Meetkunde 1: verschil tussen versies

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http://brenda-logan-85.blogskine.net brenda http://forum-avionica-29.mygreatblogs.net/blog/entry/forum-mirabel-park/ forum mirabel park http://lanzar-67.alumblogs.net/wordpress/lanzarote-playa-blanca/ lanzarote playa blanca http://metal-98.blogskine.net/wordpress/corso-disegnatore-metalmeccanico/ corso disegnatore metalmeccanico http://system-startup-38.alumblogs.net/multitasker-operating-system/ multitasker operating system http://traverso-35.blogskine.net/2008/07/ricerca-mercato-attraverso-internet/ ricerca mercato attraverso internet http://friend-99.blogskine.net/2008/08/06/my-hot-friend-mom/ my hot friend mom http://elettrochimica-cinetica-10.blogskine.net/blog/entry/cinetica-reazione/ cinetica reazione http://vincent-spano-74.alumblogs.net/2008/09/postino-vincent-van-gogh/ postino vincent van gogh http://rapid-6.blogskine.net/2008/07/27/pianta-rapida-crescita/ pianta rapida crescita http://phone-settings-5.blogskine.net/2008/08/cell-phone-reviews/ cell phone reviews http://arrivare-lanciano-32.alumblogs.net/blog/entry/arrivare-legoland-londra/ arrivare legoland londra http://nando-navarra-5.blogskine.net/blog/entry/mose-navarra/ mose navarra http://molto-34.alumblogs.net/wordpress/xp-molto-lento/ xp molto lento http://tequila-robbe-57.alumblogs.net/node/guadalajara-tequila-express/ guadalajara tequila express http://tobias-rehberger-53.blogskine.net tobias http://analisi-musicale-7.mygreatblogs.net/2008/08/scaccia-pensiero-strumento-musicale/ scaccia pensiero strumento musicale http://globi-animati-70.blogskine.net/2008/08/fractal-animation-gallery/ fractal animation gallery http://residence-39.alumblogs.net/node/residence-corones/ residence corones http://grazia-morelli-12.mygreatblogs.net/liceo-michele-morelli/ liceo michele morelli http://villa-altea-9.alumblogs.net/wordpress/villa-cortese-volley/ villa cortese volley http://outreach-worldwide-96.blogskine.net/blog/entry/internet-radio-worldwide/ internet radio worldwide http://roger-ebert-40.alumblogs.net/2008/08/28/roger-fry/ roger fry http://lankadeepa-56.blogskine.net/node/newspaper-lankadeepa/ newspaper lankadeepa http://livia-scoglio-50.alumblogs.net/wordpress/scoglio-scauri/ scoglio scauri http://giocare-lotto-92.alumblogs.net/giocare-mercante-fiera/ giocare mercante fiera http://mantua-models-95.blogskine.net/mantua-me-genuit/ mantua me genuit http://department-29.mygreatblogs.net/node/kentucky-department-of-education/ kentucky department of education http://kendra-41.blogskine.net/2008/09/17/kendra-playboy-foto/ kendra playboy foto http://spuma-liquirizia-19.alumblogs.net/spuma-di-champagna/ spuma di champagna http://mediterranean-villas-86.blogskine.net/chianti-villas/ chianti villas http://federica-fumagalli-32.blogskine.net/node/spettacolo-federica-capuanocaputo/ spettacolo federica capuanocaputo http://ecologico-78.mygreatblogs.net/2008/09/01/danno-biologico-esistenziale-ginecologico/ danno biologico esistenziale ginecologico