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About 9000 results. 1294 free access solutions
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free/or 0.5$ |
| m499 | If C is a bounded set of measure 0 and ∫ AXC exists, show that ∫ AXC = 0. |
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| m500 | If f: A ->R is integrable, show that |f| is integrable and |f A f| <f A |f| |
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| m501 | If f: [a, b] x [c, d] -> R is continuous and D2f is continuous, define F (x, y) = ∫xa (t,y) dt
a. Find D1F and D2F.
(b) If G (x) = ∫ g(x) f (t, x) dt, find G1 (x). |
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| m502 | If f: A->R is non-negative and ∫ Af = 0, show that B = {x: f (x) ≠ 0} has measure 0. |
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| m503 | If f: R2->R and D2f =0 and D2f =0, show that f is independent of the second variable. If D1f = D2f =0, show that f inconstant. |
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| m504 | If f : Rn -> R is differentiable and f (0) = 0, prove that there exist gi: Rn -> R such that f (x) = |
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| m505 | If f: Rn -> Rn, the graph of f is {(x, y): y = f (x)}. Show that the graph of is an -dimensional manifold if and only if is differentiable. |
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| m506 | If g: Rn -> Rn and detg1 (x) ≠ 0, prove that in some open set containing we can write g = to gn 0 ∙ ∙ ∙ o g1, 0.., where is of the form gi(x) = (x1, ∙ ∙ ∙ Fi (x) , ∙ ∙ ∙ Xn), and T is a linear transformation. Show that we can write g = gn o ∙ ∙ ∙ 0g1 if and only if g1 (x) is a diagonal matrix. |
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| m507 | If is continuous, show that |
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| m508 | If A is a Jordan measurable set and ε > 0, show that there is a compact Jordan measurable set C C A such that ∫ A − C1 < ε. |
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| m509 | If M C R n is an orientable (n - 1)-dimensional manifold, show that there is an open set A C Rn and a differentiable g: A-> R1 so that M = g-1 (0) and g1 (x) has rank 1 for x ЄM. |
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| m510 | If M is an -dimensional manifold-with-boundary in Rn, define μ as the usual orientation of M x = Rnx (the orientation μ so defined is the usual orientation of M. If xЄ∂M, show that the two definitions of n (x) given above agree. |
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| m511 | If M is an -dimensional manifold (or manifold-with-boundary) in R n, with the usual orientation, show that ∫ fdx1 ^ . ^ dx n, as defined in this section, is the same as ∫ M f, as defined in Chapter 3. |
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| m512 | If M is an -dimensional manifold in Rn, with the usual orientation, show that dV = dx1^ . . . ^ dxn, so that the volume of M, as defined in this section, is the volume as defined in Chapter 3. (Note that this depends on the numerical factor in the definition of w ^ n.) |
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| m513 | If M is a k-dimensional manifold with boundary, prove that ∂M is a (k - 1) -dimensional manifold and M - ∂M is a k=dimensional manifold. |
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| m514 | If M is an oriented one-dimensional manifold in RN and c: [0, 1] ->M is orientation-preserving, show that |
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| m515 | If M1CRN is an -dimensional manifold-with-boundary and M 2 C M1 - ∂M1 is an -dimensional manifold with boundary, and M1, M2 are compact, prove that |
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| m526 | If there is a nowhere-zero k-form on a k -dimensional manifold M, show that M is orientable |
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| m527 | If w is a (k- 1) -form on a compact k-dimensional manifold M, prove that ∫Mdw =0. Give a counter-example if M is not compact. |
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| m601 | Le Ei, i = 1,., k be Euclidean spaces of various dimensions. A function f: E1 X. X Ek->Rp is called multi linear if for each choice of xj € Ej, j ≠ I the function f: Ei->Rp defined by g(x) = f(x1,.,xi-1, x,xi +1, ., xk) is a linear transformation.
(a) If is multi linear and i ≠ j, show that for (h=(h1, ., hk), with hi € Ei, we have
Prove that
(b) Df (a1,., ak) x1, ., xk) = =1 f(a1,., ai-1, xi, ai+1,., ak) |
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