4. Show that the function f : X → Y defined by f(X) = X U {a} vX €x is a bijection.

Q: Let F(x, y, z) = y° z°i+ 2xyz³j+ 3xy° z²k. (a) Find the domain of F, and show that F is…

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Q: find the linearization L(x) of ƒ(x) at x = a. ƒ(x) = x3 - 2x + 3, a = 2

A: We find f(x) and f'(x) at x=2

Q: Let z = ƒ(u, y), u = ax + by, and y = ax - by. Express zx andzy in terms of fu , fy , and the…

A: Given, Let z=f(u,Y) u=ax+byY=ax-by

Q: Let f(x, y) = xcosy- y'e*y. fry at (1,0)is equal to: %3D | -2/3 -2 1

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Q: Find the linearization L(x) at x= a. f(x) = x+ a = - 1

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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: Let A={x e R|-1sx s1} = B, Show that f:A →B given by f (x)=x|x| is a bijecti on.

A: we have to show that given function is bijective

Q: The mapping $ (y) = fy' : H→H* defined by is not a linear m

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Q: 3. (a) Show all functions satisfying u"(x) = 0 are linear by integrating the expression of u"(x) %3D…

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Q: 18.7. Let f(z) = (e + e-2)/2. Evaluate f(2) -dz, where y is any %3D simple closed curve enclosing 0.

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Q: 9. If o = F(x, z) dy + G(x, y)dz is a (differentiable) 1- form on R', what can F and G be so that…

A: GIven, ω=F(x,z)dy+G(x,y)dz ⇒dω=∂F∂xdx∧dy+∂G∂xdx∧dz+∂G∂y-∂F∂zdy∧dz Required dω=zdx∧dy+y dx∧dz on…

Q: Find the linearization at x = a. f(x) = 6x®, a = 1

A: Linearization L(x) of function f(x) at a is given as L(x) = f(a) + f'(a)(x-a)

Q: Show f(x) = e and g(x) = xe linearly independent by finding its Wronskian.

A: Given: fx=exgx=xex The formula to obtain the Wronskian is Wx=f(x)g(x)f'(x)g'(x), where f(x) and g(x)…

Q: please solve this linear algebra problem.

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Q: 3. Show that every y E lị defines a continuous linear functional on l through the inner product,…

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Q: Let D, be the linear transformation from C'[a, b] into C[a, b]. Find the preimage of the function.…

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Q: Let, T:R' → R³;T(x,y,z)=(2x+ y, y- z,2y+4z) 2. Find the rank and nullity of T.

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Q: 9. Show that the function (-, -) : IR² × R² → IR defined by setting = #1#2 + 2x12 + 2y1*2 + 8y1Y2 /2…

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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: 2. Use Wronskian to show that fi = x, f2 = ln x, and f3 C° (R). x In x are linearly independent in…

A: Concept:

Q: Evaluat..f.x.dA. wha.is.the..sgiem.in.the.fiigt. R quandrant..bounded..xy=16. X..8....y.z.. y.e.x...

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Q: Show f(x) = e2= and g(x) = x e²z linearly independent by finding its Wronskian.

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Q: 3. Let C[0, 1] be equipped with the inner product (f. g) = f(x)g(x)dx, where f = f(x) and g = g(x).…

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Q: fx) =x+x, aニー Lcx) = ? (メフフ %3D

A: Given that fx=x+1x,      a=−1

Q: Find a function u = u(x, t) that satisfies the PDE Uxx = 0, subject to the boundary conditions 0 0,…

A: Consider the given differential equation, uxx=0, 0&lt;x&lt;1, t&gt;0 The auxiliary equation of the…

Q: If z = f(u) + g(u), where u = = and v = x · y. Prove that x · Zx – y · Zy = y² ·Zyy – x².Zxx using…

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Q: Let f(x, y, z) = In(x² + z) + y²e®%. Find fzym. Zyx•

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

A: Given function is: f(x,y,z) = ln(3xy-6) We need to find the domain  for this function.This can be…

Q: Let A: C [0,1]→C [0,1]. Prove that operator continuous reversible, find A. 2.11. Ax(t)= x(1)- -…

A: The given operator A such that A:C0,1→C0,1. We use the inverse operator method for solving linear…

Q: 2. Use the Squeeze Theorem to show that f(r) = 2 is in R(0, 3).

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Q: Show that the function f(x, y, 2) = (x² + y² + 2e**)4 is strictly convex on R³.

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Q: A function f(x,y) is homogeneous of degree 7 in x and y if and only if, __________.

A: Given problem is solved below.

Q: 4. Show that the map K: [1, o) → [1, ∞0) defined by 1 t dt K(x) = 3 e 2 is a contraction in the norm…

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Q: Q1:- Let T =- where W-N(0, 1) and V~xShow that T2-F(1,r).

A: We know that  The ratio of chi-square divided by its degrees of freedom is called the…

Q: X = R² and = = { $₁x} V { \\x>0} where U4 = {(x,y) | x² + y² <²} Prove (x, t) is Topology

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Q: Determine the extreme of the following functional: T/2 J(x) = | [x{(t) + * (t) + 2x1 (t)x2 (t)] dt…

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Q: Exercise 4.12 Give an cxample where f(x) - b as x → a and g(y) + c as y – b but g(f(x)) + c as x → a

A: To find an example where fx→b as x→a and gy→c as y→b but gfx→c as x→a. Let b∈ℝ be a fixed real…

Q: Compute the weak solution to the problem (a(x)u')' – u = x _for x € (0, 1), u(0) = u(1) = 0 , 1 with…

A: For the solution follow the next steps.

Q: Show that the function u = p(ry) + JFÿ¢(y/x) provides the quation „Pu = 0. dy

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Q: Prove that H(x, y) = (x, –y) provides a conjugacy between 1 X' = 1 X and Y' Y. 1 1 1 1

A: We are Given that :  H(x,y)=(x,-y) X'=11-11X , Y'=1-111Y The Matrix H is corresponding to the…

Q: fx) =x+x, aニー %3D LCx) = ? %3D

A: Our objective is to find the linearization L(x) at x=a.

Q: Are the following sets of functions, each defined on x € R, linearly independent? Use Wronskian. (a)…

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Q: If Z = f(x + g(y)) , prove: Zx * Zyx = Zy * Zxx

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Q: 1 Are the following sets of functions, each defined on x E R, linearly independent? Use Wronskian.…

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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: Let RC [0, 1] ×x [0, 1] such that (x, y) E R (i.e., xRy) whenever 4x < 6y < 9x . Deter- mine if R is…

A: R is not a function, because there can be multiple ordered pairs with the same first coordinate,…

Q: The domain of the function f (x, y, z) = In(6 – 3xy) %3D is {(x, y) E R² : xy 2} {(x, y, 2) E R³ :…

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Q: Find the Wronskian for the set of functions.{x, x2, ex, e−x}

A: Given information: x, x2, ex, e-x The Wronskian is given by the following determinant.…

Q: 5) Let z = f(x2 - y²). Show that y = 0 ду %3D ax

A: Here, ∂ represents the partial derivative, which means when differentiating with respect to x,…

Q: 2. Use Wronskian to show that fi = x, f2 = ln x, and f3 C (R). x In x are linearly independent in

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