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Questions and Answers of Numerical Analysis

Find a parametrization (É¸, E) of the ellipsoidwhich is smooth off the topological boundary Ï‘E.
a) Suppose that E is a two-dimensional region and that S = {(x, y, z) ˆˆ R3: (x, y) ˆˆ E and z = 0}. Prove thatand thatfor each continuous g : E †’ R.b) Let f : [a, b] †’ R be a Cp
Suppose that Ïˆ{B) and É¸(E) are Cp surfaces and that Ïˆ = É¸ o Ï„, where Ï„ is a C1 function from B onto Z.a) If (f, B) and (0, E)
Suppose that f: Î²3(0, 0) †’ R is differentiable with ||f(x, y)||
Suppose that ɸ(E) is a Cp surface and that (x0, y0, z0) = ɸ(u0, v0)* where (u0, v0) ∈ E°. If Nɸ (u0, v0) ≠ 0, prove that ɸ (E) has a tangent plane at (x0, y0, z0)-
Let ψ(B) be a smooth surface. Set E = ||ψu||, F = ψu and G = ||ψu||. Prove that the surface area of S is ∫B √E2G2 - F2d(u, v).
Suppose that S is a Cl surface with parametrization (ϕ, E) which is smooth at (x0, y0, z0) = ϕ(x0, v0). Let (ψ, I) be a parametrization of a C1 curve in E which passes through the point (u0, v0)
For each of the following, find a (piecewise) smooth parametrization of ϑS which agrees with the induced orientation, and compute ∫ϑS F • Tds. a) S is the truncated paraboloid y = 9 - x2 - z2,
For each of the following, compute ∫∫S F ∙ ndσ. a) S is the truncated paraboloid z = x2 + y2, 0 < z < 1, n is the outward-pointing normal, and F(x, y, z) = (x, y, z). b) S is the truncated
For each of the following, compute ∫∫S ω. a) S is the portion of the surface z = x4 + y2 which lies over the unit square [0, 1] × [0,1], with upward-pointing normal, and ω = x dy dz + y dz dx
Suppose that Ïˆ{B) and Ï•(E) are Cp surfaces and that Ïˆ = Ï• o Ï„, where Ï„ is a C1 function from B onto E.a) If {Ïˆ, B) and
Let E be the solid tetrahedron bounded by x = 0, y = 0, z = 0, and x + y + z = 1, and suppose that its topological boundary, T = Ï‘E, is oriented with outward-pointing normal. Prove for all
Let T be the topological boundary of the tetrahedron in Exercise 13.4.5, with outward-pointing normal, and S be the surface obtained by taking away the slanted face from T (i.e., S has three
Suppose that S is a smooth surface.a) Show that there exist smooth parametrizations (ϕj, Ej) of portions of S such that S = UNj=1 ϕj(Ej).b) Show that there exist nonoverlapping surfaces Sj with
For each of the following, evaluate ∫C F ∙ T ds. a) C is the topological boundary of the two-dimensional region in the first quadrant bounded by x = 0, v = 0, and y = y/4 - x2, oriented in the
Let E be a set in Rm. For each u: E †’ R which has second-order partial derivatives on E, Laplace's equation is defined bya) Show that if u is C2 on E, then Î”u = ˆ™
For each of the following, evaluate ∫C ω. a) C is the topological boundary of the rectangle [a. b] × [c, d], oriented in the counterclockwise direction, and ω = (f(x) + y) dx + xy dy, where f:
For each of the following, evaluate ∫∫S F-ndσ, where n is the outward-pointing normal. a) S is the topological boundary of the rectangle [0, 1] × [0, 2] × [0, 3] and F(x, y, z) - (x + ez, y +
For each of the following, find ∫∫S ω, where n is the outward-pointing normal. a) S is the topological boundary of the three-dimensional region enclosed by y = x2, z = 0, z = 1, y = 4, and ω =
a) Prove that if E is a Jordan region whose topological boundary is a piecewise smooth curve oriented in the counterclockwise direction, thenb) Find the area enclosed by the loop in the Folium of
a) Show that Green's Theorem does not hold if continuity of P, Q is relaxed at one point in E. [Consider P = y/(x2 + y2), Q = -x/(x2 + y2), and E = B1(0, 0).] b) Show that the Divergence Theorem does
