Let Uj, U2, Uz € R³ be an orthonormal system and let f : R³ → R be differentiable. Prove that af af af + du (2)' - (4)' (4)' af dr + dy dz

Q: Let w = xydx + 2xdy - ydz E 2'(R) and let a: R a(u, v) = (uv, u², 3u + v). Compute do, a*o, d(a*o),…

A: Given ω=xy dx+ 2x dy - y dz in Ω1ℝ3x,y,z And let α:ℝu,v2→ℝx,y,z3 be given by  αu,v=uv, u2, 3u+v. To…

Q: If r'(t) = (1,21,3r*) and r(0) = (0,1,2) then r(t) equals %3D (a) (1,r°,r'). (b) (1,r",r')+ (0,1,2)…

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Q: If α < α', show that Dα'f⊆ Dαf .

A: Let's assume that α and α' are constants such that α<α', and let f be a function that is…

Q: Let ~f : R3 → R3 and g : R3 → R. Assume ~f(x, y, z) is such that Df(0, 1, 0) = 1 −2 3
If ~g(x, y,…

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Q: Find V (V x F), if F(z, y, =) = 3e""i+7ze'j – 3e"k. V. (V x F) =

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Q: of af and Let f(x, y, A) = xy + A(5 – x – y). Find· ax' əy` af

A: f(x,y,λ)=xy+5λ-λx-λy .....(1) Differentiating (1) partially with respect to x, ∂f∂x=(1)y+0-λ-0…

Q: с) Let A :1, →1, be defined by Ах, х, ...) %3D (0, 0, х2, Хд, .). Prove that A is self-adjoint,…

A: Please find the answer in next step

Q: A. T(f(t)) = |B. T(ƒ(t)) = f(−t) from Pg to Pg = tª ƒ' (t) from P₂ to P5 = f(t)ƒ' (t) from P7 to P13…

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Q: af of H.W: Find and for the functions: b əx ду 1. f(x, y) = (xy-1)² 3. f(x, y) = (x²-1)(y + 2)…

A: Given functions are 1. f(x, y)=(xy-1)2 3. f(x, y)=(x2-1)(y+2) 4. f(x, y)=x(x2+y2)   To Find: ∂f∂x…

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: i) Assume a MA(1)-process given by Y = c+ e +6ɛ,-1, E, ~ N(0, o?), t= 1,... ,T. Compute the…

A: A time series is a sequence of observations taken sequentially in time.

Q: 1 1 1 1 ++* 1 2 1 1 -2 1 1 Which system of ODEs matches the velocity field above? Ox'= 1, y = x…

A: The velocity field is shown in the picture. Consider, x'=yy'=1 This means: y'x'=1yydy=dx Integrate…

Q: Find the differential of w = ln /x² + 3y² + 2z²

A: Here we use chain rule of differentiation to find the required differential. According to chain rule…

Q: Let y: (0, 3] → R?, y(t) = (t,?V³) and I := ran y. Sketch T and compute L(T) :!!

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Q: suppore that f(x.v,Z) = x*y +x2 and x cu,v) = sincu)t(os (u) Y(u, v) = Sincu) - (oslW find of oflav…

A: Given: f(x,y,z)=x2y+xzx(u,v)=sinu+cosuy(u,v)=sinu-cosu To determine: ∂f∂u,∂f∂v

Q: Please take turns to work on the following questions: 1. Let f(x, y) = x² + 3xy + y² Compute (a) af…

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Q: If f(x, y) = (2x? + 8y²)í, then f_(0, 0) = %3D A) 0 В) 1 C) 4 D) 2 E) Does not exist

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Q: Let S={(x,y)∈N×N|(2−x)(2−y)<2(4−x−y)}.Prove that S=TS=T, where T={(1,1),(1,2),(2,1),(1,3),(3,1)}

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Q: Let Au + u = x^2 + y for x and y be in (0, 1). u(x, O) = x^2, u(x, 1) = x, u(0, y) = 0, u(1, y) = 1.…

A: We are given the following differential equation with the boundary conditions.…

Q: Express In (4+z−i) In (4+z-i) in the form of u(x, y)+ jv(x, y)

A: The given problem is to express the given complex expression in the form of u(x,y) + iv(x,y).Given…

Q: Let F(x, y, =) = (6x° In(5y² + 9) + 4=5) i + ( 10yx + 7:)j + (20xz4 + 7y – 37 sin 72) k 5y2 + 9 and…

A: To evaluate

Q: Show that the function f(x) = x’ + 1 is invertible from R to R.

A: First prove that the  function is one -one  let f(x)=fyx7+1=y7+1x7=y7x=y therefore, the function is…

Q: Find the first partial deravatives of the function

A: Since, only question no. 21 is clearly visible in the image you have provided, that’s why we are…

Q: Find free variables in the following formulas: i. ∀x(p(x) -> ∃y~q(f(x), y, f(y))) ii. ∀x(∃yr(x,…

A: A variable used in a mathematical expression, which is neither parameter nor local variable. That is…

Q: Prove z = max (x,y) and z = min(x,y) are convex

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Q: 3. Prove F (Vr)(A→ B) → (Vx)A → (Vx)B. ->

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Q: Prove (u x v)' = u' x v + u x v

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Q: When finding the orthogonal trajectory of the family of curves y = k x³, we dy get dx 2x y Зу 2у 3x…

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Q: XU[-1,1] ve Y ~ U[0,2] iki bağımsız tekdüze (uniform) rasgele değişken ise W = X + Y rasgele…

A: Part 2: Explanation:Step 1: Determine the PDFs of X and Y- X ~ U[-1, 1] has a PDF of:\[ f_X(x) =…

Q: If x = v'u and y = 2uv, show that JJ' - 1, where J is the jacobian of u,v with respect to x and y…

A: We have find the jacobian

Q: W(s, t) = F(u(s, t), v(s, t)), where F, u, and v are differentiable. If u(3, 5) v:(3, 5) = - 2, и,…

A: Let's find.