Question
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  1. (6 boda) Provjerite da je vektorsko polje
    konzervativno, nađite mu potencijal i pomoću njega izračunajte .

Ask by Marshall Schmidt. in Croatia
Jan 21,2025

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Tutor-Verified Answer

Answer

Vektorsko polje je konzervativno. Potencijalna funkcija je . Vrijednost linijskog integrala od točke do je 19.

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Bonus Knowledge

Da bismo provjerili je li vektorsko polje konzervativno, trebamo provjeriti je li njegovo rotaciono polje jednako nuli. Izračunavamo :
Nakon izračuna, dobijamo da je , što znači da je konzervativno polje.
Sada pronalazimo potencijal tako da . Integrirajući komponente redom, dolazimo do:
gdje je C konstanta integracije.
Za izračun integral možemo koristiti teorem potencijala:
Izračunavajući na tim točkama, dolazimo do rezultata.

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