ZA/UM: FUNktsioonid

Sunday, January 30, 2011

FUNktsioonid

Selle asemel, et mõnusalt nautida oma 5 koolivaba päeva, tegelesin ma sellega, et taastada kõik need mõnusad ja nauditavad asjad oma arvutis, mis ühel kenal ööl, kui mina pahaaimamatult magasin, ära kadusid.
Niisiis, selle asemel, et siia populaarteaduslikku pikka mahlast artiklit kirjutada, hankisin ma uuesti kõik eluks vajalikud programmid, et end vähegi inimese moodi tunda. Üht konkreetset hirmus kallist (nii ma olen kuulnud) programmi paigaldades mõtlesin jagada järgmist:
*Funktsioonid: Ma usun, et kõik on koolis näinud suurt rõõmu koordinaatteljestikku trigonomeetrilisi funktsioone kritseldades. Mis tundub seksikam, kui võrrelda siinus- ja koosinusgraafikut omavahel? (y=sinx on siis punakam ja y=cosx sinakam)

Pole just eriti seksikas.
Kolmemõõtmelised graafikud aga on juba hoopis teine lugu. (Samas pean ma aga tänulik olema, et mina neid kunagi pliiatsiga paberile tegema ei pea...)
Funktsioon x^2 + y^3 - z^2 = 0 (kuradi blogspot...need ülakonksud viitavad siis astendamisele)

Siin on graafik pealtvaates, sest nii näeb ta lihtsalt parem välja. Ehk siis y-telg sõidab Sinu poole, x-telg liigub paremasse alanurka ning z-telg vasakule alanurka.

Ja nüüd siis mitu erinevat funktsiooni ühel graafikul:
Sinx + Sin9x/5=0 ning Cosx + Cos9x/5=0

Ning lihtsalt äramainimise pärast:
Funktsioon: Kui hulga X mistahes elemendile x on vastavusse seatud hulga Y mingi üks kindel element y, siis öeldakse, et on antud funktsioon f, mis toimib hulgast X hulka Y ning märgitakse:
f: X->Y

Kuna lahedad näited on mitme muutujaga, siis:

Mitme muutuja funktsioon: Kui igale (x;y) on vastavusse seatud muutuja z kindel väärtus, siis muutujat z nimetatakse kahe muutuja x ja y funktsiooniks ja tähistatakse z=f(x,y). Kahe muutuja funktsiooni graafikuks on pind ruumis.

Ehk siis kui nüüd kokkuvõtvalt märkida, et
*ühe muutuja funktsiooni graafik on joon tasandil
*Kahe muutuja funktsiooni graafik on pind ruumis
...
*n-muutuja funktsiooni graafikut saab kujutada n+1 mõõtmelises ruumis

Teise teemana mainiksin täna sõlmesid.
Sõlm on põhimõtteliselt väänatud ring 3mõõtmelises eukleidilises ruumis (mainin praegu lihtsalt ära, et eksisteerivad veel Lobatševski ja Riemanni ruumid, millest räägime teisel korral)

Matemaatiliselt erinevadki sõlmed tavaelu sõlmedest sellepoolest, et neil pole algust ja lõppu...ei toimu millegi sidumist/sõlmimist, vaid pigem väänamine.
Kõige lihtsam sõlm on selline:

Sellist sõlme saaks samamoodi kujutada graafiliselt koordinaatteljestikul ning seda kirjeldavad parameetrilised võrrandid on vastavalt:

x=(2+cos3t)cos2t
y=(2+cos3t)cos2t
z=sin3t

Kui aga rääkida sõlmede sõlmest, siis :

See sõlm on erandlik, sest on keritud ümber ringikujulise silindri.

Mul said sigaretid otsa, tänane peatükk lõppeb.
Kõik pildid on koostatud programmiga Wolfram Mathematica 7.0.
Ning neile, kellele miski ei meeldinud selline link

4 kommentaari:

Aleksander RostovJanuary 30, 2011 8:35 PM
Oh hot sweaty math, ma ei jõua seksikat pikka ja mahlast Lobatševski & Riemanni ruumide loengut ära oodata.

Also piran Mathematica ja hakkan seda kasutama maalimises ja custom brushide tegemises.
ReplyDelete
MartinJanuary 30, 2011 8:38 PM
Hm.
Ma ei ütle, et mul midagi selle värgi vastu on. See on kahtlemata mõnusalt ootamatu. Kuigi HOOLIKAS LUGEJA vaatab seda kahtlemata äratundmisrõõmuga - foreshadowing justkui andis mõista, et saab olema. Lustliku pillilooga, ma arvan, et oleks parem, aga mõttetu on mõelda, et ma teaks, kui kerge või raske ühte lustlikku pillilugu nii lühikese vahega kirjutada on. Ma arvan, et sa võiksid anda headele inimestele väikese lingi sinna, kus su vanad pininad on. Oli küll selline koht mumeelest.
Also, sa võiksid kirjutada noortele lustliku loo teemal "Siinus, koosinus, tangens - kes nad on ja kust nad tulevad ja miks nendega asju teha saab."
ReplyDelete
V.January 30, 2011 8:44 PM
On olemas link, kuhu ma vanal heal ajal usinasti postitasin, kuid viimasel ajal mitte enam eriti.
Kõik on aga seal üleval: vertigothomas.blogspot.com
ReplyDelete
V.January 30, 2011 8:46 PM
Ning olemas on ka juba Mathematica 8 aga see on veel päris värske värk...
6 on ka ok...alla selle ma väga ei soovita
ReplyDelete

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Apocalypticat mäletate? See oli selline tšellodega soome tšellode ettevõtmine, kes mängisid classy nägudega kavereid ja müüsid igast kahtlastes maades roppult albumeid.

Ja siis meil oli lahedalt rögiseva häälega ida-saksa pungitar Nina Hagen (ta oli kojn).

Ja siis meil oli vahva vasakpoolne ansambel Rammstein [ka idasaksamaalt (fuck the fall, bring back the wall!)], mis oli iseenesest normaalne ettevõtmine ja nägi hea välja, aga albumikaupa oli kuidagi üksluisevõitu kuulata.

Ja kokku sai mingi jumala vägeva asja nagu. Check it out!

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