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### hw6

Course: M 509, Fall 2009
School: UPenn
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Word Count: 404

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509 Math Assignment 6 Dr. DeTurck Due Thursday, March 26 Let's start with a couple of problems from the current College Math Journal. 1. Let f be a function defined on an open interval I containing the interval [a, b], and such that f (x) 0 for all x I. Show that 1 1 f (a + (b - a)y) dy 0 0 f 3a + b b - a + y 4 2 dy. 2. Let k be a positive integer. Show that lim n2 n 1 0 xn 1 dx = k . k + x2k + + xnk...

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509 Math Assignment 6 Dr. DeTurck Due Thursday, March 26 Let's start with a couple of problems from the current College Math Journal. 1. Let f be a function defined on an open interval I containing the interval [a, b], and such that f (x) 0 for all x I. Show that 1 1 f (a + (b - a)y) dy 0 0 f 3a + b b - a + y 4 2 dy. 2. Let k be a positive integer. Show that lim n2 n 1 0 xn 1 dx = k . k + x2k + + xnk 2 1+x m=0 (1 + km) 3. Let a, b, and c be positive real numbers with a + b + c = 3. Show that 1 1 1 a2 + b 2 + c 2 + + 1+2 . a b c 3abc 4. Evalute the integral 0 0 x2 + y 2 e-2xy dx dy 1 + (x2 - y 2 )2 using the change of variables u = x2 - y 2 , v = 2xy. 5. Evaluate T x-y x+y 2 dx dy where T is the triangle with vertices (0, 0), (0, 1), (1/2, 1/2) by using an appropriate change of variables. 6. A k-form is called decomposable if there are one-forms 1 , . . . , k such that = 1 k . (a) If , . . . , is a basis for (R4 ) , show that + is not 1 4 1 2 3 4 decomposable. (b) Show that every 2-form on R3 is decomposable. (c) Show that every n - 1-form on Rn is decomposable. 7. Let p V , where V is a vector space dimension of n and = 0. (a) Show that the set of one-forms V such that = 0 is a subspace of V . (b) Show that the dimension of this subspace is at most p. Can it be strictly less than p? What can you conclude about if the dimension is exactly p? (Try the cases p = 1, 2 first). 8. The kernel of a 2-form 2 V is defined to be ker() = {v V | (v, w) = 0 for all w V }. (a) Show that on R4 , where v1 , . . . , v4 is a basis for R4 and 1 , . . . , 4 is the dual basis for (R4 ) , that ker(1 2 ) is the span of {v3 , v4 } and ker(1 2 + 3 4 ) = {0}. (b) More generally, show that the kernel of a 2-form on any V has even codimension. (Infer that the kernel of any 2-form on an odd-dime...

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