find ∇f at the given point. ƒ(x, y, z) = x2 + y2 - 2z2 + z ln x, (1, 1, 1)

Q: Find the linearizations L(x, y, z) of the functions at the given points. ƒ(x, y, z) = xy + yz + xz…

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Q: If f(x,y,z)=e3x+4ycos5z  then satisfy laplace equation

A: Given function is: fx,y,z=e3x+4ycos5z The condition for Laplace equation is: fxx+fyy+fzz=0 Partial…

Q: Find Vf at the given point. f(x, y, z) = 32³ – 2(x² + y²) z + tan '(4xz), (0, 2, 3) Vf\(0,2,3) f(x,…

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Q: Evaluate 3xz dV where E = {(x, y, z) | 1 < æ < 3, x < y < 2æ, 0 < z < x + 3y}

A: ∬∫Ef(x,y,z) dV = ∫x=ex=f∫y=cy=d∫z=az=b f(x,y,z)dzdydx                                where , E…

Q: integrate ​f(x,y)=x2−y over the curve​ C: x2+y2=4 in the first quadrant from (0,2) to 2,2.

A: The solution are next step

Q: (a) Find the gradient and Laplacian of the scalar function T(r,0,¢) =r(cos0+sin0cosø)

A: As two questions are given I will solve only the first: We are given: T(r,θ,ϕ)=r(cosθ +sinθcosϕ)…

Q: Find d3f where f(x, y, z) = ln(x + y − z) + cos(x2y)

A: As per the question we are given a multivariate function as : f(x, y, z) = ln(x + y − z) + cos(x2y)…

Q: 10. Does the function f (x, y, z) = (x² + y2 + z?)-1/2 satisfy Laplace equation?

A: The objective is to check the given function satisfies Laplace equation or not.

Q: 5. Show that the function f(x,y,z) = 2z³ – 3z(x² + y²) satisfies the Laplace Equation, a²f + ду? əx²…

A: The given function is: f(x,y,z)=2z3-3zx2+y2f(x,y,z)=2z3-3x2z-3y2z

Q: Find the linearization L(X,Y) of the function  f(x,y)=y-1/x+1 at(0,0).

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Q: Show that a function f(x, y, z) = (x² + y² + z²) satisfies a Laplace equation.

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Q: find the linearization L(x, y) of the function ateach point. ƒ(x, y) = (x + y + 2)2 at a. (0, 0), b.…

A: Given function,  f(x, y) = (x + y + 2) 2 To find linearisation L(x, y) at i) (0, 0)  ii) (1, 2)

Q: Find the linearization z = = L(x, y) of the function z = f(x, y) = 3.x2 – y² + 5y at the point (xo,…

A: z=fx, y=3x2-y2+5y and the point is x0, y0=2, 1. Now, f2, 1=322-12+51=12-1+5=16 Now partially…

Q: The function u(x, y) = 2020xY is a solution of the Laplace's equation True False

A: The given equation is: ux,y=2020xy  The given function is a solution to Laplace's equation.

Q: Find the linearization L(x,y,z) of the function f(x,y.z) at the given points. f(x.y.z) = e* + cos (y…

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Q: Using the second fundamental theorem of calculus and chain rule, compute dz t dt. t+1 2x

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Q: dz Use chain rule to find at (u, v) = (1, 1) if dv z = z sin(y² – z + 1)4), e = 3u – v², y = vº.

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Q: Find and ди ди using the chain rule. w(x, y) = 14x²y², x = e" cos v, (u, υ) ди (u, υ)

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Q: Solve the Laplace equation subject to the boundary conditions, u,(0, y)= 0, u(2, y) = 0, u, (x, 0) =…

A: The Laplace equation is: ∂2u∂x2+∂2u∂y2=0 The given boundary conditions are: ux0,y=0, ux2,y=0,…

Q: find the linearization L(x, y) of the function ateach point. ƒ(x, y) = e2y-x at a. (0, 0), b. (1, 2)

A: Given:fx,y=e2y-x at   a.0,0    b.1,2Formula:Lx,y=fa,b+fxa,b.x-a+fya,b.x-b

Q: Show that the function f(x, y, z) = x² + y² – 222 sat isfies the Laplace equation. - Good Luck.

