Let r = ⟨x, y, z⟩ and r0 = ⟨1, 1, 1⟩. Describe the set of all points (x, y, z) such that r − r0 = 2.

Q: 3. Let r>0 and define the closed ball by B,(z) = {y € X : d(x,y) Sr} Show that B,(r) is closed set…

A: Given X be a metric space with metric d. Now define a topology τ on X as follows: τ=B⊆X:…

Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y.

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Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y. *

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Q: let (X; Z) v it piece wise comnected ?

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Q: Let X=ℝ2 and define d,:ℝ2×ℝ2→ℝ d2((x1 ,y1),(x2,y)2)) = max{|x1 - x2|,|y1 - y2|}. Draw the…

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Q: Which of the following maps from R³ to R³ are not linear? f(x, y, z)=(x, y, z + 1). f(x, y, z) =…

A: The function f is said to be linear if: f(X+Y)=f(X)+f(Y)f(kX)=kf(X) where X,Y∈ℝ3.

Q: (c) If X= R, with r={ R, ¢, Q, Irr}. Show that (R, t) is a compact space.

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Q: Which of the following maps from R³ to R³ are not linear? U □ f(x, y, z)=(x, y, z + 1). f(x, y, z) =…

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Q: Let X and Y be two T, spaces, then X×Y. :is

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Q: Let R = [0, 1] × [0, 1] . Est each with at least six rectan IR 6(x + y) dA 2

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Let r(t) = ⟨t, t, t⟩. Compute r'(t) and interpret the result.

A: rt=t, t, t

Q: Let X; ~ X"(1) i= 1,2, . - ·, n and assume they all uidependent are Y = Ź Xi %3D しニ」 ان منتتفينتمنل…

A: Given, Xi~χ12 ,i=1,2,...,n a. Y=∑Xii=1n Therefore, X1+X2+.....+Xn~χn2 above is possible using…

Q: of at (x, y, z) = (2, 3, 4). ay Let f(x, y, z) = (x + y)z. Evaluate

A: Given :fx,y,z = x+yz

Q: Find f, and f, and evaluate each at the given point. y' f(x, y) = e*y2, (In 4, 2) f,(x, y) = %3!…

A: We can find the answer as below,.

Q: Consider the sets X = {(x, y) E R²: x² + y² ≤ 1) and Y = ((x, y) = R²: x ≥ 0]. Sketch on R²the sets…

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Q: Let, T:R' → R';T(x,y,z) = (x + y+ z,2y+z,2y+3z) 2. Find the dimension of KerT and ImT

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Q: Let X= [0, 0 ), and defined d as d(x,y)= e(xy) then verify all four condition of metric space

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Q: Prove that f, uv v • dỉ = – §̟ vũu • dĩ

A: Whenever the force is conservative in nature the work done over the closed loop is always zero. i.e…

Q: Let (X, T1) and (Y, T2) be topological spaces. Show that the set {U × Y ∣ U ∈ T1} ∪ {X × V ∣ V ∈…

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Q: Let F(x, y) = (x, 0). Prove that if C is any path from (a, b) to (c, d), then F dr = 3 – a²)

A: Consider the provided information, Let Fx,y=x,0. Prove that if C is any path from a,b to c,d then…

Q: 3. Find the closure and interior of the set A = {(x, y): xy = 1} as a subset of R? (equipped with…

A: Closure and interior

Q: Find f, and f, and evaluate each at the given point. f(x, y) = eXy2, (In 5, 2) fo(x, y) f,(x, v) =…

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Q: Let f (x, y) = x² – y² . Find all all points (æ, y) such that Vf (x, y) = ±(1, 1).

A: givenf(x, y) =x2-y2∇f(x, y) =±(1, 1)

Q: (f) The set B = {(z, y) E R² : ² + y < 1} ; is disconnected. as a subset of R? with indiscrete…

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Q: Let f(x,y,z) = x² e Vy*+ Compute f(3, - 5,3) and f(5,8, - 6).

A: Given that     f(x,y,z) = x2 e√(y^2+z^2)

Q: Show that the linear map (x, y)H (x+ y, x- y) from (R?, || - |1,) to (R?, I| - ||is an isometry.

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Q: The domain of the function f(r, y, z) = In(6 2xy) is O {{z, y, z) E R³ ° : ay > 3} O {{z, y, 2) E R°…

A: see below the answer

Q: Show that fi(r) = rª and f2(r) = r² +1 are orthogonal on the interval [–1, 1).

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Q: Let F(x, y, 2) = (z, 1, y). a. Find divF and curlF.

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Q: Let r E R and --(), v- (), -- (1). 3 u = V = W = 2 For which values r ER is the set {u, v, w}…

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Q: Evaluate F dr where F(x, y, z)= (x', y’, xyz) and C is the line segment from х, у, со (1, 4, 3).

A: Use parametric to solve

Q: Let C be the rectangle in the xy-plane with vertices (0,0), (1,0), (0, 2) and (1, 2), and let F(r,…

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Q: Let w = f(u), where u = x g(y), show that w,xx = f" (x g(y)). g²(y). %3D

A: Let w=f(u), where  u=x g(y). So using the chain rule, differentiating with respect to x, we get…

Q: Let u = (1, 1), v = (3, -2), w = (4, 3), and y = ( map from R² -3, 4). Let L be a linear | to R?…

A: We know Let  Given

Q: Let F(x, y) = <3x?y3 + y", 3x°y² + y^ + 4xy³,. %3D Evaluate F. dr where C is the line segment from…

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Q: Theorem:For all Xiyiz,w EN i) X<y ^ z sw =)x+Z <ytw ii) xsy ^ zw =) x+Z< y tw )x<う tsw 1 Esw うメ+ と x…

A: Solution: We know that for x,y∈ℕ If x&lt;y then ∃k∈ℤ+,k&gt;0 such that x+k=y If x≤y then ∃k∈ℤ+,…

Q: Which of the following maps from R³ to R³ are not linear? 0 f(x, y, z)=(x, y, z + 1). f(x, y, z) =…

A: The solution is given below

Q: Use greens theorem to evaluate line interal (x2y)dx+y2dy),where c is the closed path formed be y=x…

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Q: Consider the space L'[0,1]. Let p(f) :

A: Given, pf=∫1234f212, f∈L20,1. Standard L2-norm is f2=∫01f212

Q: For normal linear spaces X, Y, prove that BL (X, Y) is complete if Y is complete.

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Q: Prove that the map || - || : L(V, W)→R defined by ||L|| = sup {|| L(x)|| | ||| < 1} is a norm on…

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Q: Sketch the set of all points (x, y) such that x + y| < e*.

A: Consider the function x+y≤ex. An absolute function is defined as x=x if x≥0-x if x&lt;0. So, there…

Q: Consider the set R. Prove: (i) VxER, x 0 = 0;

A: To prove: For any x∈ℝ, x·0=0.

Q: Find f and f, and evaluate each at the given point. y' f(x, y) = e*y, (In 4, 2) f,(x, y) = f,x, y) =…

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Q: Let f(x, y, z) = 2x(y+z), x uv, y = v w, and z = w. Find in terms of u, v, and uw.

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Q: Prove that if w = f(x,y,z), x=r-s, y=s-t, and z =t-r then ôw ôw ôw + âr' as = 0. ôt

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Q: (b) Describe graphically the set A={(x,y):d[(x,y), (0,0)]<3}for the following metric space on R²…

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Q: Let ƒ (x, y) = x² - y² . Find all all points (x, y) such that Vf(x,y) =±(1, 1).

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Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.