http://tuner-8.blogskine.net/wordpress/tv-tuner-acer/ tv tuner acer http://rates-46.alumblogs.net/pratesi-hotel-division/ pratesi hotel division http://elena-proietti-52.alumblogs.net/monologhi-di-gigi-proietti/ monologhi di gigi proietti http://flora-finlandese-15.blogskine.net/wordpress/ana-flora/ ana flora http://carrello-75.blogskine.net/node/sicurezza-carrello-elevatore-semovente/ sicurezza carrello elevatore semovente http://karin-michaelis-95.blogskine.net/2008/09/08/magda-karina/ magda karina http://quatro-zampa-46.mygreatblogs.net/2008/09/13/granuloma-zampa-cane/ granuloma zampa cane http://mappa-spoleto-31.blogskine.net/mappa-conca-tesino/ mappa conca tesino http://albergo-positano-56.mygreatblogs.net/wordpress/prenotazione-albergo-gabicce-mare/ prenotazione albergo gabicce mare http://verde-69.blogskine.net/blog/entry/branzino-pepe-verde/ branzino pepe verde http://loretta-mangili-18.mygreatblogs.net/loretta-massaggio-conegliano/ loretta massaggio conegliano http://cotton-curtains-51.alumblogs.net/2008/08/07/cotton-wool-pleat/ cotton wool pleat http://inexpensive-50.blogskine.net/2008/09/06/hosting-inexpensive-reliable-web/ hosting inexpensive reliable web http://pelosi-20.mygreatblogs.net/node/cazzo-pelosi/ cazzo pelosi http://alunni-maestrina-97.blogskine.net/2008/09/errori-linguistici-alunni-straniero/ errori linguistici alunni straniero http://addominale-laterali-6.mygreatblogs.net/2008/08/chiusura-laterali-balcone/ chiusura laterali balcone http://perama-studios-52.alumblogs.net/2008/08/13/tucson-old-tucson-studios/ tucson old tucson studios
http://brenda-logan-85.blogskine.net brenda http://forum-avionica-29.mygreatblogs.net/blog/entry/forum-mirabel-park/ forum mirabel park http://lanzar-67.alumblogs.net/wordpress/lanzarote-playa-blanca/ lanzarote playa blanca http://metal-98.blogskine.net/wordpress/corso-disegnatore-metalmeccanico/ corso disegnatore metalmeccanico http://system-startup-38.alumblogs.net/multitasker-operating-system/ multitasker operating system http://traverso-35.blogskine.net/2008/07/ricerca-mercato-attraverso-internet/ ricerca mercato attraverso internet http://friend-99.blogskine.net/2008/08/06/my-hot-friend-mom/ my hot friend mom http://elettrochimica-cinetica-10.blogskine.net/blog/entry/cinetica-reazione/ cinetica reazione http://vincent-spano-74.alumblogs.net/2008/09/postino-vincent-van-gogh/ postino vincent van gogh http://rapid-6.blogskine.net/2008/07/27/pianta-rapida-crescita/ pianta rapida crescita http://phone-settings-5.blogskine.net/2008/08/cell-phone-reviews/ cell phone reviews http://arrivare-lanciano-32.alumblogs.net/blog/entry/arrivare-legoland-londra/ arrivare legoland londra http://nando-navarra-5.blogskine.net/blog/entry/mose-navarra/ mose navarra http://molto-34.alumblogs.net/wordpress/xp-molto-lento/ xp molto lento http://tequila-robbe-57.alumblogs.net/node/guadalajara-tequila-express/ guadalajara tequila express http://tobias-rehberger-53.blogskine.net tobias http://analisi-musicale-7.mygreatblogs.net/2008/08/scaccia-pensiero-strumento-musicale/ scaccia