i) Pe(x) = 0 iff r € Ë. ii) Prove that pe is uniformly continuous.

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: Sketch a surface representing a function f of two variables x and y. Use the sketch to give…

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Q: (a) reflexive (b) irreflexive (c) symmetric

A: Note: As per our guidelines we answer up to 3 subparts.

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

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Q: Assume that u and u are continuously differentiable functions. Using Green's theorem, prove that Uz…

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Q: Prove that the circle {(x, y), x² + y? = 1} is not homeomorphic to the interval [-1, 1).

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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: 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: Calculate [(ydx – xdy)] wherein y(t) = (2cos(t), 2sin(t)),0 < t < - 27, is an circumference that…

A: Integration

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: Verify Stokes' theorem for the helicoid (1, 0) = First, compute the surface integral: = (VF)…

A: Given surface is helicoid whose vector parametric equation is \)" data-mce-style="cursor:…

Q: 1. Use Green's Theorem to evaluate |F. dr, where F(r, y) = (3y- esin, 7x + Vy + 1) and C is the…

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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: A surface is called bounded if there exists M &gt; 0 such that everypoint on the surface lies at a…

A: Explanation: The possible Quadric surfaces are Spheres, Ellipsoids, Hyperboloid of one sheet,…

Q: Prove that the angle between the tangents from (a, ß) to the circle x² + y² = a² is 2tan ¹(a/√S),…

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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: 2. Find the possible minimum value of the quantity x² + y² under the restriction y = 3x - 4. a) Set…

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Q: Use the distance formula to show that T (x,y) = (-y,x) is an isometry.

A: We have to prove that T(x, y) = (-y, x) is an isometric.

Q: Let f be a differentiable function of one variable. Show that all tangent planes to the surface z =…

A: Concept: The calculus helps in understanding the changes between values that are related by a…

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: 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: b) Consider the following metric: ds² = e²A(r) dt² - e²B(r) dr² — r² (d0² + sin² 0do²), with A(r)…

A: Solution :Consider the following 4D-curved spacetime metric.ds2 =…

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: 0. Show that C[-1,1] c C[0,1]. C[0,1]).

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Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.