Calculus o f Variation s 7 Example 2 (Sherk' s surface) . Le t u s see k a solutio n whic h ha s th e special for m u(x,y) = f(x) -f - g(y), wher e / an d g are eac h function s of a single variable. The n (5' ) become s (l + g'(y)2)f"(x) + (l + f'(x) 2 )g"(y)=0. This differentia l equatio n split s int o tw o ordinar y differentia l equa - tions b y separatin g th e variables : f"(x) g"(y) l + /'(z) 2 + l + 7'(2/) 2 ' so that ther e must b e a constant c such that f"(x) _ g"{y) i + /'(x)2~ \ + g '{ y y~c- Each o f these equation s ca n b e easil y integrated . Thu s arctan/^x) = ex (ignor e th e additiv e constant) f'{x) = ta n ex f(x) lo g | coscx| (ignor e th e additiv e constant) . Likewise, g{y) = -\og\ cos cy\. c Thus th e minima l surfac e exampl e w e obtain ha s th e for m z = u(x, y) = - lo g c Example 3 (Catenoid) . Thi s i s a surface of revolution whic h i s als o a minima l surface . Ther e ar e a coupl e o f approache s t o this . I t i s not difficul t t o assum e tha t u u(r), wher e r = \Jx 2 + y 2 , deriv e the correspondin g ordinar y differentia l equatio n fro m (5 7 ), an d the n integrate it . Th e othe r approac h i s to regar d thi s a s a single-variabl e problem, wher e th e unknow n functio n y u(x) 0 is regarded a s a curve to b e revolve d aroun d th e x-axis . The resultin g are a i s the n rb 2TT / u\/l + u' 2 dx. cos cy cos ex
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