pensiero strumento musicale http://globi-animati-70.blogskine.net/2008/08/fractal-animation-gallery/ fractal animation gallery http://residence-39.alumblogs.net/node/residence-corones/ residence corones http://grazia-morelli-12.mygreatblogs.net/liceo-michele-morelli/ liceo michele morelli http://villa-altea-9.alumblogs.net/wordpress/villa-cortese-volley/ villa cortese volley http://outreach-worldwide-96.blogskine.net/blog/entry/internet-radio-worldwide/ internet radio worldwide http://roger-ebert-40.alumblogs.net/2008/08/28/roger-fry/ roger fry http://lankadeepa-56.blogskine.net/node/newspaper-lankadeepa/ newspaper lankadeepa http://livia-scoglio-50.alumblogs.net/wordpress/scoglio-scauri/ scoglio scauri http://giocare-lotto-92.alumblogs.net/giocare-mercante-fiera/ giocare mercante fiera http://mantua-models-95.blogskine.net/mantua-me-genuit/ mantua me genuit http://department-29.mygreatblogs.net/node/kentucky-department-of-education/ kentucky department of education http://kendra-41.blogskine.net/2008/09/17/kendra-playboy-foto/ kendra playboy foto http://spuma-liquirizia-19.alumblogs.net/spuma-di-champagna/ spuma di champagna http://mediterranean-villas-86.blogskine.net/chianti-villas/ chianti villas http://federica-fumagalli-32.blogskine.net/node/spettacolo-federica-capuanocaputo/ spettacolo federica capuanocaputo http://ecologico-78.mygreatblogs.net/2008/09/01/danno-biologico-esistenziale-ginecologico/ danno biologico esistenziale ginecologico http://tuner-8.blogskine.net/wordpress/tv-tuner-acer/ tv tuner acer http://rates-46.alumblogs.net/pratesi-hotel-division/ pratesi hotel division http://elena-proietti-52.alumblogs.net/monologhi-di-gigi-proietti/ monologhi di gigi proietti http://flora-finlandese-15.blogskine.net/wordpress/ana-flora/ ana flora http://carrello-75.blogskine.net/node/sicurezza-carrello-elevatore-semovente/ sicurezza carrello elevatore semovente http://karin-michaelis-95.blogskine.net/2008/09/08/magda-karina/ magda karina http://quatro-zampa-46.mygreatblogs.net/2008/09/13/granuloma-zampa-cane/ granuloma zampa cane http://mappa-spoleto-31.blogskine.net/mappa-conca-tesino/ mappa conca tesino http://albergo-positano-56.mygreatblogs.net/wordpress/prenotazione-albergo-gabicce-mare/ prenotazione albergo gabicce mare http://verde-69.blogskine.net/blog/entry/branzino-pepe-verde/ branzino pepe verde http://loretta-mangili-18.mygreatblogs.net/loretta-massaggio-conegliano/ loretta massaggio conegliano http://cotton-curtains-51.alumblogs.net/2008/08/07/cotton-wool-pleat/ cotton wool pleat http://inexpensive-50.blogskine.net/2008/09/06/hosting-inexpensive-reliable-web/ hosting inexpensive reliable web http://pelosi-20.mygreatblogs.net/node/cazzo-pelosi/ cazzo pelosi http://alunni-maestrina-97.blogskine.net/2008/09/errori-linguistici-alunni-straniero/ errori linguistici alunni straniero http://addominale-laterali-6.mygreatblogs.net/2008/08/chiusura-laterali-balcone/ chiusura laterali balcone http://perama-studios-52.alumblogs.net/2008/08/13/tucson-old-tucson-studios/ tucson old tucson studios