This exercise is used in Section 13.6. Suppose that V is a nonempty, open set in R3 and that F: V †’ R3 is C1. Prove thatfor each x0 ˆˆ V, where n is the outward-pointing normal
Let F, G: R3 → R3 and f : R3 → R be differentiable. Prove the following analogues of the Sum and Product Rules for the "derivatives" curl and divergence. a) × (F + G) = ( × F) + ( × G) b)
Let E Š‚ R3. Recall that the gradient of a C1 function f: E †’ R is defined bya) Prove that if f is C2 at x0, then curl grad f(x0) = 0.b) If F: E R3 is C1 on E and C2 at x0 ˆˆ E, prove that
For each of the following, evaluate ∫C F ∙ T ds. a) C is the curve formed by intersecting the cylinder x2 + y2 = 1 with z = - x, oriented in the counterclockwise direction when viewed from high
Suppose that E satisfies the hypotheses of the Divergence Theorem and that S satisfies the hypotheses of Stokes's Theorem.a) If f : S †’ R is a C2 function and F = grad f on S, prove thatb)
For each of the following, evaluate ∫∫S curl F • ndσ. a) S is the "bottomless" surface in the upper half-space z > 0 bounded by y = x2, z = 1 - y, n is the outward-pointing normal, and F(x, y,
For each of the following, evaluate ∫∫S F - ndσ using Stokes's Theorem or the Divergence Theorem. a) S is the sphere x2 + y2 + z2 = 1, n is the outward-pointing normal, and ¥(x, y, z) = (xz2,
For each of the following, evaluate ∫S ω using Stokes's Theorem or the Divergence Theorem. a) S is the topological boundary of cylindrical solid y2 + z2 < 9, 0 < x < 2, with outward pointing
Prove that Green's Theorem is a corollary of Stokes's Theorem.
Let II be a plane in R3 with unit normal n and x0 ˆˆ II. For each r > 0, let Sr be the disk in II centered at x0 of radius r [i.e., Sr = Br(x0) ˆ© II]. Prove that if F: B1(x0) R3
Let S be an orientable surface with unit normal n and nonempty boundary Ï‘S which satisfies the hypotheses of Stokes's Theorem.a) Suppose that F: S †’ R3{0} is Cl, that
Suppose that E is a two-dimensional region such that if (x, y) ∈ E, then the line segments from (0, 0) to (x, 0) and from (x, 0) to (x, y) are both subsets of E. If F: E → R2 is C1 prove that the
Let Î© be a three-dimensional region and F: Q †’ R3 be C1 on Q. Suppose further that, for each (x, y, z) ˆˆ Î©, both the line segments L((x, y, 0); (x, y,
Compute the Fourier series of x2 and of cos2 x.
Prove that if f : R †’ R is integrable on [-n, Ï€], thenfor all x ˆˆ [-Ï€, Ï€] and N ˆˆ N.
Show that if f, g are integrable on [-Ï€, Ï€] and a ˆˆ R, thenand
Suppose that f: R †’ R is differentiable and periodic and that f' is integrable on [-Ï€, Ï€]. Prove that
Suppose that fN: [-Ï€, Ï€] †’ R are integrable and that fN †’ f uniformly on [-Ï€, Ï€] as N †’ ˆž.a) Prove that ak(fN)
Let E Š‚ R and suppose that f, fk: R †’ R are bounded functions. Prove that if ˆ‘ˆžk=0 fk(x) converges to f(x) uniformly on E, thenconverges to f(x) uniformly on E as N †’ ˆž.
If f: R †’ R is periodic on R and integrable on [-Ï€, Ï€], prove that the Cesaro means of Sf are uniformly bounded; that is, there is an M > 0 such thatfor all x
Let f be integrable on [-Ï€, Ï€] and L ˆˆ R.a) Prove that if {ÏƒN f)(x0) L as N †’ ˆž and if (Sf)(x0) converges, then (SNf)(x0)) †’ L.b) Prove thatconverges to ˆš2Ï€ cos
Suppose that f : [a, b] †’ R is continuous and thatfor all integers n > 0.a) Evaluate ˆ«ba P(x) f(x) dx for any polynomial P on R.b) Prove that ˆ«ba|f(x)|2 dx = 0.c) Show that f(x) = 0 for
Let Ï•N: R †’ R be a sequence of continuous, periodic functions on R which satisfyfor all N ˆˆ N, and for each 0 uniformly for x ˆˆ R.
