By Edwin Bidwell Wilson
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Extra resources for Advanced calculus; a text upon select parts of differential calculus, differential equations, integral calculus, theory of functions; with numerous exercises
C E IR there are only finitely many critical points z of ii) Moreover, if for a < /3 < 'Y the number f3 is the only critical value of E in [a, 'Y], and if Zlt ... , Zk are the only critical points of E with E(zj) /3 having Morse indeces rl, ... , is homotopically equivalent to Ma with k handles of types rlt ... , rk disjointly attached. 4 ') , m=O L:(-I)mCm = 1. 4 ") + (1 + t)Q(t) m=O where Q is a polynominal with non-negative integer coefficients, and the polynom00 inal E Rmtm == 1 is the Poincare polynominal of M, cf.
2(Bj IR") and X E C(r). = r c IRn. 10: Let r be a rectifiable Jordan curve in II('. Then there exists a solution X. 3) characterized by the condition D(X) = XEC(I') inf D(X) < 00, and X parametrizes a disc-type minimal surface of least area spanning f. 22 A. The classical Plateau Problem for disc - type minimal surfaces. 5. Regularity. 3). In order to interpret this solution geometrically we now derive further regularity properties. e. whether it is embedded (or at least locally immersed). While the first question is completely solved by Hildebrandt's regularity result  , for the second question a satisfactory answer can only be given in case n = 3 which corresponds to the physical case.
To rule out the first possibility we introduce em normalization conditions = Zm - id, e =Z - id and use the 2,.. 1 em . TJ d¢ = 0, VTJ E T;d G . o Since Zm --+ Z in L1 ,also em --+ e in L1 ,and also constraints which we may write down explicitly as follows: 2,.. 2,.. 2,.. 1 =1 e d¢ e satisfies these e sin ¢ d¢ = 1 e cos ¢ d¢ =0 . 0 0 0 But if == canst Z (mod 211") ,and is 211"-periodic and monotone, e is of the form = c - ¢, 0 ~ ¢ ~ 211" . c = 11" ,while the second and third condition give e(¢ - ¢o) The first condition now implies 1 =1 2,..
Advanced calculus; a text upon select parts of differential calculus, differential equations, integral calculus, theory of functions; with numerous exercises by Edwin Bidwell Wilson
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