`'qui- v:czpf3l= %9. ...... #e>a, #e>c; kl67 3 %9.o, .... #e>c; kl67 o !g~7 3 %w ....... #c.c'; omega %, ....... #b.a.b'; ziff7 6s =&sck( >a#d z34cli}e %7-%, #bbh 5b,3e= '- ..... #b.a.c'; dezim:trcnz34c =kxa= '- ..... #g.a'; 6fa#b.b'; pf3l n< rts t &}r#yt{ 5aft u 6f<7 spitze %; ....... #aa', #aa.c'; abl3tuspt '- ..... #b.a.b'; ziff7 zw3 =&- sck( 5b,3e= '- ..... #g.a'; 6f<7 v7tik:7 pf3l5aft '- ..... #h'; pt '- ..... #i'; bru4+f+g %;; ...... #g.a'; &}r#yt7 6f<7 v7- tik:7 pf3l5aft %: ....... #h', #ad.c'; waagrt7 }r# =m)k0ru= '- ..... #b.a.b'; ziff7 dr3 =&- sck( 5b,3e= '- ..... #e>a, #e>b; &t3lt 4, -- h`lt c z =doppypt= '- ..... #g.a'; 6f<7 h?izont:7 pf3l5aft >a#d z34cli}e %,-%: #bbi %:, ...... #h', #ad', #ad.c'; pf3l n< rts =m)k0ru= '- ..... #e>h, #g.a'; pf3l n< rts t 6f<{ 5aft u 6f<7 spitze %:: ...... #g.a'; &}r#yt7 6f<7 h?i- zont:7 pf3l5aft %::o ..... #e>h, #g.b', >a#b.b'; pf3l n< rts t 6f<{ 5aft u 6f<7 spitze %:= ...... #e>c; def*itjs&m`'~ g4 =doppypt g4hsz34c= %:=: ..... #e>c; -t15bar =doppypt g4hsz34c doppypt= %:* ...... #e>f, #ac'; n %:0 ...... #e>g; qjektiv z %/ ....... #b.a.b'; ziff7 v0r =&- sck( 5b,3e= '- ..... #h'; }7n %? ....... #b.a.b'; ziff7 f8nf =&- sck( 5b,3e= '- ..... #e>c; `, `'quiv:ct, q- p?tjal '- ..... #g.a'; 6f<7 diagon:7 pf3l5aft =l*ks obc!, rts /tc= '- ..... #h'; 5l+gcl*ie =tilde= >a#d z34cli}e %:-%? #bcj %?? ...... #e>c; /&!r g4 '- ..... #g.a'; &}r#yt7 6f<7 dia- gon:7 pf3l5aft =l*ks obc!,rts /tc= %?= ...... #e>g; kongruct =&ometr0= %+ ....... #b.a.b'; ziff7 se4s =&- sck( 5b,3e= '- ..... #e>a, #e>b, #h'; plus %+- ...... #e>b; plus!,m*us %= ....... #b.a.b'; ziff7 s0bc =&- sck( 5b,3e= '- ..... #e>a, #e>c, #h'; g4 '- ..... #g.a'; doppyt7 h?izont:7 pf3l5aft %=: ...... #e>c; def*itjs&m`'~ g4 =g4hsz34c doppypt= %== ...... #e>c; idcti5 g4, kongru- ct =zlcthe?0= '- ..... #g.a'; &}r#yt7 doppyt7 h?izont:7 pf3l5aft %==o ..... #e>h, #g.b', #ac', >a#b.b'; pf3l n< rts t doppyt{ 5aft u 6f<7 spitze =implik!nspf3l= %=0 ...... #e>g; p7spektiv z %( ....... #b.a.b'; ziff7 a4t =&- sck( 5b,3e= >a#d z34cli}e %?