Write the function f(z) = 3z2− 2z + 4 in the form A.) f(z) = u(x, y) + iv(x, y) B.) f(z) = u(r, θ) + iv(r, θ)

Q: Let f(x, y, z) = 5x³ – y + z?. Find the maximum value M for the directional derivative at the point…

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Q: Find the linearization L(x, y) of the f(x, y) = x°y® at the point (2, 2). L(x, y) = %3D

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Q: What is the directional derivative of f(x, y, z) = bx²y²z² + x² + y² + z² in the direction of v =…

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Q: The function f(x, y, z) = x² – y² = z² increases in the direction at the point (1, 1, 1).

A: We need to find if the statement is true or false.

Q: Find a function f(x, y, z) such that Vƒ = (3z, 4y°,3x).

A: Given that :-       ∇f=3z,4y3,3x  ⇒∇f=3zi^+4y3j^+3xk^  ....(1) Now we know that…

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: Exercise 4. Show that the function f is not differentiable at (0,0) f(x, y) = x³ + y³

A: We have given that ,  fx , y = x3 + y33 Here , ∂f∂x = 3x23x3 + y323 ∂f∂x = x2x3 + y323 At , x , y =…

Q: : Express the function f (x) = e") to the form u(x,y) + iv(x,y).

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Q: 3. Verify that u(x, y) = x² + 4.x – y² + 2y is harmonic and find %3D - a) the harmonic conjugate…

A: See the attachment

Q: 6) If F = (3x²y – z)i+ xz3 + y*)j – 2x³z²k, find V(V - F) at the point (2,1,0).

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Q: 2. If F(x, y, z) = 2z3 – 3(x² + y²)z, then verify that Frx + Fyy + F2 = 0. %3D

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Q: The function f(x, y) = x2 – y² at the point (0,0) is saddle point. صواب İbi

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Q: Find fr, fy, and fz. f(r, y, 2) — вуzIn(4ry) fz fy f: =

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Q: Find the linear approximation to f(x, y, z) = at the point (–2, 3, 2) f(x, Y, z) ×

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Q: (a) f :Z → Z, where f(x) = x². g : Z → Z, where g(x) = |æ|². %3D (b) f :Z × Z → Z, where f(x, y) =…

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Q: he upward flux of

A: Introduction: Let F be a function and let the region is S, then the flux of the function is given…

Q: Evaluate f, xyz dV, where Q = {(x, y, z): 0 < x < 2y, 0 <y< 1, 0 < z < x+y}.

A: According to question,Q = x,y,z : 0 &lt; x &lt; 2y, 0 &lt; y &lt; 1, 0 &lt; z  &lt; x+y

Q: Find a point zo where the function f (z) = (z – 12 – 12i) (z – 12 – 12i) is differentiable. What is…

A: Given f(z) = (z-12-12i)(z -12 -12i)¯

Q: Let f(x, y, z) = 5x³ – y + z?. Find the maximum value M for the directional derivative at the point…

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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: Let f(x, y) = x² – y?. Find (c) The linearization of f at the point (2, –3).

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Q: Find the linear approximation to f(x, y, z) = at the point (2, –1, –2): f(r, y, z) z

A: A Givenfx,y,z=xyz at the point 2,-1,-2linear approximation at the point…

Q: Evaluate fs xyz dS where S is the triangle with vertices (1,0,0), (0, 2, 0), (0, 1, 1).

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

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Q: If Wly, Y2 Y3]= 0, then (y1, y2, y3) must linearly dependent, where y1, y2 , y3 are diffferentiable…

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Q: Find the linear approximation to f(x, y, z) = " at the point (2, -1,1): f(x, y, z) z

A: Given the function

Q: Integrate f(x, y) = (x + y + 1)- 2 over the triangle with vertices (0, 0), (4 , 0), and (0, 8).

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: Find V x F for F(x, y, z)= - -xZ + y² i+ -yz x² + y² =j+√x² + y²k.

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Q: (xy(x²–y²) f(x, y) = { (x²+y²) (x, y) # (0,0) (x, y) = (0,0) ) show that fxy(0,0) f yx(0,0) Let

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Q: What does it mean for the differentiability of a function if only one of the Cauchy-Reimann…

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Q: What does it mean for the differentiability of a function if the Cauchy-Reimann equations (Ux = Vy…

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Q: (4)Find the scalar function such that F = V ø y i+ where F=- *+y x+y

A: We can find scalar function ϕ such that F→=∇ϕ ;where ∇ =i^∂∂x+j^∂∂y+k^∂∂z

Q: Let f(x, y, z) = 3xy + 2yz - xz. Find the directional derivative of f(x, y, z) in the direction of…

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Q: 2- Show that the function U(x,y)= 3x²y-y² + y is Harmonic and then find it's Harmonic conjugate V(x,…

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Q: a) If W = F(x ,y),where x =v²-g and y 2vg calculate W +W, %3D b) What is the directional derivative…

A: W=f(x,y) and x= v2-g2 &amp; y=2vg   Wx = dW/dx = (df/dx*dx/dv) + (df/dx*dx/dg) ..............From…

Q: Find the linearization of f(x,y) at the indicated point (X0,Yo): f(x,y) = In (x² + y); (1, 1) L(x,y)…

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Q: Find the linear approximation to f(x, y, z) = at the point (3, -2, -2): f(x, y, z) Z

A: To find the linear approximation of the below function at the point 3,−2,−2. fx,y,z=xyz

Q: Evaluate F.dr, where F(x,y)=[2y-3cos(x)]i + [4x +6sin(y)]j and C goes counterclockwise around the…

A: The objective is to evaluate ∫CF·dr where Fx,y=2y-3cosxi+4x+6sinyj and C goes counterclockwise…

Q: f (x, y, z) = x² + y² + z² and u = (1,0, 1) at P(1, 1, 1).

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Q: 4)Find the scalar function ø such that F = V ø where F=- y * +y x +y

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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: Find the linearization of f(x,y) at the indicated point (Xo,Yo). f(x,y) = /x +2y; (4,7) L(x,y) =

A: Given:- f(x, y) =√x+2y To find:- linearization of f(x,y) at the indicated point (xo.Yo). f(x,y)=√√x…

Q: 2. Find the directional drevative of f(x, y,z) xsinyz at the point (1,3,0) in the direction of i =i+…

A: Use projection formula

Q: Integrate the function f(x, y) = x – y over the triangle T with vertices (0,0), (1,0), and (1, 2).

A: Evaluate : ∫∫T(x-y)dA        ......(1) ∫∫Tx-ydydx T is the region enclosed by the triangle. Region…

Q: Find the analytic functiopn f (z) = u (x,y) + i ( x, y ) such that u ( x, y)=  4xy3-4x3y+x.

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Q: Find the linearization of the function z =/ at the point (-5, 1). L(1, y)

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Q: Find the linearization of f(x,y) at the indicated point (X0:Yo). f(x,y) = In (x +y); (6, 1) L(x,y) =

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Q: Find f (1, 1) S(x, y) = In( %3D Vx² + y²

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Q: Find the linearization of f(x,y) at the indicated point (X0.Yo)- f(x,y) = /x + 2y; (1,6) L(x,y) =

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