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An Introduction to Analysis 4th edition William R. Wade - Solutions

Suppose that 0 < r < 1 and that f: B1(0) R is continuously differentiable. If there is an a > 0 such that |f(x)| < ||x||a for all x ∈ B, (0), prove that there is an M > 0 such that |f(x)| < M||x|| for x ∈ Br(0).
Suppose that V is open in Rn, that f: V †’ R is C2 on V, and that fxj(a) = 0 for some a ˆˆ H and all j = I, ...,n. Prove that if H is a compact convex subset of V, then there is a constant M such thatfor all x ˆˆ H.
Let f: Rn → R. Suppose that for each unit vector u ∈ Rn, the directional derivative Duf(a + tu) exists for t ∈ [0, 1 ] (see Definition 11.19). Prove that f(a + u) - f(a) = Duf(a + tu) for some t ∈ (0, 1).
For each of the following functions, prove that f-1 exists and is differentiable in some nonempty, open set containing (a, b), and compute D(f-1){a, b) a) f(u, v) = (3u - v, 2u + 5v) at any (a, b) ∈ R2 b) f(u, v) = (u + v, sin u + cos v) at (a, b) = (0, 1) c) f(u, v) = (uv, u2 + v2) at (a, b) =
Suppose that f: = (u, v): R → R2 is C2 and that (x0, y0) = f(t0)- a) Prove that if fʹ(t0) ≠ 0, then u'(t0) and v'(t0) cannot both be zero. b) If fʹ(t0) ≠ 0, show that either there is a C1 function g such that g(x0) = t0 and u(g(x)) = x for x near x0, or there is a C1 function h such that
Let H be the hyperboloid of one sheet, given by x2 + y2 - z2 = 1. a) Use Exercise 11.6.9 to prove that at every point (a, b, c) ∈ H, H has a tangent plane whose normal is given by (-a, -b, c). b) Find an equation of each plane tangent to H which is perpendicular to the xy-plane. c) Find an
For each of the following functions, find out whether the given expression can be solved for z in a nonempty, open set V containing (0, 0, 0). Is the solution differentiable near (0, 0)? a) xyz + sin(x + y + z) = 0 b) x2 + y2 + z2 + √sin(x2 + y2) + 3z + 4) = 2 c) xyz(2 cos y - cos z) + (z cos x:
Prove that there exist functions u(x, y), v(x, y), and w(X, y), and an r > 0 such that M.V.W are continuously differentiable and satisfy the equations
Find conditions on a point (x0, y0, u0, v0) such that there exist real-valued functions u(x, y) and v(x, y) which are continuously differentiable near (x0, y0) and satisfy the simultaneous equations xu2 + yv2 + xy = 9 xv2 + yu2 - xy = 7.
Given nonzero numbers x0, y0, u0, v0, s0, t0 which satisfy the simultaneous equationsprove that there exist functions u(x, y), v(x, y), s(x, y), t(x, y), and an open ball containing (x0, y0), such that u, v, s, t are continuously differentiable and satisfy (*) on B, and such that u(x0, y0) = uo,
Let E = {(x, y): 0 < y < x} and set f(x, y) = (x + y, xy) for (x, y) ∈ E. a) Prove that f is 1-1 from E onto {(s, t): s > 2√t, t > 0} and find a formula for f-1[ (s, t). b) Use the Inverse Function Theorem to compute D(f-1)(f(x, y)) for U, y) ∈ E. c) Use the formula you obtained in part a) to
Suppose that V is open in Rn, that a ˆˆ V, and that F : V †’ R is C1 on V. If F(a) = 0 ‰ Fxj(a) and u(j): = (x1 . . . . .. . . .-xj-1, xj+1,. . . . . . . . ., xn) for j = 1,2, ...,n, prove that there exist open sets Wj containing (a1,. . . . . . . . aj, aj+1, . . . .
Suppose that f: R2 †’ R2 has continuous first-order partial derivatives in some ball Br(x0, y0), r > 0. Prove that if Î”f(x0, y0) ‰ 0, thenand
Let F : R3 †’ R be continuously differentiable in some open set containing (a, b, c) with F(a, b, c) = 0 and F(a, b, c) ‰ 0.
