课程名称︰高等微积分二 课程性质︰数学系必修 课程教师︰陈俊全 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2011/4/26 考试时限（分钟）：170分 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : Choose 5 from the following 6 problems 1. Discuss the pointwise convergence and uniform convergence of x (a) fn(x) = ─────── , 0 < x < ∞ 1 + n^2 x^2 n x (b) gn(x) = Σ ─────── , 0 < x < ∞ k=1 1 + k^2 x^2 2. Suppose {fn} is an equicontinuous sequence of functions from a compact set Ａ to Ｒ(实数), and {fn} converges pointwise on Ａ. Prove that {fn} converges uniformly on Ａ. 3. Let f: [0,1] → Ｒ(实数) be continuous. 1 (a) Assume ∫ f(x) x^k dx = 0 for k = 0,1,2,... 0 Prove that f(x) = 0 on [0,1] (b) Assume f(0) = f(1) = 0. Prove that there are polynomials Ｐn such that sup |f(x) - x(1-x)Ｐn(x)| → 0 as n → ∞ x属於[0,1] ∞ ∞ 4. Prove that Σ a_k = Ａ(Ｃ,1) implies Σ a_k = Ａ(Abel). k=0 k=0 5. (a) What is the contraction mapping principle? (b) Let Ａ = {(x,y)| 0≦x≦1, 0≦y≦1}, k(x,y) : Ａ→Ｒ(实数) be continuous, and a(x) : [0,1]→Ｒ(实数) be continuous. Assume |k(x,y)| < 1 for each (x,y) 属於 Ａ. Prove that there is a unique continuous real-valued function f(x) on [0,1] such that 1 f(x) = a(x) + ∫ k(x,y) f(y) dy. 0 6. Let f : Ｒ^2(实数^2)→Ｒ^3(实数^3) be defined by f(x,y) = (x^3 y, g(x,y), x^4 y^2). Assume f is differentiable at (1,2). (a) Prove that g(x,y) is continuous at (1,2). dg dg (b) Assume further that ─(1,2) = 5 and ─(1,2) = 1. (partial不好打 用d代替) dx dy Compute Ｄf(1,2). --

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