Exercise I f(1, y)yz (a) Let f(r,y) = rỷ - . Then deduce that f(1,y)zy (Clairaut's theorem). %3D

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

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Q: f(x,y,z) be a differentiable function the point P(2,5,-1). Let at Which of the following is the…

A: Answer is mentioned below

Q: The maximum directional derivative of the function f (x, y, z) = x'y°z° at the point (1, 1, 2) is a)…

A: Given problem:- Find the maximum directional derivative of the function f(x, y, z) = x³y³z³ at the…

Q: The maximum rate of change of the function f(x, y, z) = xyz at the point (1, 2, 3) occurs in the…

A: In this question we have to find the correct answer.

Q: then ö(x, y) he value of fC

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Q: The function f(x, y, z) = x² − y² − z ² increases in the direction 〈六〉 (1, 1, 1). at the point

A: Please check step 2 for solution.

Q: Evaluate | f(x,y) ds where Cis the portion of the curve y = In x from the point (1,0) to the point…

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Q: (B) If f(x,t) = fakt e-y²dy, then prove that kfx = ft.

A: (.)  Leibniz rule :  Given

Q: knowing that f(x,y,z) =0 ,find dx/dz if (z=x+y) (dz/dx should be interm of fx,fy,fz)

A: Given: f(x,y,z) = 0 and z = x+y

Q: EXERCISE 4 Find Vf(P) and D7f(P). (a) f(r, y, 2) = y² + z² – 4xz; U = } i – § j+ ; P(-2,1,3) %3D (b)…

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Q: 8. Find the directional derivative off at the p in the direction of v = () of f(x, y, z)=x²exx, p =…

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Q: Find the directional derivative of Du" f (2,1) in the direction of: a = 3i + 4j To f(x, y) = InGes…

A: Gradient of a function :  Let f be a differentiable function at  x , y  and let f x , y  have…

Q: Q4: Find the derivative of the function at P, in the direction of A 2 1) f(x, y) = 2xy-3y², P.…

A: According to our guidelines we can answer only one question and rest can be reposted.

Q: Integrate f(z) over unit circle C and indicate whether Cauchy's theorem may be applied (4) f(z) =…

A: With the help of Cauchy theorem of integration we solve this problem.

Q: Determine fzz, fyy, and fz for f(x,y,z) = x²y³z – y sin(xz) + e».

A: When the partial differentiation is done with respect to a specific variable, then all other…

Q: (c) If and are smooth scalar eius, show that Vx (6V) = Vox Vv (d) Let f(x, y) = 2x² + xy-y². Prove…

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Q: Find the directional derivative of f(x, y, z) = ((x+y+z)e², (x² + y²)e-²) at c = (1, 0, 1) in the…

A: The given problem is to find the directional derivative of the given vector function at point…

Q: Evaluate / = F dr where F [y+z, z+x, x+y] and C: r(t)= [t, sint, cost], from A:(0, 0, 1) to B:(T, 0,…

A: Answer

Q: if f(z)=u(x,y)+iv(x,y)=z^2+3z+4 find u & v

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Q: For the generating function A(z)=- find the corresponding r≥ 0. 1-9z

A: Given: The function generating function is Az=21-9z2. To find: The numeric function corresponds to…

Q: determine wave direction in the function f(x,y) x^y — x²y³ decreases faster at the point (2,-3)

A: Wave direction in the function f(x,y)= x4y-x2y3 that decreases faster at the point (2,-3) is…

Q: Calculate Vf, V•f, V×f f (x,y. =)= xi+ y'j-* by the function

A: Introduction: The curl of the vector f(x,y,z)=fxi+fyj+fzk is given by ∇×f=∂∂xi+∂∂yj+∂∂zk×fxi+fyj+fzk…

Q: The function f(x, y) = ¿e=y – In(y²)

A: It is partial derivative

Q: Calculate for the function 2x-9 z = f(z,y) = , z(u, v) = v'e", y(u, v) = u'e %3D %3D z+3y Select the…

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Q: of Let f(x,y) = xy° +7. Evaluate at (x,y) = (-7,1) and interpret the results. ду -

A: In this question we have find δfδy at (x, y)=(-7, 1)

Q: af Calculate at point (-1,2) where f (x, y) = 8+ x² + 3y? dv and v = i+j. 10 2 5/2 V2

A: fx, y=8+x2+3y2  &  v=i+j Partially differentiating with respect to x, we get fxx, y=0+2x+0=2x…

Q: f(x, y) = xsin²y + y² cosx, Find fxy a. sin² y - y² cosx b. 2xsinycosy-2ysinxcosy c. 2sinycosy -…

A: We need to find fxy.

Q: Let f(x, y, z) = xyz Find the direction in which the function increase most rapidly at the point…

A: Given: fx,y,z=xy2z3 The gradient of a function if given by ∇f(x,y,z) =∂f∂x,∂f∂y,∂f∂z.

Q: = xye-(a*+v'+z*). Find all points (x, y, z) such that Vf(x,y, z) Question 2. OConsider the function…

A: We have to find all point (x , y, z) such that the gradient of f(x, y, z) = (0, 0, 0).

Q: A function F is defined on R2 by F(x,y) (picture)  let f(x,y)  = (x2y+2xy2) / (x2 + y2)  Determine…

A: The partial derivative of f(x,y) with respect to x at (x0,y0)  is given by  fx(x0,y0)=limh→0…