Find all local extrema of each of the following functions. a) f(x, y) = x2 - xy + y3 - y b) f(x, y) = sin x + cosy c) f(x, y, z) = ex+y cos z d) f (x, y) = ax2 + bxy + cy2, where a ≠ 0 and b2 - 4ac ≠ 0
Suppose that points x and y are fixed in R" and seta) Prove that if for (a, b) ˆˆ R2, then the system is solved by And b) Prove that if a0 and b0 are given by part a), then the straight line whose graph is closest to the points (x1, y1),..., (xn, yn)-that is, such that is minimized -is
For each of the following, find the maximum and minimum of f on H. a) f(x, y) = x2 + 2x - y2 and H = {(x, y) : x2 + 4y2 < 4} b) f(x, y) = x2 + 2xy + 3y2, and H is the region bounded by the triangle with vertices (1,0), (1,2), (3,0) c) f(x, y) = x3 + 3xy - y3, and H = [-1, 1] × [-1, 1]
For each of the following, use Lagrange multipliers to find all extrema of f subject to the given constraintsa) f(x, y) = x + y2 and x2 + y2 = 4b) f(x, y) = x2 - 4xy + 4y2 and x2 + y2 = 1c) f(x, y, z) = xy, x2 + y2 + z2 = 1 and x + y + z = 0d) f(x, y, z, w) = 3x + y + w, 3x2 + y + 4z3 = 1 and -x3 +
Suppose that f: Rn → Rm is differentiable at a, and that g: Rm → R is differentiable at b = f(a). Prove that if g(b) is a local extremum of g, then (g o f)(a) = 0.
Suppose that V is open in R2, that (a, b) ∈ V, and that f: V → R has second-order total differential on V with fx(a, b) = fy(a, b) = 0. If the second-order partial derivatives of f are continuous at (a, b) and exactly two of the three numbers fxx(a, b), fxy(a, b), and fyy(a, b) are zero, prove
Suppose that V is an open set in Rn that a ∈ V, and that f: V → R is C2 on V. If f(a) is a local minimum of f', prove that D(2)f(a; h) > 0 for all h ∈ Rn.
Let a, b, c, D, E be real numbers with c ≠ 0. a) If DE > 0, find all extrema of ax + by + cz subject to the constraint z = Dx2 + Ey2. Prove that a maximum occurs when cD < 0 and a minimum when cD > 0. b) What can you say when DE < 0?
a) Suppose that f, g: R3 †’ R are differentiable at a point (a, b, c), and that f(a, b, c) is an extremum of f subject to the constraint g(x, y, z) = k, where k: is a constant. Prove thatand b) Use part a) to find all extrema of f(x, y, z) = 4xy + 2xz + 2yz subject to the constraint xyz =
a) Let p > 1. Find all extrema of f(x) = ˆ‘nk=1 x2k subject to the constraint ˆ‘nk=1 |xk|p = 1b) Prove thatfor all x1,..., xn ˆˆ R, n ˆˆ N, and 1
Î±) E = {(x, y) ˆˆ [0, 1] Ã— [0, 1] : x = 0 or y = 0}.Î²) E = {(x, y) ˆˆ [0, 1] Ã— [0, 1]: y Î³) E = {(x, y) ˆˆ [0, 1] Ã— [0, 1] : (2x - l)2 + (2y - l)2
a) Prove that every finite subset of Rn is a Jordan region of volume zero. b) Show that, even in R2, part a) is not true if finite is replaced by countable. c) By an interval in R2 we mean a set of the form [(x, c) : a < x < b] or {(c, y) : a < y < b) for some a, b, c ∈ R. Prove that every
Prove that every grid is a nonoverlapping collection of Jordan regions.
a) Prove that the boundary of an open ball Br(a) is given byb) Prove that Br(a) is a Jordan region for all a ˆˆ Rn and all r > 0.
