课程名称︰微积分甲下 课程性质︰必修 课程教师︰王金龙 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2011/3/24 考试时限（分钟）：40 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 2 2 A. Let u(x,y) : R → R be a C function. Express u + u in polar coordinate. xx yy 2 1 B. Let f(x,y) : R → R be a C function. d b b Show that ─ ∫ f(x,y) dx = ∫ f (x,y) dx. dy a a y Hint. Every continuous function defined on a bounded and closed set n D ㄈ R is uniformly continuous. (ㄈ：包含於) 3 C. Consider the line integral ∫ L, where L = Adx + Bdy + Cdz defined on R . Γ 3 Prove that L is exact , that is, L = df for somr f on R , if and only if the integral is independent to the path, which means that it only depends on the end points of Γ. (Note that we assume A, B, C are all continuous) ┌ x = cos t D. (a) Evaluate ∫ zdx + xdy + ydz over the arc of the helix │ │ y = sin t │ └ z = t from (1,0,0) to (1,0,2π). ydx + xdy 1 (b) Evaluate ∫ ─────── over the arc of y = sin ─ 1 + (x^2)(y^2) x from (1/2π,0) to (1/π,0). Hint. Check if the differential form is exact. --

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