Let d be a trivial metric on a non-empty set X and d₁: XXX → R such that x = y d₁(x,y) = { 2 if x #y' Show that d&d, are equivalent? vx, y EX

Q: Use Stoke's Theorem to evaluate Į F· dr where F (x, y, =) = , and C is the triangle with C vertices…

A: given Fx,y,z=z2,y2,xy ⇒Fx,y,z=z2i^+y2j^+xyk^ and a traingle C with vertices 1,0,0 ,0,1,0 ,0,0,2 to…

Q: Let O = (0,0), and a = (2, –1) be points in R2. Set G = B (O, 1) = {v = (x, y) € R² : d2(O, v) < 1}…

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Q: Show that the into R² defined by T₁ = ( 9₁ ₂1₂) — 172,9₁) and T2: (F₁, F₂) → (Y¶₁₂ Y {₂) : are…

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Q: ,A, B,C be 4 points in R², no three of which colinear. Fix A' e VA, B' e VB and C' e VC h that AB ||…

A: In the following question, concept of parallel line is applied. Parallel Line Parallel lines are…

Q: If X =R and d(x.y)30, then X metric space not metric semi metric

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Q: Let X = (1, 2, 3), Tx = (0, X, (1), (2, 3}}, then (X, Tx) is %3D %3D 1) a T1 space 2) a T2 space

A: T1 space: A topological space X has the T1 property if x and y are distinct points of X, there…

Q: ) Consider the space R^2 equipped with the sunflower metric. Identify explicitly the set of points…

A: The objective of the question is to identify the set of points in R^2 that form a ball of radius 1…

Q: Use a projection to derive a formula for the distance from a point (x,y) to the line ax+by=c, where…

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Q: fa) = b, Ab) = a, c) = C, 1d) = d is a one-to-one function, as each element is an image of exactly…

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Q: 21 - Let V = R3,S = 5 Show that = Span(S) %3D

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Q: 1 -2 4 17. Let a 4 a2 -3 and b : For what 7 value(s) of h is b in the plane spanned by aj and a2?

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Q: Suppose that Find the Jacobian Metrix for map.

A: Given y1=x1x2-x2x4y2=x12-x42y3=x1x2x3x4

Q: Homework 4. Show that the following are metric on R": d₁(x, y) = max₁sisn |X₁-yil; d₂(x, y) = -1 |x₁…

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Q: use the Fundamental Theorem of line Integrals to evaluate F. dR, where F (*.y) = < y. - X and C is…

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Q: Let u = 4 h = V= and w= 1 . For what value of h is w in the plane spanned by u am

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Q: Evaluate F.dr where F(x, v)=xyi (4, 0).

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Q: Let v = (a) 27 t (5, 2, 5) and w= (1, 5, 0). - = 4w = (b) Let 3 (t, 1, 0). For which value(s) of t…

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Q: / (sec. sec u tan u -1) du

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Q: Name the preimage of R'. T' T V' R' R Your answer: O T O V O R

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Q: Let = (0,0), and a = (2,-1) be points in R². Set G=B¹2 (0,1)={v = (x, y) = R²: d₂(0,v) <1} H = B¹¹…

A: Let, and  be points in  SetTo find:Sketch  and The coordinates system,, and  be points in  Set

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: O Let ( X, T) be topdogical space and A,B CX- ldentify the Condition when a (Int(A)u FrlB) Ul F CA)…

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Q: 24). Consider the frinction fa)= 2. :) t the Find the equatien ") Eaplenin why the intersectio…

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Q: #4. Set XyZ F(x4a)= | 4 3ゃex xy tyz+zX メ*ytを Show that F is 11 an open set ontaining (1,0,-1). in O…

A: Given - Let Fx, y, z = xyzxy + yz + zxx + y + z To Show -  Show that F is 1 : 1 in an open set…

Q: graph the inequality on a coodinate plane x &gt; with line under it 2

A: To graph the given inequality x≥2.The equation x=2 has a graph that is a vertical line with all…

Q: Which of the following maps from R² to R³ are R-linear? 2 f(x, y) = (x. x - 2. x+y+z) f(x, y) = (x…

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Q: C is the line segment from (-1,0, 3) to (1, 3, 4), and F(x, y, z) = zi+xj+yk, evaluate Jo F.dr.

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Q: Which of the following is the least distance from the point (2, –1,1) to the plane x + y – z = 2?

A: Given: The plane equation is x+y-z=2and the distance from the point is 2,-1,1 To find the least…

Q: Evaluate e [ f(x, y, z) ds, where f(x, y, z) = 2 + 2y + (z + x) and C is the line segment from (2,…

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