Let E be a Jordan region in Rn. a) Prove that E° and are Jordan regions. b) Prove that Vol(Eo) = Vol() = Vol(E). c) Prove that Vol(E) > 0 if and only if E° ≠ 0. d) Let f: [a, b] → R be continuous on [a, b]. Prove that the graph of y = f(x), x ∈ [a, b], is a Jordan region in R2. e) Does
Suppose that E1, E2 are Jordan regions in Rn.a) Prove that if E1 Š‚ E2, thenb) Prove that E1 ˆ© E2 and E1 E2 are Jordan regions. c) Prove that if E1, E2 are nonoverlapping, then Vol(E1 ‹ƒ E2) = Vol(E1) + Vol(E2). d) If E2 Š‚ E1, prove that Vol(E1 E2) =
Let E ⊂ Rn. The translation of E by an x ∈ Rn is the set x + E = {y ∈ Rn: y = x + z for some z ∈ E}, and the dilation of E by a scalar a > 0 is the set aE = {y ∈ Rn: y = az for some z ∈ E}. a) Prove that E is a Jordan region if and only if x + E is a Jordan region, in which case Vol(x +
A set E ⊂ Rn is said to be of measure zero if and only if given ε > 0 there is a sequence of rectangles R1, R2,... which covers E such that ∑∞k=1 |Rk| < ε a) Prove that if E ⊂ Rn is of volume zero, then E is of measure zero. b) Prove that if E ⊂ Rn is at most countable, then E is of
Show that if E ⊂ Rn is bounded and has only finitely many cluster points, then E is a Jordan region.
for 7 = 1,2. Prove that
Suppose that E is a Jordan region and that f: E → R is integrable. a) If f(E) ⊂ H, for some compact set H, and ɸ : H → R is continuous, prove that ɸ o f is integrable on E. b) Show that part a) is false if ɸ has even one point of discontinuity.
Prove the following special case of Theorem 12.29i. Suppose that E and E0 are a Jordan regions in Rn, and that f: E → R is bounded. If f is continuous on E \ E0, then f is integrable on E.
Let E be a Jordan region in Rn with E Š‚ [0, 1] Ã— ˆ™ ˆ™ ˆ™ Ã— [0, 1]. If f, g are integrable on E withand if g(x)
Let E be an open Jordan region in Rn and x0 ˆˆ E. If f: E †’ R is integrable on E and continuous at x0, prove that
a) Suppose that E is a Jordan region in Rn and that fk: E †’ R are integrable on E for k ˆˆ N. If fk †’ f uniformly on E as k †’ ˆž, prove that f is integrable on E andb) Prove that exists, and find its value for any Jordan region E in R2.
If E0 ⊂ E are Jordan regions in Rn and f: E → R is integrable on E, prove that f is integrable on E0.
Let H be a closed, connected, nonempty Jordan region and suppose that f: H †’ R is continuous. If g: H †’ R is integrable and nonnegative on H, prove that there is an x0 ˆˆ H such that
Suppose that Q: = {(x, y) ˆˆ R : x > 0 and y > 0} and that f is a continuous function on R2 whose first-order partial derivative satisfies |fx|for (x, y) ˆˆ Q, prove that F is bounded on Q. [You may use polar coordinates to change variables in F.]
Suppose that E is a Jordan region in Rn and that f, g: E → R is integrable on E. a) Modifying the proof of Corollary 5.23, prove that fg is integrable on E. b) Prove that f ⋁ g and f ⋀ g are integrable on E (see Exercise 3.1.8).
Suppose that V is open in Rn and that f: V †’ R is continuous. Prove that iffor all nonempty Jordan regions E Š‚ V, then f = 0 on V.
