Fyll02anal_07_1 - FULLADIO 2H ANALUSH 2006-07 DIAFORISIMES...

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Unformatted text preview: FULLADIO 2H ANALUSH 2006-07 DIAFORISIMES SUNARTHSEIS 1. DÐnontai oi C 2 -tˆxhc sunart seic f (x, y), u = u(x, y) kai v = v(x, y), ìpou o metasqhmatismìc T (x, y) = (u, v) pou orÐzoun oi u, v eÐnai antistrèyimoc. An eÐnai ux = vy kai uy = −vx deÐxte ìti isqÔei fx2 + fy2 = (fu2 + fv2 )(ux vy − uy vx ) 2. An h sunˆrthsh f (x, y, t) eÐnai C 2 -tˆxhc kai ikanopoieÐ thn exÐswsh fx2 − 1 fyt = 0, deÐxte ìti h sunˆrthsh g(x, y, t) = t− 2 f (u, v, w), ìpou u = x , t x2 1 ,w=− ikanopoieÐ thn gx2 − gyt = 0. 4t t 3. An h diaforÐsimh sunˆrthsh h(x, y) epalhjeÔei thn exÐswsh hx + hy = 0 deÐxte ìti kai h φ(x, y) = h(x + g(x − y), x), ìpou g paragwgÐsimh sunˆrthsh (miac metablht c), epalhjeÔei epÐshc thn φx + φy = 0. v=y− 4. 'Estw F = F1 i + F2 j + F3 k, ìpou Fi (tx, ty, tz) = tFi (x, y, z). DeÐxte ìti eÐnai: ∂Fi ∂Fi ∂Fi +y +z = Fi , i = 1, 2, 3. ∂x ∂y ∂z An eÐnai epÐ plèon kai ×F = rotF = 0, deÐxte ìti F = f , ìpou f (x, y, z) = 1 (xF1 + yF2 + zF3 ). 2 5. H olik antÐstash R tri¸n antistˆsewn R1 , R2 , R3 se parˆllhlh sÔndesh dÐnetai apì ton tÔpo: 1 1 1 1 = + + R R1 R2 R3 i) Na deiqjeÐ ìti x ∂R ∂R ∂R + + = ∂R1 ∂R2 ∂R3 1 1 1 2 + 2 + R1 R2 R3 2 / 1 1 1 + + R1 R2 R3 2 ii) An R1 = R2 = 100Ω kai kˆpoia qronik stigm oi timèc touc auxˆnoun katˆ 1Ω/sec kai R3 = 200Ω kai thn Ðdia qronik stigm h tim thc ellat¸netai katˆ 2Ω/sec na breÐte thn antÐstoiqh metabol thc R, me grammik prosèggish. 6. Na upologisteÐ h kateujunìmenh parˆgwgoc parˆgwgoc thc sunˆrthshc √ f (x, y, z) = x2 + 2xz + y 2 sto shmeÐo P (0, 23 , 3 ) katˆ thn kateÔjunsh tou 2 kajètou dianÔsmatoc sto P sthn epifˆneia pou apoteleÐtai apì ta tm mata epifanei¸n me exis¸seic x2 + y 2 = 2z 2 , 0 ≤ z ≤ 1 kai x2 + y 2 + z 2 = 3, √ 1 ≤ z ≤ 3. 7. Na lujoÔn epÐshc oi ask seic tou biblÐou: 14, 15/sel.173 kai 20, 21/sel 174. Parˆdosh mèqri 29-8-2007 ...
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This note was uploaded on 10/02/2009 for the course G 001 taught by Professor Shmmygr during the Spring '07 term at National Technical University of Athens, Athens.

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