Q.2. Let t, be the particular point topology on X (a, b, c, d), and f:(X,T)-(X, r.) is defined by f(a) = f(b) = c, f(c) a, f(d) = a. Is f continuous or not?

Q: 4) For which one of these linear maps T:R → R? is det T > 0? a) T(1, y) = (y, -r) b) T(1, y) = (x,…

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Q: (a) Assume that g(x, y) is a function of the form g(x, y) = 0 if (x, y) = (0,0). (i) Show that g(x,…

A: As per the policy, we are allowed to answer only one question at a time. So, I am answering the…

Q: X= R² and T = { $₁,x} V {₂\x>0} where U₁ = {(x, y) | x² + y ² < 2² } Prove (XT) is Topology

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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: If f: (a,b]-R has a local minimum at pe (a,b) and f is differentiable at p, then prove that f'(p) =…

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Q: f(z) = Im z is continuous 0 صواب ihi O

A: True

Q: The domain of the function f (x, y, z) = In(6 – 2æy) is O {(z, y) E R² : zy > 3} O {(z, y, 2) E R° :…

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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: 4. Let f(x, y) be a function defined on a disk D not containing the origin which is given by f(r, y)…

A: See the answer below

Q: 6. Consider S = [a, b] with p(x, y) = |x – y|. Verify that T : S – S with T(x) = - and a > 1 is a…

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Q: t BLX,X) denoted set f al incod bounded operates an defined on ow hat fex every TE BLXX) for Hhe…

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Q: (a) Assume that f(x, y, z) and F (x, y, z) are arbitrary differentiable functions such that f(x, y,…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

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A: Let Ω∈Rn be open with smooth boundary. (a)To prove the Green's function is uniquely determined:…

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Q: Given the vecor-valued function R(t) = cosht i ln(t+1) j+et k, find lim h→0 R(0+ h) - R(0) h O…

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Q: (10) Define the metric d on R² by d( (x₁, y₁), (x2, y2)) = max{ |x₁ − x2\,|y₁ - y2|}. Verify that…

A: The given problem is to verify that the given metric d on R2. Given metric…

Q: Let A = Z. Let R be defined on A x A where a Ry if 4 (x- y). Prove that R is transitive.

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Q: 2. Use the Divergence Theorem to compute ffs F. dS for F = (xy, yz, x² z + z² ) and S the boundary…

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Q: How to solve this?

A: We are given that x + y + z = 5 where x, y, z ∈ R. Now, we need to prove that x2 + y2 + z2 ≥ 5/3.…

Q: Let F(x, y, z) = (x-1z, y¬lz, In(xy). (a) Verify that F = V ƒ, where ƒ(x, y, z) = z In(xy). (b)…

A: Given Fx, y, z=x-1z, y-1z, lnxy (a) Verify for F=∇f where fx, y, z=z lnxy The gradient of a function…

Q: (b) Show that the function defined by IN if (x, y) (0,0), √√√z²+y² f(x, y) = 0 if (x, y) = (0,0) is…

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Q: 7. let E be the reyion bounded by curve 2+2 in the yex? fint quadant let u xy, Vzy-x d Czry)…

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Q: (5) Determine whether Fř is independent of the path when C is any closed curve and (i) F(x, y, z) =…

A: Here we use the curl of vector function to determine its dependency on the closed curve as follows

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: - Let f(z) = cosh z. (1) Find the image of S := {z = x+iye C: x < 0,0 < y < } under f.

A: Follow the steps.

Q: (a) Let Ty if +y +0, g(x, y) = 0 if r = y = 0. Prove that g(r, y) is continuous at (0,0).

A: To Prove: g(x,y) is continuous at (0,0)

Q: If f(x, y) is continuous on [a, b] × [c, d], and g(x, y) = Sª S' f(s,t)dsdt, prove that, for a < x <…

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Q: What is the minimum value of the function f(z, y) =ey on the set D= {(z,y) | 16z2 + y <1}? %3D…

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Q: i) True or False: T = {0, R}U {(-x, a] : a € R} is a topology on R.

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Q: Suppose f(x, y) is continuous and non-negative over a non-empty open region R in the plane with a…

A: Based on given data,For Non empty open region R,fx, y is continuous,andfx, y is non -…

Q: Let f (z)=z. Find zpl 2) from z =0 to z =4+2/ alongthe curve C given z 21 to z =2i.

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Q: Define T: (C¹ [0, 1], || · ||…) → (℃ [0, 1], || ||) by Tf = f'. Show that T is not continuous. Is…

A: T:C'[0,1], .∞→C'[0,1], .∞ T(f)=f' Let f(x)=xn then T(x)=nxn-1 T(x)=nxn-1 as n→∞,   T(x)→∞ So T is…

Q: If f(z) = u + iv is entire and u(x, y) is bounded above, apply Liouville's Theorem to exp(f(z)) to…

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Q: The domain of the function f (x, y, z) = In(6 – 3ry) is - o {(x, y) E R² : xy > 2} O {(x, y, z) E…

A: Please refer to the image below

Q: Evaluate the limit: r(t + h) – r(t) lim h→0 for r(t) = (1-6, sin t, 8) h r'(1) = ( ...

A: We have to find components of r't

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A: We have to prove that   √a+√b≤ √(a + b) for all a, b ≥ 0.

Q: At what points (x,y,z) in space are the functions continuous? a. h(x,y,z) = In (2z2 – x² – 3y³ – 4)…

A: we want to find the domain of given  function

Q: (3) Suppose that T: V F is a linear map. Prove that if u E V such that u 4 Ker(T) then V = Ker(T)…

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Q: Let f : C[0,1] → P be given byf (x ) = x (1)Vx e C[0,1]. Show that f is continuous w.r.t the supnorm…

A: f is continuous with respect to sup norm. Let xn be a sequence of continuous functions in 0,1 such…

Q: 4- Test the continuity of following at (0, 0) and if di type of discontinuity. fx.y) y + 0 (i) 0, y…

A: (i) Given that, f(x,y)=xexy2yy≠00y=0 Take x=y2, f(x,y)=y2e1y=ye So f(x,y)→0 as y→0.  And y=0 then…

Q: If f:[a, b] →R has a local minimum at p E (a, b) and f is differentiable at p, then prove that f'(p)…

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Q: Let T : R³ → R³ be a surjective linear map. What can you say about the dimension of ker(T) ?

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Q: Let F(r, y) = yeyi+xe*vj. Evaluate the line integral | 1 F. dr where C is the closed path consisting…

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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: . Suppose f(a, y) is continuous and non-negative over a non-empty open region R in the plane with a…

A: According to question,f(x, y) is continuous,fx, y ≥ 0        non - negative,∫∫Rfx, y dA = 0