Evaluate each of the following iterated integrals.a) ˆ«10 ˆ«10 (x + y) dx dyb)c)
If aconverges uniformly on [c, d], prove that is improperly integrable on (a, b) and
Evaluate each of the following iterated integrals. Write each as an integral over a region E, and sketch E in each case.a)b) c) d)
For each of the following, evaluate ∫E f dV. a) f(x, y) = (1 + x2)-1 and E is bounded by x = 1, y = 0, and y = x3. b) f(x, y) = x + y and E is the triangle with vertices (0, 0), (0, 1), and (2, 0). c) f(x. y) = x2exy and E is the triangle with vertices (0, 0), (1, 0), and(l, 1). d) f(x, y, z) = x
Compute the volume of each of the following regions. a) E is bounded by the surfaces x + y + z = 3, z = 0, and x2 + y2 = 1. b) E lies under the plane z = x + y and over the region in the xy-plane bounded by the curves x = √y/2, x = 2√y, x + y = 3. c) E is bounded by z = y2, x = y2 + z2, x = 0,
a) Verify that the hypotheses of Fubini's Theorem hold when / is continuous on R. b) Modify the proof of Remark 12.33 to show that Fubini's Theorem might not hold for a nonintegrable f, even if f(x, y) is continuous in each variable separately; that is, if f(x, ∙) is continuous for each x ∈ [a,
a) Suppose that fk is integrable on [ak, bk] for k = 1,..., n, and set R = [a1, b1] Ã— ˆ™ ˆ™ ˆ™ Ã— [an, bn]. Prove thatb) If Q = [0, 1]n and y : = (1, 1,..., 1), prove that
The greatest integer in a real number x is the integer [x]: = n which satisfies n < x < n + 1. All interval [a, b] is called Z-asymmetric if b + a ≠ [b] + [a] + 1. a) Suppose that R is a two-dimensional Z-asymmetric rectangle (i.e., that both of its sides are Z-asymmetric). If ψ(x, y): = (x -
Let E be a nonempty Jordan region in R2 and f: E †’ [0, ˆž) be integrable on E. Prove that the volume of Î© = {(x, y, z): (x, y) ˆˆ E, 0
Let R = [a, b] Ã— [c, d] be a two-dimensional rectangle and f: R †’ R be bounded.a) Prove thatfor X = ‹ƒ or X = L. b) Prove that if f is integrable on R, then c) Compute the two iterated integrals in part b) for and R = [0, 1] Ã— [0, 1]. Prove that f is
Evaluate each of the following integrals.a)b) c)
a) Prove that the improper integral ˆ«£° e-x2 dx converges to a finite real number.b) Prove that if f is the value of the integral in part a), thenc) Show that d) Let Qk represent the n-dimensional cube [-k, k] Ã— ˆ™ ˆ™ ˆ™
Let H Š‚ V Š‚ Rn, with H convex and V open, and suppose that É¸: V †’ Rn is C1.a) Show that if E is a closed subset of H° andthen ˆˆh(x)/||h|| †’ 0 uniformly on E, as h †’ 0. b) Show that if R is a closed rectangle in H° and S := (DÉ¸(x))-l exists for some x ˆˆ R, then
For each of the following, find ∫∫E f dA. a) f(x, y) = cos(3x2 + y2) and E is the set of points satisfying x2 + y2/3 < 1. b) f(x, y) = y√x - 2y and E is bounded by the triangle with vertices (0, 0), (4, 0),and (4, 2).
For each of the following, find ∫∫∫E f dV. a) f(x. y, z) = z2 and E is the set of points satisfying x2 + y2 + z2 < 6 and z > x2 + y2. b) f(x, y, z) = ez and E is the set of points satisfying x2 + y2 + z2 < 9, x2 + y2 < 1, and z > 0. c) f(x, y, z) = (x - y)z and E is the set of points
a) Prove that the volume bounded by the ellipsoidis 4Ï€abc/3.b) Let a, b, c, d be positive numbers and r2 c) Show that for any a > 0, the volume of the region bounded by the cylinders x2 + z2 = a2 and y2 + z2 = a2 is 16a3/3.
a) Compute ˆ«ˆ«E ˆšx - y ˆšx + 2y d A, where E is the parallelogram with vertices (0 0), (2/3, -1/3), (1, 0), (1/3, 1/3).b) Compute ˆ«ˆ«E 3ˆš2x2 - 5xy - 3y2 dA, where E is the parallelogram bounded by the lines y = x/3, y =
Suppose that V is nonempty and open in Rn and that f: V †’ Rn is continuously differentiate with Î”f ‰ 0 on V. Prove thatfor every x0 ˆˆ V.
