<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Osder, Laplacen DY, harmoniset fkt. (Lis\303\244n\303\244 "yht\303\244l\303\266iden muokkausta")</Font></Text-field>

a) Osoita, ett\303\244 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toteuttaa Laplacen yht\303\244l\303\266n.

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

simplify(%);

Yhdistelm\303\244n subs, eval sijasta voidaan komentaa pelkk\303\244 eval. Mutta argumentit vastakkaisessa j\303\244rjestyksess\303\244 kuin subs-komennossa.

Ajatus: eval(lauseke, ehdot) # evaluoi lauseke n\303\244iden ehtojen vallitessa.

eval(LapDy,u(x,y)=u);simplify(%);

(Joskus Maple p\303\244\303\244tt\303\244\303\244 siirty\303\244 "document moodista" "worksheet moodiin", mik\303\244 n\303\244kyy punaisina sy\303\266ttein\303\244. T\303\244ll\303\266in p\303\244\303\244tet\303\244\303\244n puolipisteeseen (Maple kyll\303\244 huomauttaa.))

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIilGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdRJmZhbHNlRicvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSJ+RidGOS9GPEY/Rj0vRkFGP0ZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlVGT0ZPRk9GOQ==Oletetaan, ett\303\244 funktioilla u(x,y) ja v(x,y) on jatkuvat 2. osittaisderivaatat ja ne toteuttavat Cauchy-Riemannin yht\303\244l\303\266t. (N\303\244in on erityisesti, jos u ja v ovat analyyttisen

funktion Re- ja Im-osat.)

Osoita, ett\303\244 u ja v ovat harmonisia.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTA==

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVElZGlmZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJENSMUYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieEYnRi9GMkZBLUY+Ni1RIn5GJ0ZBRkMvRkdGRUZIRkpGTEZORlBGUi9GVkZURmVuRmVuRmVuRmVuLUY+Ni1RIiNGJ0ZBRkNGaG5GSEZKRkxGTkZQRlJGaW4tRiw2JVFgb35kaWZmfnNvdmVsbGV0dHVuYX55aHR8Xnd8X3VsfF53fGF2fF53fGF2bn5vcGVyb2l+cHVvbGl0dGFpbn4odXVzaXNzYX52ZXJzaW9pc3NhKUYnRi9GMg==

JSFH

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2I1EhRictSShtYWN0aW9uR0YkNiVGKy8lK2FjdGlvbnR5cGVHUTptYXBsZXNvZnQuY29tOmxhYmVsKEw5OTgpRicvJSV2aWV3R1EmbGFiZWxGJy1JI21vR0YkNi1RIitGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkEvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGUC1GMDYlRisvRjNROm1hcGxlc29mdC5jb206bGFiZWwoTDk5NylGJ0Y1LUY5Ni1RIjtGJ0Y8Rj8vRkNRJXRydWVGJ0ZERkZGSEZKRkwvRk9RJjAuMGVtRicvRlJRLDAuMjc3Nzc3OGVtRictRjk2LVEifkYnRjxGP0ZCRkRGRkZIRkpGTEZmbi9GUkZnbkZqbi1GOTYtUSIjRidGPEY/RkJGREZGRkhGSkZMRmZuRl1vLUYsNiVRRn5NeXxed3xhdnN+eWh0ZWVubGFza3V+dG9pbWlpfnB1b2xpdHRhaW4uRicvJSdpdGFsaWNHRmVuL0Y9USdpdGFsaWNGJw==

Alempana tutkitaan, miten yleisesti laskutoimitukset ja muut operaatiot toimivat puolittain. Osoittautuu, ett\303\244 t\303\244ss\303\244 on puutteita. Siksi voitaneen suositella tinkimist\303\244 eleganssista ja pit\303\244ytymist\303\244 lhs-rhs-tyyliin (vrt. seuraava kohta:"Yht\303\244l\303\266iden muokkausta"). Toki toimiviksi tiedetyt voi hoitaa yll\303\244 olevalla eleganssilla. Hyv\303\244 puoli on, ett\303\244 toimimattomista tulee huutavat punaiset virheilmot.

c) Sekaderivaattojen yhtyminen.

T\303\244llaisilla laskuilla voidaan vahvistaa uskoa yleiseen lauseeseen, joka kertoo sekaderivaattojen yhtymisest\303\244 tietyill\303\244 (varsin yleisill\303\244) oletuksilla.

PkkiZkc2IiwoKiZJInhHRiQiIiRJInlHRiQiIiMiIiIqJkYnIiIlLUkkc2luR0YkNiNGKUYrRistSSRjb3NHRiQ2IyomRidGK0YpRitGKw==

PkklZnh4eUc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiZJImZHRiRJInhHRiRGKkkieUdGJA==

PkklZnh5eEc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiZJImZHRiRJInhHRiRJInlHRiRGKg==

PkklZnl4eEc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiZJImZHRiRJInlHRiRJInhHRiRGKw==