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m11  1. Let f, g: A >R be integrable.
a. For any partition of and any subrectangle of , show that and and therefore and . 
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m13  14. If is a closed rectangle, show that is Jordan measurable if and only if for every there is a partition of such that , where consists of all subrectangles intersecting and consists of allsubrectangles contained in . 

m54  (a) Let A C Rn be an open set such that boundary A is an (n  1) dimensional manifold. Show that N = AU boundary A is an dimensional manifold with boundary. (It is well to bear in mind the following example: if A = {x ЄRn}: x < 1 or 1 < x < 2}, then N = AU boundary A is a manifold with boundary, but ∂ N ≠ boundary A.
(b) Prove a similar assertion for an open subset of an ndimensional manifold. 
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m159  a. If c: [0, 2π] x [1, 1] > c: [0 , 2π] x [1, 1] > R3 is defined by c (u,v) = (2 eos (u) + vsin (u/2) eos (u),
2sin (u) + vsin (u/2) sin (u), veos (u/2)). 
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m160  a. If C is a set of content 0, show that the boundary of C also has content 0.
b. Give an example of a bounded set C of measure 0 such that the boundary of C does not have measure 0. 
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m161  a. If f: a, b> R is nonnegative and the graph of f in the x,y plane is revolved around the axis in R3 to yield a surface M, show that the area of M is 
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m162  a. If f is a differentiable vector field on M C Rn, show that there is an open set AЭM and a differentiable vector field F on A with F(x) = F (x) for xЄM.
b. If M is closed, show that we can choose A = Rn. 
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m163  a. If f : R > R satisfies f1 (a) ≠ 0 for all a €R, show that f is 11 on all of R. 
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m164  a. If M is a kdimensional manifold in Rn and k < n, show that M has measure 0.
b. If M is a closed dimensional manifold with boundary in Rn, show that the boundary of M is ∂M. Give a counterexample if M is not closed.
c. If M is a compact dimensional manifold with boundary in Rn, show that M is Jordanmeasurable. 
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m165  a. Let Ί: Rn > Rn be selfadjoint with matrix A = (aij), so that aij = aji. If f (x) = <Tx, x> =Σ aij xixj, show that Dkf (x) = 2 Σj = 1 akjxj. By considering the maximum of <Tx, x>on Sn1 show that there is xЄSn1 and ^ ЄR with Tx = ^x.
b. If V = {yЄRn: <x, y> = 0}, show that Ί(v) CV and Ί: V and Ί: V > V is selfadjoint.
c. Show that Ί has a basis of eigenvectors. 
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m166  a. Let f : R > R be defined by
F (x) = {x 2 sin 1/x) x ≠ 0,
0 x = 0. 
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m167  a. Let f: Rn > R be a continuously differentiable function. Show that f is not 11.
b. Generalize this result to t the case of a continuously differentiable function f: Rn > Rm with m < n. 
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m168  a. Let g: Rn > Rn be a linear transformation of one of the following types: 
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m169  a. Show that an unbounded set cannot have content 0. 
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m170  a. Show that the set of all rectangles [a1, b1] x . x [an, bn] where each ai and each bi are rational can be arranged into a sequence (i.e. form a countable set).
b. If A C Rn is any set and O is an open cover of A, show that there is a sequence U1, U2, U3,. of members of O which also cover A. 
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m171  a. Show that Theorem 55 is false if M is not required to be compact.
b. Show that Theorem 55 holds for noncompact M provided that w vanishes outside of a compact subset of M. 
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m172  a. Show that this length is the least upper bound of lengths of inscribed broken lines. 
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m173  a. Suppose that f: (0, 1) > R is a nonnegative continuous function. Show that ∫ (0, 1) exists if and only if lim Є> ∫ c 1c f exists.
b. Let An = [1  1/2n, 1  1/2n +1] Suppose that f: (0, 1) >R satisfies ∫Arf = (1)n/n and f(x) = 0 for all x Є Un An. Show that ∫(0,1)f does not exist, but limЄ>∫(Є, 1  Є)f = log 2. 
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m175  An absolute ktensor on v is a function Vk >R of the form w for w Є Ak (V). An absolute kform on M is a function such that n (x) is an absolute ktensor on Mx. Show that ∫Mn can be defined, even if M is not orientable. 
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m219  Applying the generalized divergence theorem to the set M = {XЄrN: x < a} and , find the volume of Sn – 1 = {xЄRn: x = 1} in terms of the dimensional volume of Bn = {x Є Rn: x <1}. (This volume is π n/2 / (n/2)! if is even and 2(n+1)/2π (n1)/2 1∙3∙5∙. ∙ N if is odd.) 
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