Let X = (0,00). -> a) Show that X is not complete with the standard metric d: X2 → [0, 0o) defined by d (x, y) = |x - yl b) Use #1 to find a metric p on X so that X is complete with respect to p.

Q: Consider the closed rectangle R = [a,b] × [c,d] in ℝ2 (under the Euclidean metric). i. Explain why…

A: Consider the closed rectangle R = [a, b] × [c, d] in ℝ2 (under the Euclidean metric). For (i), We…

Q: Find d(A) in the context of R with the standard metric when i) A = (0, 2) %3D ii) A= the set of…

A: d denotes standard metric on R which is  d(x, y) = |x-y|  , for all x, y in R.  For any non empty…

Q: The complex function fla)=ulx.y) + ¿elx.y) Con be differentiated ony point Tin complex opoce The…

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Q: Define p: R^2 x R^2 to [0, infinity) by the rule p(x1,y1),(x2,y2)=squareroot((x1-x2)^2+(y1-y2)^2)…

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Q: ii. Use the basic definition of complex integration to evaluate the integral I $, z¹ dz, where the…

A: Step 1: Step 2: Step 3: Step 4:

Q: Q2: In usual topological space (R.T), answer the following: 4- Is [0,1] is arcwise in R? Why?

A: Given that ℝ, Tu is usual topological space. To show: 0,1 is arcwise in ℝ.

Q: The metric for a three-sphere is ds² = R² (d² + sin² § (d0² + sin² 0d²)) where ranges from 0 to 27,…

A: We are given that the metric for a three-sphere is ds2=R2dξ2+sin2ξdθ2+sin2θdϕ2 where ϕ ranges from 0…

Q: Let f (z) = 33O = Q A = 10and B= 10 + 10iet C be the contour composed of the line segments OAand…

A: The given problem is to evaluate the complex integral of the given function over the given semi…

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: Integrate h(x, y) = (x+y)i + (y²-x)j over the closed curve that begins at (-7,0), goes along the…

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Q: d, d': xxx-R be two metrices on x that Satisfy following for every nEX and >0 7>0 st. Bai (x, S) C…

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Q: a) Use Green's theorem to compute f F dr where c is a circle of radius R oriented CCW and F(x, y) =…

A: Green's theorem:∫C[Pdx+Qdy]=∫∫T∂Q∂x-∂P∂ydxdy where T is the region bounded by a simple closed curve…

Q: Let C be the space curve represented by represented by & (+) = [(4-2+, 36-2, 61) a.) Reparametrize R…

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Q: Let (t) = (t, t, tº, tª) for all t € R. Define C = Þ(R). Let p = (1, 1, 1, 1). Derive the tangent…

A: Let  for all Let the point be We need to calculate the tangent and normal spaces to C at p.

Q: 8. let C be the curue space ret) = (e*,2et ,2et ),0s ts lnca) the rength of C is

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Q: Explain why or why not Determine whether the following statementsare true and give an explanation or…

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Q: 5. Let X = R? and define d, : R? ×R² →R by d (x,,y,),(x,,y2)) = max {x, – x,|-\y, -y}} . 13 a)…

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Q: Draw the curve and find the length in range [0,27]: r(t) = acosti + asentj + btk

A: Given: rt=acosti+asintj+bt

Q: Let f (z) = 3O = 0 A = 10and B= 10 + 10iet C1 be the contour composed of the line segments OAand…

A: Solution  Given  A=10   ,  B=10+10i

Q: Let S' denote the set of (x, y) E R × R such that x? + y² This space is called the circle. Define a…

A: S1={(x,y)∈ℝ2/x2+y2=1} and p:ℝ→S1 by pt=cos2πt,sin2πt. To show: p is continuous, open,closed.

Q: Sketch the set of points (r, θ) that satisfy the following conditions: {(r, θ) : −2 ≤ r < 2, 0 ≤ θ ≤…

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Q: Show that 7(u, v)= (²-²,2ue) maps the triangle-((u, v):05≤SS 3) to the domain D bounded by z = 0, y…

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Q: The concept of the linear approximation builds from the fact that differentiable functions are…

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Q: Sketch the following sets of points in the x-y plane. {(x, x^2): x is in R}

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Q: is that The necessary and sufficient condition that a curve is a plane cury 4 r 00 T 004 r = 0.

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Q: (-3). 6 What is the equation of the plane tangent to f at the point a? Let f: R² → R. At some point,…

A: Let  f: ℝ2→ℝ,at some  point  a⇀=-96 and  ∇f(a⇀)=37  , fa⇀=3 The  equation  of  the tangent  plane…