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Condition |
free/or 0.5$ |
| m602 | Led f, g: A -> R be integrable and suppose f < g. Show that fA f < f Ag. |
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| m604 | Let: A ->R and P be a partition of A. Show that f is integrable if and only if for each subrectangle S the function f / s, which consists of f restricted to S, is integrable, and that in this case f A f = ∑S f S f | S |
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| m605 | Let A= {x, y}: x <0, or x ≥ 0 and y ≠ 0}.
a. If f: A->R and D1f =D2f = 0, show that f is a constant.
b. Find a function f: A->R such that D2f =0 but f is not independent of the second variable. |
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| m606 | Let An be a closed set contained in (n, n + 1). Suppose that f: R ->R satisfies ∫Arf = (−1)n/n and f = 0 outside Un An.. Find two partitions of unity Φ and Ψ such that ∑ǿЄΦ∫Rǿ∙f and ∑ǿЄΦ∫Rψ∙f converge absolutely to different values. |
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| m607 | Let be a continuous real-valued function on the unit circle {x € R2 : |x| =1} such that
f (0, 1) = g(1, 0) = 0 and g(- x )= - g(x).
Define f: R2->R by
F (x) = {|x| . g (x/|x| x ≠0,
0 x = 0.
(a) If x € R2 and h: R ->R is defined by h (t) = f (tx) show that h is differentiable.
(b). show that f is not differentiable at (0, 0) unless g = 0. |
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| m608 | Let A be the set of Problem 1-18. If T = ∑i = 1 (bi − ai) <, show that the boundary of A does not have measure 0. |
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| m609 | Let C C [0, 1] x [0, 1] be the union of all {p/q} x [0, 1] where p/q is a rational number in [0, 1] written in lowest terms. Use C to show that the word ``measure" in Problem 3-23 cannot be replaced with ``content". |
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| m610 | Let C C A x B be a set of content 0. Let A1 C A be the set of all x Є A such that {y Є B: (x, y) Є C} is not of content 0. Show that A1 is a set of measure 0. |
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| m611 | Let A C Rn be an open set and f : A-> Rn a continuously differentiable 1-1 function such that det f1 (x) ≠ 0 for all . Show that f (A) is an open set and f -1: f (A) ->A is differentiable. Show also that f (b) is open for any open set B C A. |
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| m612 | Let A C Rn be an open set and f : A-> Rn a continuously differentiable 1-1 function such that det f1 (x) ≠ 0 for all . Show that f (A) is an open set and f -1: f (A) ->A is differentiable. Show also that f (b) is open for any open set B C A. |
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| m614 | Let f: [0, 1] x [0, 1] -> R be defined by |
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| m615 | Let f: [0, 1] x [0, 1] -> R be defined by |
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| m616 | Let f: A -> Rp be as in Theorem 5-1.
a. If x Є M = g-1(0), let h: U -> Rn be the essentially unique diffeomorphism such that goh (y) = (y n – p + 1 . yn) and h (0) = x. Define f: Rn- p -> f: Rn-p ->Rn by f (a) = h (0, a). Show that is 1-1 so that the vectors f * ((e1)0). f* ((en-p)0) are linearly independent.
b. Show that the orientations μ can be defined consistently, so that M is orientable.
c. If P = 1, show that the components of the outward normal at are some multiple of D1g (x). . . Dng(x). |
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| m617 | Let f: [a, b] -> R be integrable and non-negative, and let Af = {(x, y): a < x < b and 0 < x < f (x)}. Show that Af is Jordan measurable and has area ∫ ba f. |
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| m618 | Let f: [a, b] be an increasing function. Show that {x: f is discontinuous at x} is a set of measure 0. |
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| m619 | Let f: [a, b] x [c, d] -> R be continuous and suppose D 2 f is continuous. Define f (y) = ∫ ba f (x, y) dx. Prove Leibnitz Rule: f1 (y) = ∫ ba D2 f (x, y) dx. |
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| m620 | Let f: A->R be integrable and let g = except at finitely many points. Show that is integrable and f AF = f Ag. |
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| m621 | Let f be defined as in Problem 2-4. Show that Dxf (0, 0) exists for all , but if g ≠ 0, , then Dx + yf (0,0) =Dxf (00 Dx + y f (0, 0) = Dx f (0 , 0) + Dyf (0, 0) Is not true for all x and all y. |
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| m622 | Let f, g: [0, 1] -> R3 be nonintersecting closed curves. Define the linking number l (f, g) of and g by (cf. Problem 4-34 |
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| m623 | Let f : R x R -> R be differentiable. For each x € R defined gx: R -> R By gx (y) = f(x,y). Suppose that for each x there is a unique y with gx1(y) =0; let c(x) be this y. |
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