-%( #bca '- ..... #e>b, #ab'; mal =kr2z= '- ..... #h'; kr2z =5r`g= =m)- k0ru= %* ....... #b.a.b'; ziff7 n2n =&- sck( 5b,3e= '- ..... #h', #aa', #aa.c', #ab'; }r# =m)k0ru!,abl3tu= '- ..... #g.a'; 6f<7 diagon:7 pf3l5aft =l*ks /tc!, rts obc= %*o, ..... #e>c; n !g~7 3 %*&e ..... #e>e, #ab'; } n y{ct v %*9. ..... #e>c; n kl67 3 %*? ...... #e>c; n `, n `'quiv:ct, n qp?tjal %*?= ..... #e>g; *kongruct =&ome- tr0= %*= ...... #e>c; /g4 %*== ..... #e>c; n idcti5 g4, *kon- gruct =zlcthe?0= %** ...... #g.a'; &}r#yt7 6f<7 dia- gon:7 pf3l5aft =l*ks /tc!,rts obc= %*"l ..... #e>d; t3lt n %) ....... #b.a.b'; ziff7 nuq =&- sck( 5b,3e= '- ..... #aa', #aa.a'; -kn8pft t >a#d z34cli}e %(-%) #bcb =kr3s, kuq7= '- ..... #e>b; -kn8pft t =kuq7, -kettusz34c, kr3sop7a- tor= '- ..... #h'; kr3s, kuq7 =m)k0ru= %| ....... #aj', >a#b.a'; ob77 *- de'x =h*tc= o xponct '- ..... #aa', #aa.a', #aa.c'; ob7e grcze =h*t77 ob77 *de'x= '- ..... #aj'; ,?d77 ob77 *de'x %|c ...... #ab'; ho4&,yt% 'c 3 m)- k0ru f kompl{ct`re mc- gc %|. ...... #e>e, #ab'; s'yxetri5e diff7cz %0 ....... #h'; d< =m)k0ru= %0, ...... #e>f, #ac'; u %0: ...... #e>e; et =-b+dsthe?0= %0?? ..... #e>c; ~spr#t /&!r %0= ...... #e>c; ~spr#t %0. ...... #e>e, #ab'; &5nittc t %# ....... #b.a.a'; zlz34c %#l ...... #aa', #aa.c'; lim% %#l: ..... #aa', #aa.c'; lim% sup7- ior %#l<: .... #aa', #aa.c'; lim% *- >a#d z34cli}e %)-%# #bcc f7ior %#% ...... #aa', #aa.a'; /cd =d zw3( ,qz34c } t3l . s'ymbols.= %#{ ...... #f', #f.c'; e$>e spezi- eqe 9ffncde braiqe5t- klax7 %#} ...... #f', #f.c'; e$>e spezi- eqe 5~cde braiqe5t- klax7 %#1 ...... #aj'; ,?d77 ,/7 *de'x %#2 ...... #f', #f.c'; r/de spezi- eqe 9ffncde braiqe5t- klax7 %#| ...... #aj'; ,?d77 ob77 *de'x %#` ...... #f', #f.c'; r/de spezi- eqe 5~cde braiqe5t- klax7 %#. ...... #e>g, #ad'; sckrt 1 %#...2...` #b.a.d'; p7iodi57 dezi- m:bru4 %#!{ ..... #f', #f.c', >a#b.a'; &- 5w3f( spezieqe 9ffncde braiqe5tklax7 %#!} ..... #f', #f.c', >a#b.a'; &- 5w3f( spezieqe 5~cde braiqe5tklax7 >a#d z34cli}e %# #bcd %` ....... #f'; r/de 5~cde klax7 %`, ...... #e>e, #ab'; ~h`lt, } ob7mc& v %`= ...... #e>e, #ab'; ~h`lt o } g4 %. ....... #b.a.c'; dezim:trcnz34c =pt= * |nme!fqc '- ..... #b.a.e'; gl0d7usz34c '- ..... #e>a, #e>b; mal =pt= %- ....... #e>a, #e>b, #h'; m*us '- ..... #g.a'; }r# 4 e pf3l5aft %-+ ...... #e>b; m*us!,plus %" ....... #a.b'; z3lctrcnz34c zw zw3 /mtbar ;na#b.a'; r/d% 'd =f p)tieqe abl3tu= %"h ...... #c.e'; 'h-'qu7, reduz0r( pl+$5e kon}+( %"l ...... #f', #f.e', >a#b.a'; sckrt7 }r# =+ ac ,yc= '- ..... #ab'; sckrt7 }r#, p ~ '- ..... #e>d; t3lt '- ..... #aa', #aa.c'; *(gr:}r# %"p ...... #c.e'; w37}ra'~5% 'p >a#d z34cli}e %`-%" #bce %"% ...... #e>g, #ad'; p)aqy z =d zw3( ,qz34c } t3l . s'ymbols.= '- ..... #f', #f.e', >a#b.a'; sckrt7 doppy}r# =+ ac ,yc= =d zw3( ,qz34c } t3l . s'ymbols.= %"%= ..... #e>g; p)aqy u g4 =d zw3( ,qz34c } t3l . s'ym- bols.= %> ....... #c.a', #c.b'; 6 o ,=7e g~bu4}abc '- ..... #h'; +k8nd>usz34c f 6fa#b.b'; kl67 'qu7}r# 6% ,z?dnus- pf3ls %>~ ...... #aa', #aa.c'; *(gral ;7 )t %>:, ..... #e>h, #g.a'; ,z?dnuspf3l % ....... #h', #aj'; +k8nd>usz34c f zsfa~cde ,/e m)k0ruc '- ..... #c.a', #c.d', #ad.c'; #; ;e t'ypografi5e |z34nu '- ..... #d.a'; kcnz34c f 6hc- s'ymbole %ct ..... #d.f'; 2ro-'cct =2rozo- >a#d z34cli}e %"-% #bcf ne= %dkr .... #d.f'; krone =d`nem)k= %e>v .... #d.d'; yektroncvolt %i>r .... #d.f'; rupie =*dic= %k$hz ... #d.d'; kiloh7tz %k<>w ... #d.d'; kiloohm %m ...... #d.d'; met7 %min .... #d.d'; m*u( %mm ..... #d.d'; miqimet7 %m>a .... #d.d'; miqiamp7e %rad .... #d.c', >a#b.a'; radi+t =rad= %rad|; #d.c', >a#b.a'; 'qua- dratradi+t %s ...... #d.d'; sek/de %sec .... #d.d'; sek/de %#j) .... #d.b'; qzct %#j)) ... #d.b'; qmiqe %"c ..... #d.f'; 'cct =? a{ >usa= %"e ..... #d.f'; 2ro =2rozone= %"l ..... #d.f'; pf/d =? a{ g~bri- t+nic= %"s ..... #d.f'; doqar =? a{ >usa= %"s>a ... #d.f'; doqar =1}r:ic= %"y ..... #d.f'; 'yc =japan= %"y ..... #d.f'; 'yuan =4*a= %"* ..... #d.c', >a#b.a'; m*u( >a#d z34cli}e % #bcg =}r#= %"** .... #d.c', >a#b.a'; sek/de =doppy}r#= %") ..... #d.c', >a#b.a'; grad =kr*gy= %>aud ... #d.f'; doqar =1}r:ic= %>chf ... #d.f'; fr+kc =5w3z= %>cny ... #d.f'; 'yuan =4*a= %>czk ... #d.f'; krone =t5e4i5e rk= %>dkk ... #d.f'; krone =d`nem)k= %>eur ... #d.f'; 2ro =2rozone= %>gbp ... #d.f'; pf/d =g~brit+nic= %>inr ... #d.f'; rupie =*dic= %>jpy ... #d.f'; 'yc =japan= %>mv .... #d.d'; megavolt %>m'e>v #d.d'; megayektroncvolt %>nzd ... #d.f'; doqar =n2seel+d= %>nz"s #d.f'; doqar =n2seel+d= %>tl .... #d.f'; pf/d!,lira =t8r- k3= %>trl ... #d.f'; pf/d!,lira =t8r- k3= %>usd ... #d.f'; doqar =>usa= %>v ..... #d.d'; volt %>"a .... #d.d'; "ang}r9m %$fr. ... #d.f'; fr+kc =5w3z= >a#d z34cli}e % #bch %$hz .... #d.d'; h7tz %$kc .... #d.f'; krone =t5e4i5e rk= %$me>v #d.d'; megayektroncvolt %w ... #d.d'; mikrowatt %w .... #d.d'; ohm %! ....... #aj'; zw3t% qjektiv- v7}`rkusz34c '- ..... #c.a', #c.d', #ad.c'; #, ;e t'ypografi5e |z34nu '- ..... #g.a'; 6fa#b.a'; &- 5w3f( 9ffncde klax7 %!} ...... #f', #ab', >a#b.a'; &- 5w3f( 5~cde klax7 %!5 ...... #h'; zw3t% abk8nd>usz34c f -}`rk( zsfa~cde m)- k0ruc '- ..... #aj'; 5lu~z34c f -}`rk( qjektive %!, ...... #a.a', #a.a.b'; +k8n- d>usz34c f 6e pa~a& * >a#d z34cli}e %-%! #bci math{atik5t %!: ...... #e>h, #g.a', #h'; pf3l n< l*ks t 6f<{ 5aft u 6f<7 spitze %!:, ..... #e>h, #g.a'; pf3l n< l*ks u rts t 6f<{ 5aft u 6fb, >a#b.a'; mal =}7n= %!>{ ..... #f', >a#b.b'; g1'~5e 9ffncde klax7 =ob7e grcze= %!>} ..... #f', >a#b.b'; g1'~5e 5~cde klax7 =ob7e grc- ze= %!! ...... #g.a'; doppy( spitze n< l*ks o /tc %!<{ ..... #f', >a#b.b'; g1'~5e 9ffncde klax7 =,/e grcze= %!<} ..... #f', >a#b.b'; g1'~5e 5~cde klax7 =,/e grc- ze= %$ ....... #c.a', #c.b'; 6 g~bu4}a- be, &fgt v 6{ o ,=7c kl6bu4}abc '- ..... #h', #aj'; +k8nd>usz34c f zsfa~cde ob7e m)- >a#d z34cli}e %!-%$ #bdj k0ruc '- ..... #h'; -}`rkusz34c f zs- fa~cde m)k0ruc '- ..... #aj'; qjektivv7}`rkusz3- 4c %${ ...... #f', >a#b.a'; spitze 9ffncde klax7 %$} ...... #f', >a#b.a'; spitze 5~cde klax7 %$2 ...... #ad'; zsfa~cde m)k0ru f bogc =bogc 8 ,=7c s'ymbolc= %$5 ...... #h'; abk8nd>usz34c f -- }`rk( zsfa~cde m)k0ruc '- ..... #aj'; 5lu~z34c f -}`rk( qjektive %$: ...... #ad'; zsfa~cde m)k0ru f }re$e =waagrt7 }r# 8 ,=7c bu4}abc= %$:, ..... #ad'; zsfa~cde m)k0ru f vektor =pf3l 8 ,=7c s'ymbolc= %$!{ ..... #f', >a#b.b'; z3lczs- fa~usklax7: ,=7e z3lc zsfa~cde g~e l*ke &- 5w3f( klax7 %$$c ..... #c.e', #ab'; mc& r kom- >a#d z34cli}e %$ #bda ple'xc zlc %$$h ..... #c.e', #ab'; mc& r 'qua- t7nionc %$$n ..... #c.e', #ab'; mc& r nc zlc %$$p ..... #c.e', #ab'; qjektive &- rade %$$q ..... #c.e', #ab'; mc& r r!n:c zlc %$$r ..... #c.e', #ab'; mc& r reeqc zlc %$$z ..... #c.e', #ab'; mc& r gzc zlc %< ....... #c.a', #c.c'; gr04i5e bu4}abc '- ..... #h'; +k8nd>usz34c f 6fusz34c f szz34c '- ..... #a.b'; z3lctrcnz34c + r ,ye 6% leerz34cs %'{ ...... #f', >a#b.a'; }umpfw*k- l>e 9ffncde klax7 >a#d z34cli}e %$-%' #bdb %'} ...... #f', >a#b.a'; }umpfw*k- l>e 5~cde klax7 %'. ...... #a.a', #a.a.b'; abk8n- d>usz34c f 6e pa~a& * math{atik5t %'. ...... #a.a', #a.a.c'; +k8n- d>usz34c f 6e pa~a& * te'xt5t '- ..... #a.a', #a.a.c'; abk8n- d>usz34c f 6e pa~a& * te'xt5t '- ..... #a.a', #a.a.b'; abk8n- d>usz34c f 6e pa~a& * math{atik5t %'<= ..... #a.c', #f'; 9ffncde u 5~cde klax7 f +m7kuc zr braiqe5t,8tgu >a#d z34cli}e %' #bdc >a#d z34cli}e %' #bdd >a#e :pha;ti5% s}c 6tg h*. >a abk8nd>u: s0he +- u abk8nd>ustk abl3tus}r#: s0he }r#e abl3tuspt: s0he pt `, `'quiv:ct, qp?tjal: #e', #e>c'* +:'ysis: #aa +- u abk8nd>ustk: '- br84e: #i.c '- doppyleerz34ctk: #a.a.d '- h?izont:e zsfa~uc t 7l|t7u 3 te'xt: #ae.b '- math{atik5t: #a.a'*, #a.a.b'*, #a.a.e', #c.g', >a#b.aa '- te'xt5t: #a.a'*, #a.a.c'*, #a.a.e', >a#b.aa '- m)k0ruc, 6fa#b.aj "ang}r9m: #d.d'* +m7kuc zr braiqe5t,8tgu: #a.c'*, >a#e sa #bde #b.a.c', #c.c', #c.d', #c.e', #c.f', #e', #f'*, #f.c', #g.a', #ae.a', >a#b.f', >a#b.ab apo}roph =b zlc=: #b.a.a', #b.a.e `'quiv:ct, qp?tjal, `: #e', #e>c'* )abi5e ziff7n: s0he zlc )gumct =f/ktj=: #aa'*, #aa.b'* )gumc(: s0he f/ktjc >b bogc: #h'*, #h.b', #ad.a'* braiqe5tt5e +m7kusklax7n: s0he +m7- kuc zr braiqe5t,8tgu braiqe5tklax7n, spezieqe: #f'*, #f.c'* br84e: #i '- +f+gs- u cdez34c: #i'*, #i.c '- |fhe 5b,3e: #i'*, #i.c '- dezim:br84e: #b.a.c '- 6fa#e sa->b #bdf '- g~- u kl65bu: #b.b', #c.a', #c.b', #c.c', #c.d', #c.f', #c.h', (,?w?t'- kompakth v7sus kon('xt/abh`ng>k), >a#b.f '- kzw?ts'ymbole: #b.f '- lat6i5e: #c.b '- t'ypografi5e |z34nuc: #c.d bu4}abc`e s'ymbole: s0he bu4}abc >c 'c)t%i5% qdukt =m:kr2z=: #e', #e>b'*, #ab'* >d 'd, r/d% =p)tieqe abl3tu=: #c.e'*, #aa'*, #aa.c'* datum: s0he zlc dyta: #c.c'* '- g~% 3 diff7czz34c: #c.e'*, #aa'*, #aa.c'* dezim:br84e: s0he br84e dezim:kla~ifikat?c: s0he zlc dezim:trcnz34c: s0he zlc diff7czz34c, dyta =g~%=: #c.e'*, #aa', #aa.c doppyleerz34ctk: #a.a.d dr3e$: #ad.a'* >a#e sb->d #bdg 4, &t3lt 4, -h`lt c z =doppypt=: #e', #e>a'*, #e>b'*, >a#b.ab ,4me~7: #ad.a'* >e 6hcs'ymbole: #d '- 1s bu4}abc: #d.d '- kcnz34nu: #d.a y{ct v: #e', #e>e'*, #ab'* ~h:tc * o