Let [a, b] be a nondegenerate, closed, bounded interval.a) Prove that given any polynomial P on R and any ε > 0, there is a polynomial Q on R, with rational coefficients, such that P(x) - Q(x)\
If f is integrable on [-Ï€, -] and a ˆˆ R, prove that
Prove that there is no continuous function whose Fourier coefficients satisfy |ak(f)| > 1/√k for k ∈ N.
Prove that if f: R → R belongs to C2(R) and f, f' are both periodic, then Sf converges to f uniformly and absolutely on R. (See also Exercise 14.4.5.)
If f: R †’ R belongs to Cˆž(R) and f(j) is periodic for all j > 0, prove that Sf is term-by-term differentiable on R. In fact, show thatuniformly for all j ˆˆ N.
Suppose that f: R → R is periodic on R, integrable on [-π, π], and that ak(f) > 0 for k = 0, 1,....a) Prove that (Skf)(0) > (Sjf)(0) for all k > j > 0.b) Prove that SNf(0) <
Suppose that f : R †’ R is continuous and periodic. The modulus of continuity of f is defined bya) Show thatfor k ˆˆ N.b) Prove thatfor k ˆˆ N.c) Use part b) to give a different proof the
a) Compute the Fourier coefficients of f(x) = x.b) Prove that
Suppose that f is continuous and of bounded variation on [-π, π]. Prove that SNf → f pointwise on (-π, π) and uniformly on any [a, b] ⊂ (-π, π).
a) Prove thatpointwise on (-Ï€, Ï€) and uniformly on any [a, b] Š‚ (-n, Ï€).b) Prove thatuniformly on [-Ï€, Ï€].c) Find a value for
Prove that if f is continuous, odd, and periodic, then ∑∞k=1 bk(f)/k converges.
Let L ˆˆ R. A series ˆ‘ˆžk=0 ak is said to be Abel summable to L if and only ifa) Let Sk = ˆ‘ˆžj=0 ak. Prove thatprovided any one of these series converges for all 0 b) Prove that if
Let f : R †’ R be periodic and a > 0. Suppose that f is Lipschitz of order a; that is, there is a constant M > 0 such thatfor all x, h ˆˆ R. a) Prove that holds for each h
Suppose that f: R → R is periodic and of bounded variation on [-π, π]. Prove that SN f → f almost everywhere as N → ∞ (see Exercise 14.2.8).
Suppose that F: R → R has a second symmetric derivative at some x0. Prove that if F(x0) is a local maximum, then D2F(x0) < 0, and if F(x0) is a local minimum, then D2F(x0) > 0.
Prove that if F: R → R is periodic, then there exists at most one trigonometric series which converges to f pointwise on R.
Suppose that f: R → R is periodic, piecewise continuous, and of bounded variation on R. Prove that if S is a trigonometric series which converges to (f(x+) + f(x-))/2 for all x ∈ R, then S is the
Suppose that F: {a, b) → R is continuous and D2F(x) > 0 for all x ∈ (a, b). Prove that F is convex on (a, b).
In any metric space the following three definitions of compactness are equivalent: 1. Every sequence has a convergent subsequence. 2. Every open cover has a finite sub cover. 3. Every collection of
Every bounded sequence of real numbers has a convergent subsequence. [Construct a Cauchy sequence by successively dividing the interval containing the bounded sequence. Then use the completeness of
A conventional assumption in production theory is free disposal, namelyy ∈ Y ⇒ yʹ ∈ Y for every yʹ < yA technology is said to be monotonic ifx ∈ V(y) ⇒ xʹ ∈ V(y) for every xʹ
Use the Bolzano-Weierstrass theorem to show that R is complete.The following proposition is regarded as the most important theorem in topology. We give a simplified version for the product of two
If X1 and X2 are linear spaces, then their product X = X1 × X2 is a linear space with addition and multiplication defined as follows: (x1, x2) + (y1, y2) = (x1 + y1, x2 + y2) α(x1, x2) = (ax1,
Use the definition of a linear space to show that 1. x + y = x + z ^ y = z 2. ax = ay and a ≠ 0 ⇔ x = y 3. ax = βx and x ≠ 0 ⇔ a = fi 4. (α - β)x = αx - βx 5. α(x - y) = αx - αy 6.