Show that Vol is rotation invariant in R2; that is, if ɸ is a rotation on R2 (see Exercise 8.2.9) and E is a Jordan region in R2, then Vol(ɸ(E)) = Vol(E).
a) Compute the Jacobian of the change of variables from spherical coordinates to rectangular coordinates.b) Assuming that Vol is translation and rotation invariant (see Exercises 12.1.7 and 12.4.7), verify the following classical formulas: the volume of a sphere of radius r is 3/4πr3, and the
Let vj = (vj1,. . . . . . . . .,Vjn) ˆˆ Rn, j = 1,. . . . . . ., n, be fixed. The parallelepiped determined by the vectors vj is the setand the determinant of the vj's is the numberdet(v1...,vn) := del[vjk]n Ã— n.Prove thatVol(P(v1...,vn)) = |det(v1,...,vn)|.Check this formula for n = 2 and n
If f, g : Rn → R, prove that spt(fg) ⊂ spt f ∩ spt g.
Prove that if f, g ∈ Cc∞(Rn), then so are fg and af for any scalar a.
Prove that if f is analytic on R and f(x0) ≠ 0 for some x0 ∈ R, then f ∉ C∞c(R).
Suppose that V is a bounded, nonempty, open set in Rn and that ɸ: V → Rn is 1 - 1 and continuously differentiable on V with Δɸ ≠ 0 on V. Let W = {Wj}j∈N be an open covering of V and {ɸj}j∈N be a Cp partition of unity on V subordinate to W, where p > 1. Prove that {∅,j o ∅-1}j∈N is
Let V be open in Rn and V = {Vj}j∈N, W = {Wk}k∈N be coverings of V. If {ɸj}j∈N is a Cp partition of unity on V subordinate to V and {ψk}k∈N is a Cp partition of unity on V subordinate to W, prove that {ɸj ψk)j,k∈N is a Cp partition of unity on V subordinate to the covering {Vj ∩
Show that, given any compact Jordan region H Š‚ Rn, there is a sequence of Cˆž functions É¸j such that
a) Prove that r is differentiable on (0, ˆž) withb) Prove that rÊ¹ is Cˆž and convex on (0, ˆž).
Show that
Show that
Show that the volume of a four-dimensional ball of radius r is π2r4/2, and the volume of a five-dimensional ball of radius r is 8π2r5/15.
Suppose that n > 2 and define an n-dimensional ellipsoid byProve that
Suppose that n > 2 and define an n-dimensional cone byProve that
Find the value offor each k ˆˆ N.
If f: Î²1(0) †’ R is differentiable with f(0) = 0 and || f(x)||
Let ψ(t) = (a sin t, a cos t, σ(t) = (a cos 2t, a sin 2t, I = [0, 2π), and J = [0, π). Sketch the traces of (ψ, I) and (σ, J). Note the "direction of flight" and the "speed" of each parametrization. Compare these parametrizations with the one given in Example 13.4.
Trie absolute curvature of a smooth curve with parametrization (Ïˆ, I) at a point x0 = Ïˆ(t0) is the numberwhen this limit exists, where 0(t) is the angle between Ïˆ'(t) and Ïˆ'(t0), and ï¬(t) is the arc length of if Ïˆ(I) from
Let C be a smooth C2 arc with parametrization (É¸, [a, b]), and suppose that s = ï¬(t) is given by (2). The natural parametrization of C is the pair (v, [0, L]), wherev(s) = (É¸ o ï¬-1)(s) and L = L(C).a) Prove that ||v'(s) = 1 for all s
Let a, b ∈ Rm, b ≠ 0, and set ɸ(0 = a + tb. Show that C = ɸ(R) is a smooth unbounded curve which contains a and a + b. Prove that the angle between ɸ(t1) - ɸ(0) and ɸ(t2) - ɸ(0) for any t1, t2 ≠ 0 is 0 or π.
Let I be an interval and f: I †’ R be continuously differentiable withfor all Î¸ ˆˆ I. Prove that the graph r = f(Î¸) (in polar coordinates) is a smooth Cl curve in R2.
Show that the curve y = sin(1/x), 0 < x < 1, is not rectiflable. Thus show that Theorem 13.17 can be false if C is not an arc.