A: Given function f(x,y,z)=x2+y2-2z2. We have to show function satisfies the Laplace equation. A…

Q: Solve by theorems of linearity and translation. f(t) = cos t + sin t

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Q: dw Use the Chain Rule to find dt w = x²e²y cos(32), x = cos t, y = In (t + 2), z = t

A: The chain rule states that the derivative of f(g(x)) is f'(g(x))⋅g'(x).

Q: d use the Chain Rule to calculate f(r(t)). dt f(x, y) = x² – 3xy, r(t)= (cos t, sin t), 1 = }

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Q: Find the linearizations L(x, y, z) of the functions at the given points. ƒ(x, y, z) = (sin xy)/z at…

A: Linearization of a function of three variable[say f(x,y,z)] at (a,b,c) is…

Q: Let f(x, y, z) = ln(x2 + z) + y2exz − cos(2yz). Find fzyx.

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Q: Find the linearizations L(x, y, z) of the functions at the given points. ƒ(x, y, z) = xy + yz + xz…

A: Function: f(x,y,z)=xy+yz+zx, at(0,0,0) We need to find fx,fy,fz. at the point (0,0,0)

Q: Show that the function f(x, y, z) = x² + y? – 2z² satisfies the Laplace equation.

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Q: Show that the function satisfies the Laplace equation f (x, y, z) = (/x2 – 2y² + z?)*

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Q: Show that the function z = 5xy satisfies Laplace’s equation                                ∂2z/∂x2 +…

A: Given function: z=5xy

Q: 1. (a) Let f(r,y)- cos(z + 2y). Compute the diectional derivative Df(r,9), where u- (b) Compute ff,…

A: Solve the following

Q: 1) f(t) =t cos(@t) 2) f(t)=tsin(@t) 3) f(t) =t cosh(at)

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Q: Find fa, fus and fz. f (x, y, z) = (5z)) 3z In (x*y cos

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Q: 9. Demonstrate that the functions Y1 = x2, y2 = sin(x), y3 = xe are linearly independent on R.

A: Recall: If y1,y2 and y3 are the given function and         Wy1,y2,y3x = y1y2y3y1'y2'y3'y1''y2''y3''…

Q: Find the linearizations L(x, y, z) of the functions at the given points. ƒ(x, y, z) = x2 + y2 + z2…

A: To find the linearization Lx,y,z of the function at the given points fx,y,z=x2+y2+z2 at 1,0,0

Q: Find fa, fy, and fz. f(x, y, z) = 5z ln(x³y³ cos(z)) fx = fy fz f₂ = =

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Q: Using an appropriate linearization, the local linear approximation of f() = T+ cos z at COS I is -…

A: Use linear approximation

Q: Find the linearization L(x,y) of the function f(x,y) = e* cos (9y) at the points (0,0) and ...

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

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Q: If x'+y' +z-9, dx/dt = 5, and dy/dt = 4, find dz/dt when (x,y,z) (2,2,1)

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Q: (c) Show that the function z = e* sin y + ev cos x satisfies Laplace equation ax? a2z = 0.

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Q: The two functions y1(t) = sin t cos t and y2 (t) = sin 2t are linearly independent. O True False

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Q: Let h(r,y) = r'e. Find its linearization at the point (1,0) and use this to approximate f(2,0). %3D

A: Given function is h(x,y)=x2ey we have to find linearization at point (1,0). then to approximate…

Q: siven f(x,y,z) = sin (3xy) + 3xyz3. Find fxzy(3,2,2).

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Q: x + Y f(x, y, z) = x sin + e*% – 1

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Q: E(z, t) = ex cos(@t – kz + Þ1) + ey cos(wt – kz + $2) = €;E× +e,Ey %3D

A: Consider the given equation: Where,

Q: Let z = f(x, y) = 2xy + x. Write the differential dz in terms of x, y, dx, and dy. %3D

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Q: Suppose x, y, and z are positive functions of t. If z x + y?, dx/dt 1, and dy/dt 2, find dz/dt when…

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Q: 6x Find the linearization L(x,y) of the function f(x,y) = e cos (y) at the points (0,0) and

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Q: Find the linearization of the function f(x, y, z) = xy+ yz + zx at the point (1,1,1).

A: Linearization of the function f(x,y,z) at (a, b, c) is given by…