УСЛОВИЕ ЭКСТРЕМАЛЬНОСТИ ПОВЕРХНОСТИ ВРАЩЕНИЯ ДЛЯ ФУНКЦИОНАЛА ТИПА ПЛОЩАДИ

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<ul><li><p> 517.957 + 514.752</p><p>.. , .. </p><p> , - . - - .</p><p> R3, M C2 ,S = . , R3 C2- (x) (x). , [3]</p><p>J1(M) =</p><p>S</p><p>(|x|)dS, J2(M) =</p><p>(|x|)dx,</p><p> x = (x1, x2, x3), dS . - M , </p><p>J(M) = J1(M) J2(M).</p><p> - J1(M) , J2(M) = const. -, [1] - , .</p><p>1. </p><p> M , x3 -, = (t), P t(. 1). P t . - , t (a, b), 0 2, = (t) . </p><p> . 1. . 11. 20072008</p><p>.</p><p>.</p><p>,..,</p><p>20072008</p><p>39</p></li><li><p>r(t, ) = {(t) cos , (t) sin , t},rt = { cos , sin , 1},r = { sin , cos , 0}.</p><p>E = (rt, rt) = 2 + 1, F = (rt, r) = 0, G = (r, r) = </p><p>2.</p><p>dS2 = (2 + 1)dt2 + 2d2.</p><p>. 1</p><p>J1(M) =</p><p>S</p><p>(|x|)dS =2</p><p>0</p><p>d</p><p>b</p><p>a</p><p>EG F 2dt = 2b</p><p>a</p><p>2 + 1dt,</p><p> = ((t)).</p><p>J2(M) =</p><p>(|x|)dx =b</p><p>a</p><p>dt</p><p>2</p><p>0</p><p>d</p><p>0</p><p>d = 2</p><p>b</p><p>a</p><p>h((t))dt,</p><p>40 .. , .. . </p></li><li><p> = ((t)) </p><p>h() =</p><p>0</p><p>(y)ydy.</p><p> - . M J2(M), (t), </p><p>(a) = A, (b) = B,</p><p> , J1(M). , - J(M) = J1(M) J2(M). (t) , , </p><p>J [(t)] =</p><p>b</p><p>a</p><p>(</p><p>2 + 1 + h())dt. (1)</p><p> t, F = F (, ), [2] </p><p>d</p><p>dt(F F) = 0.</p><p>d</p><p>dt</p><p>(</p><p>2 + 1 h() 2</p><p>2 + 1</p><p>)= 0. (2)</p><p> = u. - </p><p>(u)u</p><p>u2 + 1 h(u) (u)uu2</p><p>u2 + 1</p><p>= </p><p>(u)uu2 + 1</p><p> h(u) = .</p><p> = const. </p><p>u2 =</p><p>((u)u</p><p> + h(u)</p><p>)2 1. (3)</p><p>dt = du((u)u</p><p>+h(u)</p><p>)2 1</p><p>.</p><p> . 1. . 11. 20072008 41</p></li><li><p>t = </p><p>du((u)u</p><p>+h(u)</p><p>)2 1</p><p>+ C. (4)</p><p> , C.</p><p> 1. M R3 , </p><p>t = </p><p>du((u)u</p><p>+h(u)</p><p>)2 1</p><p>+ C </p><p> (2) (1).</p><p> M .</p><p>2. </p><p> 1. () = 1, () = 0. </p><p>J [(t)] =</p><p>b</p><p>a</p><p>2 + 1dt.</p><p> (4) </p><p>t = </p><p>du(u</p><p>)2 1</p><p>+ C = archu</p><p>+ C.</p><p> = u = ch</p><p>(t C</p><p>)</p><p> , .</p><p> 2. () = u32 , () = 0. </p><p>J [(t)] =</p><p>b</p><p>a</p><p>1</p><p>2 + 1dt.</p><p>t =</p><p>du1</p><p>2u 1</p><p>+ C.</p><p>42 .. , .. . </p></li><li><p> 1/2 = k, u = k, = sin2 2 </p><p>t = k</p><p>1 d + C =k</p><p>2</p><p>(1 cos )d + C = k</p><p>2( sin ) + C.</p><p> = u = k sin2</p><p>2=</p><p>k</p><p>2(1 cos )</p><p> .</p><p>3. </p><p> - , -. , () = () = 1. , - , , . , .</p><p> 2. (t) (2) t0 (a, b) . t = t0 , .</p><p>. , , t0 = 0 , , t &gt; 0 (t) , t &lt; 0 . </p><p>F (u) =1(</p><p>(u)u+h(u)</p><p>)2 1</p><p>.</p><p> (3) </p><p>u2 = F2(u).</p><p> t1 = , t2 = , &gt; 0. u1 = u(t1),u2 = u(t2), u0 = u(0). </p><p>u1</p><p>u0</p><p>F (z)dz = t1 = ,</p><p>u0</p><p>u2</p><p>F (z)dz = t2 = .</p><p> . 1. . 11. 20072008 43</p></li><li><p> , </p><p>u2</p><p>u1</p><p>F (z)dz = 0,</p><p> F (u) : u1 = u2. - (2).</p><p>Summary</p><p>EXTREMALITY CONDITION OF SURFACE OF REVOLUTIONFOR AREA-TYPE FUNCTIONAL</p><p>V.A. Klyachin, T.V. Tkacheva</p><p>Present article is devoted to investigation of extremal rotation surfaces for squaretype functional. The solutions of differentional Euler-Lagrange equation are obtained.Also, the symmetry property of these surface is proved and demonstrated examplesfunctionals and its corresponding solutions are constructed.</p><p>1. . . . .:, 1989. 312 .</p><p>2. .., .., .. : . .: , 1986.</p><p>3. .. // . . . . 2006. . 70, 4. C. 7790.</p><p>4. .. // .. 260. 1981. 2. . 293295.</p><p>44 .. , .. . </p></li></ul>

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