2. Does d(x, y)= (x-y)? define a metric on the set of all real numbers?

Q: Let d: R? x R² →R, defined as d (x, y) = |x1 - yı|+ |#2 – y2]| where x = (x1, x2), y = (y1, y2) ,…

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Q: In Exercises 27 and 28, determine the maximum curvature for the graph of each function.

A: You have asked multiple unrelated questions in the same post. I have addressed the first one. Please…

Q: Show that d given by d(x, y) = |x-ylis a metric on R by using definition 4.2.1. You may want to use…

A: we knew  that given metric d(x,y) = |x-y| is always ≥0.  now d(x,y) = 0 ⇔ |x-y| = 0…

Q: Let d be metric on X, and de f ined di on X as di = max (d (r, ).0}, then dis semi metric but not…

A: Given d be an metric on X and d1 is defined on X by  d1=maxdx,y,0                     (i) To verify…

Q: Let d: R? × R² → R, defined as d (æ, y) = |æ1| where a = ("1, #2), y = (y1, 42) , then d is metric…

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Q: Determine the maximum curvature for the graph of f(x) = 6 In (3x). The maximum curvature is at x =

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Q: determine the maximum curvature for thegraph of each function.ƒ(x) = x/x + 1 for x >-1

A: Consider the function f(x)=xx+1 for x>-1. f'(x)=x+1-xx+12=1x+12f''(x)=-2x+13

Q: Let C denote the circle of radius 1 in R2 centered at the origin, oriented counterclockwise. for…

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Q: The curvature of any line with slope m in 2-space at any point on it is TTL

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Q: determine the maximum curvature for thegraph of each function. ƒ(x) = ln x

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Q: Let X=R2 and let d be the Euclidean metric.  Define d3:  R2 x R2 to R by d3(x, y) = min{1, d(x, y)}…

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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: 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: Let d: R x R R. defined as d (z, y) = |#1 - ml wherez= (21,2), y= (y.), then d is %3D Osemi metric…

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Q: Suppose that Di is the open disk with center 0 and radius 1. Let f(z) be analytic on Di \ {0} and…

A: Given- Suppose that D1 is the open disk with center 0 and radius 1. Let fz be analytic on D1\0 and…

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

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Q: \(d\) is a metric on \(X\). Show that \(\rho : X^2 \to [0,\infty)\) defined by \[ \rho(x,y) =…

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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: 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: Exercise. a) For every ,r, y E R, let p(r, y) =| x² – y? |. Is pa metric U..... on R? b) For every…

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Q: Let X=R2 and defined d2:  R2 x R2 to R by d2((x1, y1)) = max{|x1-x2|, |y1-y2|} Verify that d2 is a…

A: Given a function d2: R2×R2 → R, defined by d2x1,y1,x2,y2=maxx1-x2,y1-y2. To verify that d2 is a…

Q: Exercise. a) For every ,x, y E R, let p(x, y) =| x? – y? |. Is p a metric on R? b) For every r, y E…

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Q: å:RR→R be defined by dcx.y)=|xl+1yl, HxixR (R,d)is Than a metric space.

A: As per our guidelines we supposed to solve the very first question when the multiple questions are…

Q: True or false: the curvature of,k, for x^2+y^2=100 is larger than for x^2+y^2=1/4

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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: Suppose d is Show that p: X?> [0,00) defined by a metric on X, p Cxy)= 3d (x,y) %3D 2+3d(x, u) is…

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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: Please do no 3 for me using definition of Metric spaces

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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 C denote the positively oriented boundary of the square whose sides lie along the lines x =+ 2…

A: The general form of a complex number is given by z=x+iy, where x,y are two real numbers and i=-1.…

Q: 2. Assume that (X, dX) and (Y, dY ) are complete spaces, and give X × Y themetric d defined byd((x1,…

A: To prove that every Cauchy sequence in the product space XxY converges (under the metric defined in…

Q: If X =R and d(x,y)= 1, then X not metric semi metric metric space

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Q: Show the equivalence of the three definitions for the following 2-dimensional manifold: implicit…

A: Given: The given statement is shown as follows, implicit surface x2+y2=z2.

Q: Determine the maximum curvature for the graph of f(x) = 9 In (8x). The maximum curvature is at x=

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Q: Let R be the region in the first quadrant bounded by x = = 1, (x − 2)² + y² = 1, and x² + y² = 4.…

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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: 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 Gauas and mean curvatures on , X = (u + v,u –- v,uv) at u = 1, v = 1 are K = 116 and H…

A: Given, X=u+v,u-v,uv u=1,v=1 , k=116 and H=182 We know that, E=xu,xuF=xu,xvG=xv,xv So, we need to…

Q: The curvature of any line with slope m in 2-space at any point on it is 0

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Q: b. Draw the image of the W-plane into which the region bounded by the straight lines in the Z-plane…

A: The solution is given below

Q: .3 Find the radius of curvature of y=x' at x=1,

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Q: Q4: State the Fundamental Theorem of Independent of Path and Morera's Theorem. Why can't apply these…

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Q: Determine the maximum curvature for the graph of f(x) = 7 In (2x). The maximum curvature is at x=

A: We are given function f(x)=7ln(2x) of the graph Formula for curvature is κ(x)=|f′′(x)(1+(f′(x))2)32|…

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: Let X=ℝ2 and define d2,:ℝ2×ℝ2→ℝ by d2((x1 ,y1),(x2,y2)) = max{|x1 - x2|,|y1 - y2|}. a) Verify that…

A: Given that X=ℝ2 and defined by d2:ℝ2×ℝ2→ℝ Part(a): Now from given d2:ℝ2×ℝ2→ℝ by…

Q: *6. Let f: R' →R be given by =+4x'z-6xyz+4y-3x'y. Is M=f-((0}) a smooth surface (2-dimensional…

A: Given, fxyz=z2+4x3z-6xyz+4y3-3x2y2 Now, writing the given function in matrix form.…

Q: Let X = {2,3, 4,..} and de fined metric on X as d (r, y) = |- yl %3D %3D Find орen ball B (2,2) =…

A: Follow the definition: B(2,2) = { x in X : d(x,2) &lt; 2 }

Q: Let X = {1,2,3, 4, ...} and de fined metric on X as d (r, y) |r-yl %3D Find open ballB (2,2) = o {2}…

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