Let G : R? → R³ and F : R³ → Rª be defined by G( (x, y)) F( (a,b, c) ) (x +y, x – y, 2y) (a + b+ c, 2a + c, b + 2c, a + b). (a) Compute F o G( (x, y) ). (b) Let E, E', E" be the standard basis for R², R³, Rª, respectively. Find the matrices of G, F, and Fo G relative to these bases; that is, find [G]E, [F]E", and [F o G]E". (c) Verify that [F]E GE = [F 0 G]E".

Q: Let z=x+iy ∈ C where x∈R\{1}, y∈R, and |z|=1. If w = (z+1)/(z-1), then show that Im(w) = y/(x-1).

A: Use the following formulae, to obtain the required result. Real part of a+ib=aImaginary part of…

Q: 16: Let f(t,x) be piecewise continuous in rand Lipschitz in x on [t,,t,]x W with a Lipschitz…

A: The given statement is as follows. Let ft,x be piecewise continuous in t and Lipschitz in x on…

Q: The adjoint operator of the operator L(u): d'u ;xE[0, 1] %3D dx2 with boundary conditions u(1)=2…

A: We are given   The boundary conditions at x=1 is given as u(1)=2u(0); u'(1)=u'(0)

Q: Z. F(x, у, 2) %3D y z x/

A: Concept:

Q: Two functions u and v of x and y are said to satisfy the Cauchy- Riemann eguations if u, = v, and u,…

A: Thje given functions are,u(x,y)=y(x2+y2) , v(x,y)=x(x2+y2)

Q: Let W = F(x , y , z) be a function such that Fx(0 , 0 , 0) = -1 , Fy(0 , 0 , O) = 3 , F2(0 ,0,0) = -…

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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 F(x, y,z)=xi + yj+zk, f(x,y,z)=||F(x,y,z)||. Prove that: Vf" = nf"-²F F v(In f) =

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

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Q: In Exercises 23-28 explain why fis continuous. 26. f(x, y, z) = sin (xyz – 1) 27. f(x, y, z) = ln(e"…

A: The function is continuous if all partial derivatives exist.

Q: Let f(x, y, z) = ln(x² + z) + y²e®z – cos(2yz). Find fzyn. Zyx ·

A: Given function is  f(x, y, z)=ln(x2+z)+y2exz-cos(2yz)

Q: 3. Show that F(x,y, z) = (1+ yz) i+ (z + xz)j+ (y + xy) k is conservative. Hence find its potential…

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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: 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: Let T: R2→R2 such that T(1, 0) = (0, 1) and T(0, 1) = (1, 0).(a) Determine T(x, y) for (x, y) in…

A: Given,         We have to find the transformation.

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: Show that the functions e*, cos x, and sin x are linearly independent using the Wronskian.

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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…

Q: (c) Define f on C[0,1} by : f (x) = ] x(t)dt, xeC([0,1]. Prove that f is a bounded linear functional…

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Q: Let f(x, y) xy, x = 1- Vt, and y = 1+ Vt. Find df using the chain rule. dt

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Q: Let F(x, y, z) = z* +x+y sin x +x In z 0, where z =f(x, y), then at (0,0,1) and %3D are equal to:

A: Given,          Fx,y,z=z4+x+ysinx+xlnz=0 , where z=fx,y

Q: Let T: R2→R2 such that T(1, 0) = (1, 0) and T(0, 1) = (0, 0). (a) Determine T(x, y) for (x, y) in…

A: Given Data T(1,0)=(1,0). T(0,1)=(0,1).   (a) Let, The general expression for the linear…

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: Prove that the function f(x1,x2) = In (e*i+= + eV=i+1) is convex over R².

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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: let f (x, y) = x² + xy + y². Using the chain rule, calculate %3D af -(u(x, y), v(x, y)) being u = v…

A: Given that                                   f(x, y)=x2+xy+y2 And also we have…

Q: Let z = F(x , y), where x = u ev - 4 and y = ut e -V + 3. Use the chain rule %3D az to find du at (u…

A: Concept- using the chain rule of differentiation

Q: Prove that the function f(z2) defined by x*(1+ i) – y°(1 – 1) , (z ± 0) x² + y? f(z) = [0, (z = 0)…

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Q: Compute the Wronskian of a set of functions {vı, y2} = {e* cos(/), e* sin(v¤)} cos (Væ), e* sin(va)}…

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Q: Find g: R R, continuous in [0. ) and position in (0, ) satisfying g1) %3D )dt

A: Here we have,  g:R→R, continuous in [0,∞) and position in (0,∞) Also given g(1)=1 and…

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: Show that e, sin x, cos x are linearly independent using Wronskian Method.

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Q: Let F(x,y,z) = z* + x+ y sin x + x In z 0, where z fx.y), then at (0,0,1) and are dy equal to: ax

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Q: д. L. = a(x, y)+b(x, y) +c(x, y). – d(x, y) -

A: Proof: Consider V be any vector space of all functions of two variable x  and y whose partial…

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: 2. Let z = sin(xy) + x siny,x=u² + w² and y = uw. 2a. Compute Vz= when u = 0 and w = 1.

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Q: Consider p(x, y, z) = 2xyz - x²y³ v(x, y, z) = xy² + y sin(z) F(x, y, z)= F₂(x, y, z)i + F₂(x, y,…

A: Let F→=f1, f2, f3. The gradient of the scalar field: g is defined as: ∇g=∂g∂x i^+∂g∂yj^+∂g∂zk^. The…

Q: Show that e", sin x, cos a are linearly independent using Wronskian Method.

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Q: Let f(x,y, z) = In(x²2+ z) + y²e*z – cos(2yz). Find fzyx·

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Q: Consider the functions z = - e* In y, x= In (u cos v), and y = u sin v. dz dz (a) Express du and dv…

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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: Compute the value of f.dr If F= -yi+xj and c is a from (-1,1Y to CI,1) %3D SemICirele

A: Proceed as shown below

Q: 41. Let f(x, y, z) =x?y+ y²z. Use the Chain Rule to calculate af/as and af/at (in terms of s and f),…

A: Since f is function of x, y and z.  For partial derivative f with respect to s and t we using chain…

Q: [0, 1]. If f(1), g () e V then show that (f(t), g (t)) = [ƒ(t) g (f) dt defines an inner product on…

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Q: Let f(x, y, z) = x2y + y2z. Use the Chain Rule to calculate af /as and af /at (in terms of sand t),…

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Q: Let z = f(x, y) and r =r cos 0, y = r sin 0. Use the chain rule to express: af %3D 11. af dy Te and…

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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.

Q: (a) Let F z sin y, where r(s,t) = st and y(s, t) = st. Use the chain rule to find %3D %3! OF

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