Sketch the trace and compute the arc length of each of the following. a) ɸ(0 = (et sin/, et cost, et), t ∈ [0, 2π] b) y3 = x2from(-l, 1) to (1,1) c) ɸ(t) = (t2, t2, t2), t ∈ [0, 2] d) The astroid of Example 13.5
For each of the following, find a (piecewise) smooth parametrization of C and compute ∫Cgds. a) C is the curve y = √9 - x2, x > 0, and g(x, y) = xy2. b) C is the portion of the ellipse x2/a2 + y2/b2 = 1, a, b > 0, which lies in the first quadrant and g(x, y) = xy. c) C is the intersection of
Let C be a smooth arc and gk : C → R be continuous for n ∈ N. a) If gk → g uniformly on C, prove that ∫C gk ds → ∫C g ds as k → ∞. b) Suppose that {gk} is pointwise monotone and that gk → g pointwise on C as k → ∞. If g is continuous on ɸ(I), prove that ∫C gk ds → ∫C g
Suppose that (É¸, I) is a parametrization of a smooth arc in Rm, and that function, 1 - 1 from J onto I. If for all but finitely many u ˆˆ J, is continuous, prove that
Let C be the piecewise smooth curve É¸(I1 U I2), where I1 = (-ˆž, -1), I2 = (-1, ˆž), and
For each of the following, sketch the trace of (ɸ, R), describe its orientation, and verify that it is a subset of the surface S. a) ɸ(t) = (3t, 3 sin t, cos t), S = {(x, y, z) : y2 + 9z2 = 9} b) ɸ(t) = (t2, t3, t2), S = {(x, y, z) : z = x] c) ɸ(t) = (t, t2, sin*), S = {(x, y, z):y = x2} d)
For each of the following, find a (piecewise) smooth parametrization of C and compute ∫C F ∙ Tds. a) C is the curve y = x2 from (1, 1) to (3, 9), and F(x, y) = (xy, y - x). b) C is the intersection of the elliptical cylinder y2 + 2z2 = 1 with the plane x = -1, oriented in the counterclockwise
For each of the following, compute ∫C ω. a) C is the polygonal path consisting of the line segment from (1, 1) to (2, 1) followed by the line segment from (2, 1) to (2, 3), and ω = y dx + x dy. b) C is the intersection of z = x2 + y2 and x2 + y2 + z2 = 1, oriented in the counterclockwise
a) Let c ∈ R, δ > 0, and set τ(K) = δu + c for u ∈ R. Prove that if (ɸ, I) is a smooth parametrization of some curve, if J = τ-1(I), and if ψ = ɸ o r, then (ψ, j) is orientation equivalent to (ɸ, I). b) Prove that if (ɸ, I) is a parametrization of some smooth arc, then it has an
Let (É¸, I) be a smooth parametrization of some arc and Ï„ be a C1 function, 1-1 from J onto I, which satisfies Ï„'(u) > 0 for all but finitely many u ˆˆ J. If Ïˆ = É¸ o r, prove thatfor any continuous F : É¸(I)
Let f : [a, b] → R be C1 on [a, b] with fʹ(t) ≠ 0 for t ∈ [a, b]. Prove that the explicit curve x = f-1(y), as y runs from f(a) to f(b), is orientation equivalent to the explicit curve y = f(x), as x runs from a to b.
Let V ‰ Î¸ be open in R2. A function F: V †’ R2 is said to be conservative on V if and only if there is a function f : V †’ R such that F = f on V. Let (x, y) ˆˆ V and let F = (P, Q) : V †’ R2 be continuous on V.a) Suppose that C(x) is
Suppose that f: [0, 1] → R is increasing and continuously differentiable on [0, 1]. Let T be the right triangle whose vertices are (0, f(0)), (1, f(0)), and (1, /(l)). If c represents the hypotenuse of T, a and b represent the legs of T, and L represents the arc length of the explicit curve y =
For each of the following, find the surface area of S. a) S is the conical shell given by z = √x2 + y2, where a < z < b. b) S is the sphere given in Example 13.31. c) S is the torus given in Example 13.32.
For each of the following, find a (piecewise) smooth parametrization of S and of ϑS, and compute ∫∫S g da. a) S is the portion of the surface z = x2 - y2 which lies above the xy-plane and between the planes x = 1 and x = -1, and g(x, y, z) = √1 + 4x2 + 4y2. b) S is the surface y = x3, 0 < y

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