== Theorievragen ==
http://promotion-10.alumblogs.net/2008/09/mr-promotion-com/ mr promotion com http://interactive-phisic-64.alumblogs.net/wordpress/free-interactive-sex-movie/ free interactive sex movie http://dentro-38.blogskine.net/2008/09/14/pene-dentro-figa/ pene dentro figa http://riflusso-gastrico-95.mygreatblogs.net/wordpress/linfoma-gastrico-recidivante/ linfoma gastrico recidivante http://charles-88.alumblogs.net/node/king-charles-spaniel-breeders/ king charles spaniel breeders http://video-93.alumblogs.net/2008/09/11/niky-belucci-video/ niky belucci video http://encoder-video-18.blogskine.net/2008/09/11/dragostea-din-tei-video/ dragostea din tei video http://sport-calciomercato-48.alumblogs.net/wordpress/fornitore-articolo-sportivo/ fornitore articolo sportivo http://dedica-alunni-43.alumblogs.net/wordpress/alunni-straniero-ospedale/ alunni straniero ospedale http://cancellare-definitivamente-99.blogskine.net/2008/09/01/cancellare-indirizzo-msn/ cancellare indirizzo msn http://rifiuto-urbani-99.blogskine.net/2008/08/12/rifiuto-ferroso-siena/ rifiuto ferroso siena http://toilet-81.blogskine.net/blog/entry/femdom-toilet/ femdom toilet http://ginger-66.mygreatblogs.net/node/ginger-lang/ ginger lang http://funzionano-servlet-58.mygreatblogs.net/blog/entry/funzionano-intercettazioni-telefonica/ funzionano intercettazioni telefonica http://nevada-81.mygreatblogs.net/blog/entry/nevada-worldwide-s/ nevada worldwide s http://motore-ricerca-89.mygreatblogs.net/supereva-risultato-ricerca/ supereva risultato ricerca http://vigor-lamezia-82.alumblogs.net/comune-di-lamezia/ comune di lamezia http://archivio-fotografico-72.alumblogs.net/blog/entry/archivio-notarile-palermo/ archivio notarile palermo http://cotonine-americane-25.blogskine.net/blog/entry/auto-truccate-americane/ auto truccate americane http://movie-attack-36.blogskine.net/2008/08/03/celebrity-movie-archives-com/ celebrity movie archives com http://lascuola-64.alumblogs.net/node/sdid-lascuola/ sdid lascuola http://pellicola-antiproiettile-57.alumblogs.net/2008/08/fornitore-pellicola-trasparente/ fornitore pellicola trasparente http://fattoria-vincitore-73.blogskine.net/wordpress/due-tre-stalla-vincitore/ due tre stalla vincitore http://metal-icons-18.blogskine.net/2008/09/01/ferie-contratto-metalmeccanici/ ferie contratto metalmeccanici http://robert-palmer-65.alumblogs.net/wordpress/david-roberts-egypt/ david roberts egypt http://etrusca-8.alumblogs.net/blog/entry/agriresidence-costa-etrusca/ agriresidence costa etrusca http://antiche-79.alumblogs.net/2008/08/10/civilta-antiche/ civilta antiche http://scuola-normale-33.blogskine.net/bruno-mondadori-scuola/ bruno mondadori scuola http://domiciliazione-telefonica-45.mygreatblogs.net/wordpress/risoluzioni-contratto-compagnia-telefonica/ risoluzioni contratto compagnia telefonica http://cipriani-hotel-66.alumblogs.net/2008/09/hotel-pineto/ hotel pineto http://event-target-91.blogskine.net/blog/entry/club-seventeen-pic/ club seventeen pic http://savior-9.alumblogs.net/2008/08/ristorante-saviore-dell-adamello/ ristorante saviore dell adamello http://arcuri-93.blogskine.net/blog/entry/fotografia-hard-emanuela-arcuri/ fotografia hard emanuela arcuri http://gendarmeria-vaticana-58.alumblogs.net/vaticana-vamuseo-vaticani/ vaticana vamuseo vaticani http://arena-quotidiano-53.alumblogs.net/2008/08/14/tryp-macarena-sevilla/ tryp macarena sevilla http://nissan-murano-43.alumblogs.net/wordpress/lampadario-moderno-murano/ lampadario moderno murano http://sublime-romanticismo-29.mygreatblogs.net/wordpress/sublime-thumbs/ sublime thumbs http://scaccia-insetto-7.alumblogs.net/scaccia-animale-ultrasuono/ scaccia animale ultrasuono http://louise-witton-74.alumblogs.net/wordpress/saint-louise/ saint louise http://alfredo-annese-14.mygreatblogs.net/wordpress/alfredo-liguori-giovanni-capodanno/ alfredo liguori giovanni capodanno http://credible-information-29.alumblogs.net/information-warfare/ information warfare http://murano-giglio-26.alumblogs.net/blog/entry/murano-giglio/ murano giglio http://calice-ligure-4.blogskine.net/wordpress/hotel-riva-ligure/ hotel riva ligure http://cognome-85.blogskine.net cognome http://istituto-nautici-69.blogskine.net istituto http://eliminare-42.alumblogs.net/node/eliminare-fast-track/ eliminare fast track http://retribuzione-apprendistato-87.alumblogs.net/apprendistato-e-maternita/ apprendistato e maternita http://score-music-61.blogskine.net/2008/09/07/remixare-musicaalbergo-san-remo/ remixare musicaalbergo san remo http://innocente-17.blogskine.net innocente http://aiuti-spaces-46.alumblogs.net/2008/08/08/lives-spaces/ lives spaces
Een lijst met theorievragen die Professor Dillen reeds gesteld heeft, afkomstig uit de examenvragenbundel van Cudi.
 
