Let X =  R x R  x R = R3 and define the metric d: R 3 x R 3 to R  by d((x1, y1, z1), (x2, y2, z2)) = |x1 - x2| + |y1 - y2| + |z1 - z2| Describe the neighborhood N(0, 1) where 0 is the origin in R3.

Q: Consider the neighborhood of the real interval [0, 1]: U.= UD.(x). ZE(0,1) For a fixed pe (0, 1),…

A: Given: The ε-neighborhood of the real interval[0,1]: Uε=Ux∈[0,1]Dε(x) To proof: limε→0f'(p)=0

Q: Show that each function in satisfies a Laplaceequation. ƒ(x, y, z) = 2z3 - 3(x2 + y2)z

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Q: (B) Prove that the stream line of b are counterclockwise circles, for that calculate F = Vx (-y ěx+…

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Q: f(x,y) = 4xy² – x²y² – xy³

A: Given f(x,y)=4xy2-x2y2-xy3 and a triangular  region D with vertices (0,0) ,(6,0) and (0,6) Case 1 To…

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Q: Consider the following mappings on RXR: * d1(1, y) = e²=y, d2 = |2r - 2y). and d3 = V12r – yl. %3D…

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Q: Q2: (a) Let M = {x,y,z) and define d: MXM → R by Determine whether d is a metric on M. d(x,x) = d(y,…

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Q: 1. Determine which of the following function define a metric on R : (i) d(r, y) = V ly° + |x|° (ii)…

A: Metric: A metric, d on X is a function defined on X × X such that for all x, y, z ∈ X, which…

Q: 4. Let f(x, y) be a function defined on a disk D not containing the origin which is given by f(r, y)…

A: See the answer below

Q: Consider 2-dimensional hyperbolic space with metric ds ds'+dy and y > 0. The length of a line…

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Q: Let d: Rx R*→[0, ∞) such that d(XF)=√(x, y)² show that (R.d) metric space.

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Q: Let F(x, y) = yeyi+xe"yj. Evaluate the line integral F. dr where C is the closed path consisting of…

A: The answer is found in step-2

Q: Let F(x, y) = ye"vi + xe"Vj. Evaluate the line integral F of line segments from (0,3) to (0, 0),…

A: Given function is, Fx,y=yexyi+xexyj

Q: Prove that the circle {(x, y), x² + y² = 1} is hot homeomorphic to the interval [-1, 1].

A: We will prove this result by contradiction: Suppose there exists a homeomorphism between -1,1 and S…

Q: Sketch the space curve r(t) = −ti + 4tj + 3tk and find its length over the given interval [0, 1] .

A: Given space curve is rt=-ti+4tj+3tk. Sketch the above space curve as shown below.

Q: Let g, h: -R and /:R'+ R be defined by /(x, y) = (g(x, y), h(x, y). Assume curl /(x, y) = 7x – 7y,…

A: Let g,h:ℝ2→ℝ and f:ℝ2→ℝ2 and fx,y=g(x,y),h(x,y). Assume that ∇×f=7x-7y, ∇f=7y-8x and…

Q: Suppose that (z) is entire and that the harmonic function u(x, y) = Re[fz)] has an upper bound uo;…

A: As requested following is the answer for question 12.

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A: Let Ω∈Rn be open with smooth boundary. (a)To prove the Green's function is uniquely determined:…

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Q: Suppose that D is the unit disk in the xy-plane, which is contained in the unit square S = [0, 1] ×…

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Q: evaluate $ A. dš over the sphere R = 3 and show whether Divergence theorem holds or not. %3D

A: Calculate surface integral over R=3 or dr=0 ∮A→×ds→=∮R cos2θ,π,0×r2sinθdθdϕ,rdϕdr,rsinθdrdθ…

Q: (b) Let X = {r € R: -1 <1<0or 0 <1< 1} with the usual metric.Then X is disconnected.

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Q: (10) Define the metric d on R² by d( (x₁, y₁), (x2, y2)) = max{ |x₁ − x2\,|y₁ - y2|}. Verify that…

A: The given problem is to verify that the given metric d on R2. Given metric…

Q: Find T, N, and k for the space curve r(t) = - ti - (a cosh (t/a))j, a > 0.

A: Unit Tangent. Unit Normal. Curvature.

Q: Let X = R and x, y E R. Admit for all x and y the metric d(x, y) = |arctan x arctan y|. | Compare…

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Q: Let F(x, y, z) = (x-1z, y¬lz, In(xy). (a) Verify that F = V ƒ, where ƒ(x, y, z) = z In(xy). (b)…

A: Given Fx, y, z=x-1z, y-1z, lnxy (a) Verify for F=∇f where fx, y, z=z lnxy The gradient of a function…

Q: (b) Show that the function defined by IN if (x, y) (0,0), √√√z²+y² f(x, y) = 0 if (x, y) = (0,0) is…

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Q: -) Let -y M(x, y) = x2 and N(x, y) = + y? x2 + y? how that ƏN ƏM dx ду ) Show that -y F(x, y) = x² +…

A: Given :-  A vector field for conservative field in the plane ,  ∂N∂x = ∂M∂y To Determine :-  M x,y =…

Q: Consider 2-dimensional hyperbolic space with metric ds = dr'+dy and y > 0. The length of a line…

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Q: Let P be the tangent plane to the surface z=2x2+y2 at the point Q=(1,1,3). Which of the following…

A: tangent planes to a surface are planes that just touch the surface at the point and are “parallel”…

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Q: Let W = Span{u} where u -2. Let y = |1, find the distance from y to W L3]

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Q: et X=R^(2) and C={(x y)in R^(2):x>=0 y=0}. Show that C is not a generating cone.

A: Given: Let   and .  To show that C is not a generating cone.

Q: Let d : R x R → R be defined to be d(x, y) = | arctan(x) – arctan(y)|. Show that d is a metric on R.

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Q: „Consider the space (R,t). .where t={R,Ø}U{ACR:[0,1)CA} f A=[0,1], then the interior of A is (0,1) O…

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Q: Evaluate the limit: r(t + h) – r(t) lim h→0 for r(t) = (1-6, sin t, 8) h r'(1) = ( ...

A: We have to find components of r't

Q: Verify that (S, d) is a metric space. 10.2 Prove that d(x, y) < d'(x, y)s/2 d(x, y) for all points'…

A: To prove dx,y≤d'x,y≤2dx,y for all points x,y∈ℝ×ℝ where d is the usual metric and d' is the…

Q: ) Show that the function defined by if (x,y) = (0,0), f(x, y) = { 0 if (x, y) = (0,0) is not…

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Q: „Consider the space (R,t). .where t={R,Ø}U{ACR:[0,1)CA} :lf A=(0,1), then the closure of A is (0,1)…

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Q: Let f : C[0,1] → P be given byf (x ) = x (1)Vx e C[0,1]. Show that f is continuous w.r.t the supnorm…

A: f is continuous with respect to sup norm. Let xn be a sequence of continuous functions in 0,1 such…

Q: d(r, y) 1+d(x,y) 5. Let (X, d) be a metric space. Define f : X x X → R by f(r, y) = Show that f is a…

A: (χ , d)  be a metric space , define f :χ×χ→ℝ by f(x , y)=d(x , y)1+d(x , y) Positive definite…

Q: Let X = R², x, y EX where x = (x₁, x2), y = (y₁.Y2) with the metric function (4 pts) (x, y) =…

A: This is a question of topology and metric space.

Q: (a) Let D be the region in ryz-space defined by the inequalities 1 <r < 2,0 < ry < 2, 0 < z< 1.…

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