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Capitolul 1 Ecuatii diferentiale de ordinul ˆıntâi rezolvabile prin ...

Capitolul 1 Ecuatii diferentiale de ordinul ˆıntâi rezolvabile prin ...

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188 CAPITOLUL 6<br />

iar ∂Q<br />

∂n<br />

este <strong>de</strong>rivata normalǎ <strong>de</strong>finitǎ pe ∂Ω <strong>prin</strong>:<br />

∂Q<br />

∂n<br />

∂Q<br />

= · n1 +<br />

∂x1<br />

∂Q<br />

∂x2<br />

· n2 =<br />

= ∂Q<br />

· cosα1 +<br />

∂x1<br />

∂Q<br />

∂x2<br />

· cosα2<br />

cosαi =< n, ei >= ni, i = 1, 2; ∇P ¸si ∇Q sunt funct¸iile vectoriale (gradient¸ii<br />

funct¸iilor P ¸si Q) <strong>de</strong>finite <strong>prin</strong>:<br />

∇P = ∂P<br />

∂x1<br />

∇Q = ∂Q<br />

∂x1<br />

· e1 + ∂P<br />

∂x2<br />

· e1 + ∂Q<br />

∂x2<br />

Pentru <strong>de</strong>monstrat¸ie se calculeazǎ membrul stâng al egalitǎt¸ii (6.10) t¸inând<br />

seama <strong>de</strong> formula <strong>de</strong> integrare <strong>prin</strong> pǎrt¸i (6.9) ¸si se obt¸ine:<br />

<br />

<br />

∂ ∂Q<br />

P · ∆Qdx1dx2 = P<br />

Ω<br />

Ω ∂x1 ∂x1<br />

<br />

= P · cosα1 ·<br />

∂Ω<br />

∂Q<br />

∂x1<br />

<br />

+ P · cosα2 ·<br />

∂Ω<br />

∂Q<br />

∂x2<br />

<br />

∂Q<br />

= P ·<br />

∂x1<br />

· e2<br />

· e2.<br />

+ ∂<br />

∂x2<br />

<br />

ds −<br />

<br />

ds −<br />

<br />

∂Q<br />

dx1dx2 =<br />

∂x2<br />

Ω<br />

Ω<br />

∂P<br />

∂x1<br />

· ∂Q<br />

dx1dx2 +<br />

∂x1<br />

∂P<br />

·<br />

∂x2<br />

∂Q<br />

dx1dx2 =<br />

∂x2<br />

<br />

· cosα2 ds −<br />

· cos α1 +<br />

∂Ω<br />

∂Q<br />

∂x2<br />

<br />

∂P<br />

− ·<br />

Ω ∂x1<br />

∂Q<br />

dx1dx2 +<br />

∂x1<br />

∂P<br />

·<br />

∂x2<br />

∂Q<br />

<br />

dx1dx2 =<br />

∂x2<br />

<br />

∂Q<br />

= P · · cos α1 +<br />

∂Ω ∂x1<br />

∂Q<br />

<br />

· cosα2 ds−<br />

∂x2<br />

<br />

− ∇P · ∇Qdx1dx2 =<br />

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