=== Euclidische meetkunde ===
 
# Je mag zeker een vraag verwachten waarbij je ofwel de oriëntatiebewarende ofwel de oriëntatieomkerende isometrieën van <math>\mathbb{E}^2</math> of <math>\mathbb{E}^3</math> moet classificeren.
# Definieer rotatie in <math>\mathbb{E}^2</math> en <math>\mathbb{E}^3</math>.
# Definieer spiegeling en bewijs dat het een isometrie is.
#
#* Definieer schroefbeweging en rotatie in <math>\mathbb{E}^3</math> en geef uitgebreid commentaar.
#* Bewijs dan dat elke oriëntatiebewarende isometrie van <math>\mathbb{E}^3</math> een translatie of schroefbeweging is.
# Zij <math>F : \mathbb{E}^n \to \mathbb{E}^n</math> een willekeurige afbeelding. Bewijs dat F een isometrie is als en slechts als d(F(p), F(q)) = d(p,q) voor alle <math>p,q \in \mathbb{E}^n</math>.
# Bewijs de volgende stelling: Zij F een isometrie. Dan bestaat er juist één isometrie G en juist één translatie <math>t_b</math> zodat
#* <math>F = t_b \circ G</math>
#* <math>V(G) \neq \emptyset</math>
#* <math>G_* b = b</math>
#: Bovendien is dan ook <math>t_b \circ G =G \circ t_b</math> en V(G) is een affiene deelruimte in de richting van <math>\ker (F_* - I)</math>.
 
=== Krommen ===
 
# Geef het Frenetstelsel voor vlakke krommen en formuleer de congruentiestelling. Vergelijk het Frenetstelsel van congruente krommen met dat van booglengtegeparamatriseerde krommen.
# Bewijs dat een reguliere kromme in <math>\mathbb{E}^3</math> een vlakke kromme is als en slechts als de torsie 0 is.
#
#* In <math>\mathbb{E}^2</math>. Toon aan dat de kromming de kromme volledig bepaalt.
#* Definieer gesloten krommen en de rotatie-index en verklaar hoe de rotatie-index bepaald kan worden uit de kromming.
#
#* Definieer de cirkelschroeflijn en de cilinderschroeflijn en verklaar de naamgeving.
#* Geef de nodige en voldoende voorwaarden voor de kromming en torsie zodat een reguliere kromme een cilinderschroeflijn is.
#
#* Definieer het Frenet-referentiestelsel voor ruimtekrommen en bewijs de formules van Frenet.
#* Formuleer en bewijs de congruentiestelling voor ruimtekrommen.
# Formuleer en bewijs de congruentiestelling voor booglengtegeparametriseerde vlakke krommen.
# Rechten realiseren de kortste afstand tussen twee punten. Behandel uitvoerig.


== Tussentijdse toetsen ==
== Tussentijdse toetsen ==

Versie van 28 sep 2008 11:09

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Tussentijdse toetsen

2006-04-??

  1. Geef een basis voor de richting van de volgende affiene deelruimte van 𝔸4:
    x1−x3−x4=−1
    x1+x2−2x3=1
    x2−x3+x4=2
    x1+3x2−4x3+2x4=5
  2. Zij S een niet-lege deelverzameling van 𝔸n. Toon aan: S is een affiene deelruimte van 𝔸n als en slechts als voor elk tweetal punten p, q van S de verbindingsrechte pq tot S behoort.
  3. Zij b, c twee vaste punten in 𝔸2 en L een vaste rechte die bc snijdt in p. Toon aan dat de zwaartepunten van de driehoeken abc, waarbij a varieert op L \ {p}, steeds tot een vaste rechte behoren. Doe dit
    • analytisch;
    • synthetisch.
    (Voor het synthetisch bewijs mag je alle eigenschappen uit de cursus en de oefenzittingen gebruiken, maar zeg er wel bij welke eigenschap je gebruikt.)
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