What is the linear hull of the vectors {(1, 0), (0,2)} in R2?
Given a fixed set of players N, each coalition T Š‚ N determines a unanimity game uT (example 1.48) defined by1. For each coalition S c N, recursively define the marginal value of a
If S ⊂ X is a subspace of a linear space X, then 1. S contains the null vector 0 2. for every x ∈ S, the inverse -x belongs to S
Give some examples of subspaces in Rn.
The linear hull of a set of vectors S is the smallest subspace of X containing S.
A subset S of a linear space is a subspace if and only if S = lin S.
If S1 and S2 are subspaces of linear space X, then their intersection S1 ∩ S2 is also a subspace of X.
If S1 and S2 are subspaces of linear space X, their sum S1 + S2 is also a subspace of X.
Give an example of two subspaces in R2 whose union is not a subspace. What is the subspace formed by their sum?
Show that a set of vectors S ⊂ X is linearly dependent if and only if there exists distinct vectors x1, x2, . . . , xn ∈ S and numbers a1, a2, . . . , an, not all zero, such that a1x1 + a2x2
Is the set of vectors {(1, 1, 1), (0, 1, 1,), (0, 0, 1)} ⊂ R3 linearly dependent?
Let U = {uT: T ⊂ N; T ≠ ∅} denote the set of all unanimity games (example 1.48) playable by a given set of players N. Show that U is linearly independent.
Every subspace S of a linear space is linearly dependent.
Show that the representation in equation (6) is unique, that is, if x = α1x1 + a2x2 +..........+ anxn and also if x = β1x1 + β2x2 +............+ βnxn then αi = βi for all i.
Every linear space has a basis. [Let P be the set of all linearly independent subsets of a linear space X. P is partially ordered by inclusion. Use Zorn's lemma (remark 1.5) to show that P has a
Is {(1, 1, 1, (0, 1, 1), (0, 0, 1)} a basis for R3? Is {(1, 0, 0), (0, 1, 0), (0, 0, 1)}? A linear space which has a basis with a finite number of elements is said to be finite dimensional.
Any two bases for a finite-dimensional linear space contain the same number of elements.
The following facts about bases and dimension are often used in practice.
A linearly independent set in a linear space can be extended to a basis.
Any set of n + 1 elements in an n-dimensional linear space is linearly dependent. The next two results highlight the dual features of a basis, namely that a basis is both • A maximal linearly
A set of n elements in an n-dimensional linear space is a basis if and only if it is linearly independent.
A set of n elements in an n-dimensional linear space X is a basis if and only if it spans X.
As a linear space in its own right, a subspace has a unique dimension. The dimension of a subspace cannot exceed that of it parent space. Furthermore a proper subspace of a ®nite dimensional
A proper subspace S ⊂ X of an n-dimensional linear space X has dimension less than n.
What are the coordinates of the vector (1, 1, 1) with respect to the basis {(1, 1, 1), (0, 1, 1,), (0, 0, 1)}? What are its coordinates with respect to the standard basis {(1, 0, 0, (0, 1, 0,), (0,
In any linear space every subspace is an affine set, and every affine set containing 0 is a subspace.
For every affine set S there is a unique subspace V such that S = x + V for some x ∈ S.
Let X be a linear space. Two affine subsets S and T are parallel if one is a translate of the other, that is, S = T + x for some x ∈ X Show that the relation S is parallel to T is an equivalence
Let H be a hyperplane in a linear space X. Then H is parallel to unique subspace V such that 1. x0 ∈ V ⇔ H = V 2. V ⊂ X 3. X = lin{V, x1} for every x1 ∉ V 4. for every x ∈ X and x1 ∉ V,
Show thatis an affne subset of Rn.
The set S = {x1, x2, . . . , xn} is affinely dependent if and only if the set {x2 - x1, x3 - x1, . . . , xn - x1} is linearly dependent. Exercise 1.157 implies that the maximum number of affinely
The set S = {x1. x2. . . . , xn} is affinely dependent if and only if there exist numbers a1, a2, . . . , an, not all zero, such that a1x1 + a2x2 +...........+ anxn = 0 with a1 + a2 +..........+ an =

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