g4: #e', #e>e'*, #ab'* ~h:tc *, t3lmc& v: #e', #e>e'*, #ab'* ~spr#t: #e', #e>c'* xponctc: (zm &,| 0% regyw7ks'- $la$te>x), #b.a.b', #e', #h.a', #aj'*,#aj.c.a', >a#b.a', >a#b.d xponcti:f/ktj: #aa'*, #aa.b'* >f fettd$: s0he t'ypografi5e |z34nuc, ;e f/ktjc: #aa', #aa.a', #aa.b '- )gumc(: #c.f', #aa.b', #ad.b', >a#b.ab '- xponcti:f/ktj: #aa'*, #aa.b'* >a#e sd->f #bdh >g g e uhrz3g7s*n: #ad.a'* gyd: #b.a.c', #b.a.e', #d.f', >a#b.c &mi5( zlc: s0he zlc &ometr0: #c.f', #e>g'*, #ad.a'* &t3lt 4: s0he 4 gl0d7u lg7 zlc: s0he zlc gl0d7usz34c: s0he zlc goti5e bu4}abc: s0he t'ypografi5e |- z34nuc, ;e grczc, ob7e u ,/e =h*t7e *diz%=: #aa'*, #aa.a'*, #aa.c gr04i5e bu4}abc: #c.c'* g~bu4}abc: s0he bu4}abc >h h?izont:7 }r# =m)k0ru=: s0he }r#e 'h-'qu7, reduz0r( pl+$5e kon}+(: #c.e'* h'yp7byf/ktjc: #ad.b'* h'yp7boli'cus: #ad.b >i idcti5 g4, kongruct =zlcthe?0=: #e', #e>c'* - uhrz3g7s*n: #ad.a'* >a#e sg->i #bdi *diz%: '- aq&m6: #aj'*, #aj.c', >a#b.d '- 3 grczc b *(gr:c: #aa.c'*, >a#b.d '- 3 grczc b qduktc u suxc: #aa.a'*, >a#b.d '- 1s gzc zlc: #b.a.b', #aj.c.c', >a#b.d '- ,?d7e: #aj'*, #aj.c.b', >a#b.d '- h*t7e: #aj'*, #aj.c.c', >a#b.d *kongruct: '- op7!ns- u ry!nsz34c =&ometr0=: #e', #e>g'* '- zlcthe?0, n idcti5 g4: #e', #e>c'* *(gr:e: #aa'*, #aa.c'* *(gr:}r#: #aa'*, #aa.c'* >k k3l =m)k0ru=: #h'*, #h.a kcnz34c f 6hcs'ymbole: #d.a'* kcnz34nu v bu4}abc: s0he bu4}abc kilo-: #d.e klax7n: '- e$>e klax7n: #f'* '- e$>e spezieqe braiqe5tklax7n: #f'*, #f.c'* >a#e si->k #bej '- 6fe klax71s!d$e: #f.d '- r/de klax7n: #b.a.d', #f'* '- r/de spezieqe braiqe5tklax7n: #f'*, #f.c'* '- spezieqe braiqe5tklax7n: #f'*, #f.c'* '- }umpfw*kl>e klax7n: #f'* '- te'xtklax7n: #f.f '- vekt?klax7n: #f.d', #ad.c '- z3lczsfa~usklax7: #f'*, #f.d kl6 g: #e', #e>c'* kl6bu4}abc: s0he bu4}abc kl67 3: #e', #e>a'*, #e>c'* kl67 o g4: #e', #e>c'* kl67 o !g~7 3: #e', #e>c'* kl67, g4 o !g~7: #e', #e>c'* kompl{ct`re mcgc: #ab'* >a#e sk #bea kongruct =&ometr0=: #e', #e>g'* koppa: #c.c'* kr3s: '- &ometr0: #ad.a'* '- m)k0ru: #h'*, #h.a kr2z, 5r`g%: '- m:kr2z: #e', #e>b'* '- m)k0ru: #h'*, #h.a kursivd$: s0he t'ypografi5e |z34nuc, ;e kzw?ts'ymbole: s0he bu4}abc >l $la$te>x, &,| * 0{ w7k: (zm &,| 0% regyw7ks'- $la$te>x) la'yout, 