(d) Show that the function defined by h: N → N: h(x) = 2*, Vx € N is one-to-one but not onto. Now consider the function defined by j: R → R*: j(x) = 2*,Vx € R. Is j bijective? Justify your answer. If so, find j~1.

Q: Find the divergence of the field. F = (2x+7y-8z)i + (5x +3y+6z)j + (-6x-y+2z)H div F =

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A: Solution

Q: c) For x e ll n2, define || x ]| = || x || 1+ || x ||2: Show that (11 n2, ||· || is a Banach space.

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A: It is provided that the Wronskian of f and g is tcost-sint. We need to determine the Wronskian of u,…

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: Z. F(x, у, 2) %3D y z x/

A: Concept:

Q: Exercise 0.3.23. (a) Prove that the function f(x) : = vex over [0, ∞). 1+ex is strictly con-

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

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A: Wronskian of the function

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Q: Let f(x) = sin x and g(x) = cos x, and W = span{f, g}. 10.1 Use the Wronskian to show that f and g…

A: As per our norms we have to write one question at a time kindly repost remaining questions.

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A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

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Q: 4. Show that for any bounded linear operator T on H, the operators T₁=1/√(T+T*) and T₁=1 (1 ==…

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A:

Q: Let A(x) = S° f(t) dt, with f(x) as in figure. %3D 1,8 - 0,5 -1 A(x) has a local minimum on (0, 6)…

A: Given: A(x)=∫0x f(x)

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Q: 1. Determine whether f is Riemann integrable over (0, 1] and evaluate S f(z) dr whenever fE R(0, 1])…

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Q: Define f : X → K by f(x) = x'(1), x € X. Again, the linear functional f is not continuous. By 6.4,…

A: Given: f:X→K by fx=x'1, x∈X To prove f is not continuous using the given corollary.

Q: Exercise 3. Show that f : Z → N so that f (z) = z² + 1 for all z E Z is neither one-to-one %3D nor…

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Q: 5) Determine the entire function f(2) = u+ jv satisfying Re(f(2)) = u(x, y) = ae" cos(y) + r? – y²,…

A: That's easy. Thumb up. Have a good day!!! We use Milne Thomsan's theorem

Q: 3. Find a field F(x,y) = (M(x, y), N(x, y)) in the xy-plane with the property that at each point

A: Let f = xi + yj  Then F = ff = unit vector = xi+yjx2+y2

Q: 2. Consider the function f(z) = ( -3) on (0, 1]. (a) Prove that / for r, ye 0, 1], (b) Prove that /…

A: Contraction mapping theorem: Let (M, d) be a metric space. A mapping f:X→X is contraction mapping,…

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: fi (x) = x, f2(x) = cos x, fa(x) = 5

A:

Q: Show that the Dirichlet function f defined on [0 , 1] by f(x) = { 1 if x is rational 0 if x is…

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

A: We have given operator A such that A: C[0,1]-C[0,1].Also we must know that the inverse operator…

Q: Find the divergence of the field F. F = 8x 5 i + 3y 5 j + 2z 5 k 8x4 + 3y4 + 2z4 O 40x4 + 15y4 +…

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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).

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Q: Find the divergence of the field. F = (- 6x + 8y – 4z)i + (- 8x – 5y + 3z)j+ (x + 5y + 2z)k div F =

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Q: Let A: C [0,1]→C [0,1]. Prove that operator A is continuous reversible, find A. Ax() = x() + ja-m}xt…

A: Solution:- We have given operator A such that A: C[0,1]- C[0, 1] Also we must know that the inverse…

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

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Q: (a) Let U be an open subset of R" and consider two continuously differentiable scalar valued…

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Q: If the Wronskian of f and g is t cos t - sin t, and if u = f+ 2g, v = f - g, find the Wronskian of u…

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Q: Let f(x) = sin x and g(x) = cos x, and W = span{f,g}. Use the Wronskian to show that f and g are…

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Q: Find the divergence of the field F. yj - xk F= 1/2 +: 2 OA. 3/2 2 -X O B. 3/2 2 X -y OC. 3/2 O D. 0

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Q: Let r; = ro + hj for j = 0,1,2. Use the quadratic polynomial interpolating f at those nodes %3D %3D…

A: The solution is given below

Q: Let T: C* (R) → C (R), T(f) = f' – 2f be a (linear) differential operator on the space of infinitely…

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Q: Find the divergence of the field F. F = yj-xk (y2, x2 1/2 x2- y2 (y2+ x23/2 -x2 ( 2+ x33/2 x2…

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Q: Find the Wronskians of the given sets of functions and determine whether the functions are linearly…

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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: ANSWER:

Q: Let c(t) Use the chain rule to compute (foc)'(t). (cos(t), sin(t), t²) and f(x, y, z) = x² + y² +…

A: The objective is to find the derivative.