Which of the following transformations are linear? (a) T₁: R² R², T₁ R², T₂ I Y = Y = xy I (b) T₂: R² Y (c) S₁: F→ R, S₁(f) = f²(0) + f f(t)dt (d) S2: F→ R, S2(f) = f(0) + f f(t) dt (e) M₁: F→ F, M₁ (f) = (t² +1) f(t)

Q: B. Find the gradient of the function at the given point. S(x, y, z) = (x² + y² + z?)i+ In (x y z)

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Q: (a) z = x² + y² – 4x in P(1,2) (b) z = yln(x +y) in P(-3,4) %3D

A: a) Compute the gradient of z=x2+y2-4x in P1,2 as follows: It is known that the gradient is…

Q: Explain why the function is differentiable at the given point. Then find the linearization L(x, y)…

A: Note that : Since you have posted multiple questions, we will provide the solution only to the first…

Q: 4. Solve the following initial value problem using Laplace transforms: y" + 3y + 2y = g(t) with y(0)…

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Q: (a) Show conclusively that the map f : R² → R defined by f(x, y) = x cos(y) is not linear.

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Q: Find the gradient of the function at the given point. f(x, y) = 2x + 5y² +6, (3, 1) Vf(3, 1) =

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Q: et r be a three-dimensional real vector-valued function such that r′′ exists. Show that d/dt [r(t)…

A: Given: r is a real vector-valued function in three dimensions, and r'' exists. Our aim is to show…

Q: Q.4 (a) Calculate the Gradient, . F (x, y, z) =- x+ y+ z ·j+ x+ y + z x+ y+ z

A: The gradient of a vector function F→(x,y,z)=f1(x,y,z)i^ +f2(x,y,z)j^+f3(x,y,z)k^ is given by…

Q: Let f(x, y, 2) =x² + 3r°y? + 2°, g(u, v) = (u² + v², uv, 3u – 2v). Find the derivative of f(g(u, v))…

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Q: Compute the gradient of each of the following functions. а. f(х, у, 2) %3D х sin(yz) b. f(x, y, z) =…

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Q: Consider the function: f(z,y) = 1 – xy coS TY (a) Find the linearization L(x, y) of ƒ at (1, 1) (b)…

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Q: (b) Consider the function F : R³→R³ such that 2x1 2x2 cos x3 – e²2r –x3 sin x3 – 2e"2 X3 +4 F(x) =…

A: Definition of Jacobian:        A determinant which is defined for a finite number of functions of…

Q: Find the partial derivatives of the function -7x – 4y f(x, y) 8x – 5y f:(x,y) fy(x, y) =|

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Q: Use the Chain Rule to calculate the partial derivatives д(т, у) — sin (т? + у?), х— 4и — v, у — —…

A: Given g(x, y)=sin(x2+y2) ,x= 4u-v and y=-(2u+2v)

Q: 1. Find the gradient of the function f(x, y, z) = ln(xy) - zx² at the point (1, 1,-1). 2. Find the…

A: The given function is f(x,y,z)=ln(xy)-zx2 We have to find the Gradient of the f(x,y,z)=ln(xy)-zx2 at…

Q: Find the gradient of the function at the given point. f(x, y) = 2x + 4y² + 4, (3, 4) Vf(3, 4) =

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Q: Find the second-order partial derivatives of f(x,y,z) with respect to x, y, and z, respectively.

A: Here we find second order derivative.