5twe4sy 4: #a.a.a leere mc&: #ab'* leerz34c, ,/!d$u: #a.b l04tc}6: '- fr+kc;!tge: #b.a.c', #b.a.e '- zlcgl0d7u: #b.a.e lim%: #aa'*, #aa.c'* log)ithmusf/ktjc: #aa'*, #aa.b'* logik: #e>f'*, #ac'* >a#e sk->l #beb >m mal '- 'c)t%i5% qdukt =m:kr2z=: #e', #e>b'*, #ab'* '- kr2z: #e', #e>b'* '- pt: #e', #e>a'*, #e>b'* '- }7n: #e', #e>b'* m)k0ruc: #h '- 6fk) '- we4sy z te'xt5t: s0he 5twe4sy matrizc: #f.d mega-: #d.e ,=7e z3lc zsfa~cde g~e l*ke &5w3f( klax7 =z3lczsfa~usklax7=: #f'*, #f.d ,=z3l>e klax71!d$e: #f.d mc&: '- r gzc zlc: #c.e'*, #ab'* '- r komple'xc zlc: #c.e'*, #ab'* '- r nc zlc: #c.e'*, #ab'* '- r 'quat7nionc: #c.e'*, #ab'* '- r r!n:c zlc: #c.e'*, #ab'* >a#e sm #bec '- r reeqc zlc: #c.e'*, #ab'* '- leere: #ab'* mcgcl=re: #e', #e>e'*, #ab'* met7: #d.d'* mikro-: #d.e miqi-: #d.e m*us: #b.a.b', #e', #e>a'*, #e>b'*, >a#a.b '- 3 m)k0ru: #h'*, #h.a '- * *diz%: #aj.c.c', >a#b.d m*u(: #d.c'*, #d.d'* >n nabla: #ad.c'* n: #e', #e>f'*, #ac'* n `, n `'quiv:ct, n qp?tjal: #e', #e>c'* n `'quiv:ct, n qp?tjal, n `: #e', #e>c'* n y{ct v: #e', #e>e'*, #ab'* n !g~7 3: #e', #e>c'* n idcti5 g4, *kongruct =zlcthe?0=: #e', #e>c'* n kl67 3: #e', #e>c'* n qp?tjal, n `, n `'quiv:ct: #e', #e>c'* num7us: #aa'*, #aa.b'* >a#e sm->n #bed >o ob7mc& v, ~h`lt: #e', #e>e'*, #ab'* o: #e', #e>f'*, #ac'* ohm: #d.d'* op7!nsz34c: #e', #e>b'* ?dnuszlc: s0he zlc >p p)aqy u g4: #e', #e>g'* p)aqy z: #e', #e>g'*, #ad.a'* p)aqyograx: #ad.a'* p)tieqe abl3tu, r/d% 'd: #c.e'*, #aa'*, #aa.c'* pa~a& * te'xt5t: #a.a', #a.a.c p7iodi5e dezim:br84e: s0he zlc p7spektiv z: #e', #e>g'* pf3le: #g'* '- ;5tu v: #g.c'* '- def*0r(: #e>g'*, #g', #g.b'* '- h?izont:e: #e>g'*, #g', #g.a'* '- logik: #ac'* '- m)k0ru: #h'*, #h.a', #h.b', #ad.c'* '- modul)e: #g.a'* '- op7!ns- u ry!nsz34c: #e>h'* '- 5l8~yz34c: #g.a'* pf/d: #d.f'* >a#e so->p #bee pi: #c.c'* '- 3 qduktz34c: #c.e'*, #aa'*, #aa.a'* pl+$5e kon}+(: #c.e'* pzh:t7: #ae.a plus: #e', #e>a'*, #e>b'* '- 3 m)k0ru: #h'*, #h.a '- * *diz%: #aj.a', #aj.c.c plus!,m*us: #e', #e>b'* qduktz34c: #c.e'*, #aa'*, #aa.a'* qjektiv z: #e', #e>g'* qjektive &rade: #c.e'*, #ab'* qjektivtk: #aj '- cde s`mt7 qjektive: #aj'*, #aj.b', >a#b.aj '- 6fc'* qzct: #d.b'* pt: '- abl3tuspt: #h'*, #h.a', #aa'*, #aa.c'* '- m:pt: #e', #e>a'*, #e>b'* '- m)k0ru: #h'*, #h.a '- szpt: #c.g >a#e sp #bef '- t1scd7-trcnz34c =gl0d7usz34c=: #b.a.e'* pt #d': '- akzctz34c: #a.b', #d.d'* '- z3lctrcnz34c: #a.b'* '- leerz34c7sz: #a.b'*, #aj.a '- s'ymbol;,}t3l: #a.b '- zsh:(pt: #a.b'* pt #f': '- szz34c, +k8nd>u f: #c.g'* '- z3lctrcnz34c: #a.b'* >q 'quadrat: #ad.a'* 'quadratradi+t: #d.c'* >r radi+t =rad=: #d.c'* r+dm)k0ru: #a.a.a rte$: #ad.a'* ry!nsz34c: #e', #e>c'* rhombus: #ad.a'* r9mi5e zlc, ziff7n: #b.b'* rot, 'curl =rot!n=: #ad.c'* r/d% 'd =p)tieqe abl3tu=: #c.e'*, #aa'*, #aa.c'* >a#e sp->r #beg >s szz34c: '- kcnz34nu: #c.g '- 5twe4sy: #a.a.b', #a.a.d', #a.a.e 5l+gcl*ie, tilde =m)k0ru=: #h'*, #h.a', #h.b 5l8~yz34c: '- 6hc: >a#b.h '- &ometri5e s'ymbole: #ad.a '- kzw9rt7: #c.f'*, >a#b.ab '- pf3le: #g.a'* '- -we4slu t: #e 5r`g7 }r#: s0he }r#e 5te re4cv7,rc: #b.a.a', >a#a 5twe4sy =math{atik u te'xt=: #a.a', #c.h', >a#b.aa 5r`g% kr2z: s0he kr2z, 5r`g% 5w3z: '- fr+kc;!tge: #b.a.c', #b.a.e '- zlcgl0d7u: #b.a.e sek/de: #d.c'*, #d.d'* sckrt 1: '- op7!ns- 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#aj'*, #aj.d >z zlc: #b'* '- )abi5e: #b.a '- datum: #b.a.f '- dezim:br84e: #b.a.c '- dezim:kla~ifikat?c: #b.a.f '- dezim:trcnz34c: #b.a.a', #b.a.c'*, #b.a.f', >a#a.e '- gze zlc: #b.a.b >a#e sv->z #bfb '- &mi5( zlc: #i.a '- &sck( 5b,3e: #b.a.b'*, #i.a '- gl0d7u lg7 zlc: #b.a.a', #b.a.e'*, >a#a '- gl0d7usz34c: #b.a.a', #b.a.e'*, >a#a '- mcgc =gze, reeqe zlc usw.=: #c.e'*, #ab'* '- ?dnuszlc: #b.a.f '- p7iodi5e dezim:br84e: #b.a.d'* '- r9mi5e: #b.b'* '- ,})d5b,3e: #b.a.a'* '- uhrzt: #b.a.f '- -!kz( 5b,3e 1s r te'xt5t: #b.a.f '- zlcbr84e: s0he br84e '- zlz34c: #b.a.a'* zlcbr84e: s0he br84e zlz34c: s0he zlc z34c: '- suxcz34c: #c.e'*, #aa'*, #aa.a'* '- qduktz34c: #c.e'*, #aa'*, #aa.a'* '- diff7czz34c: #c.e'*, #e>e'*, #aa'*, #aa.c'* z3lctrcnz34c =braiqe5t=: #a.b'* >a#e sz #bfc z3lcumbru4: '- braiqe5tz3lcumbr84e: #a.b'* '- kcnz34nu v 5w)z5tz3lcumbr84c: #f'*, #f.d', #f.e'*, #ad.c'* z3lczsfa~usklax7: #f'*, #f.d ziff7n: #b'* '- ,})d5b,3e: #b.a.a'* '- &sck( 5b,3e: #b.a.b'* ,z?dnuspf3l: #e>h'*, #g.a'* zsfa~uc, h?izont:e: #ae.b'* zsh:(pt: #a.b'* :::::::::::: cde . bu4% ====== >a#e sz #bfd