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Condition |
free/or 0.5$ |
| 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 . |
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| 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 n-dimensional 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 non-negative 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 1-1 on all of R. |
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| m164 | a. If M is a k-dimensional 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 counter-example if M is not closed.
c. If M is a compact -dimensional manifold with boundary in Rn, show that M is Jordan-measurable. |
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| m165 | a. Let Ί: Rn -> Rn be self-adjoint 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 Sn-1 show that there is xЄSn-1 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 self-adjoint.
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 1-1.
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 5-5 is false if M is not required to be compact.
b. Show that Theorem 5-5 holds for noncom-pact 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 non-negative continuous function. Show that ∫ (0, 1) exists if and only if lim Є-> ∫ c 1-c 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 k-tensor on v is a function Vk ->R of the form |w| for w Є Ak (V). An absolute k-form on M is a function such that n (x) is an absolute k-tensor 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 (n-1)/2 1∙3∙5∙. ∙ N if is odd.) |
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