IS a function.. (b) Let R= {(x,y) € R × R : 1 < ¤ < 4 or x = 6}, %3D and S = {(x, y) E R × R : –1 < y < 1}. i. Find the relations S-1 and R-1o S. ii. Represent the relations R, S, RUS and RNS in the (x, y)- plane.

Q: 4) For which one of these linear maps T:R → R? is det T > 0? a) T(1, y) = (y, -r) b) T(1, y) = (x,…

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Q: dw dw at the point (u,v) = dw dw For the functions w= xy + yz + xz, x= 3u + v, y = 3u - v, and z=…

A: Solution:-

Q: Is the map T: R² → R² given by T(x, y) = (x,y + 1) linear?

A: T: V-&gt; W  mapping from V to W is a linear map if and only if  -&gt;(u+v)=T(u)+T(v) -&gt;T(au) =…

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Q: Let f (x, y) = (x − y)e^x . Use the Chain Rule to calculate ∂f/∂u and ∂f/∂v (in terms of u and v),…

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Q: The mapping $ (y) = fy' : H→H* defined by is not a linear m

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Q: Let f (2) be an analytic function, which is represented by f (2) = u (x, y) + iv (x, y) where z = x…

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Q: Given the map f: R3 R4 defined by f(x,y,z) = (0,0,x,0) then rank of f is: O A. 3- O B. 2- OC.- OD.…

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Q: Find the linearization L (x, y) of the function f (x, y) = y/-x at (-9,4). 2 L(x, y) -x+ 3y – 6 3…

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Q: Let f (z)be an analytic function, which is represented by f (2) = u (x, y) + iv (x, y) where z = x +…

A: The general form of a complex number is given by z=x+iy, where x,y are two real numbers and i=-1.…

Q: (b) Let R= {(x,y) E R × R : 1 < x < 4 or x = 6}, %3D and S = {(x, y) E R × R: –1< y < 1}. i. Find…

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Q: The domain of the function f (x, y, z) = In(6 - 3xy) %3D is {(x, y) E R? : xy 2} {(x, y, z) E R :…

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Q: Consider the functions f(x, y, z, t) z?, t?) and g(u, v) = (uv, u +v, u – v). Use the chain rule to…

A: I have solved in general and also using chain rule. Composition  of several variable functions:  Let…

Q: 2. - For each map below, either prove that it is linear or demonstrate that it is not linear. (a) T…

A: Consider a map T:U→V is linear if    (i) T(u+v) =T(u)+T(v)   (ii) T(αu )   = αT(u)   , for all u,v…

Q: Which of the following maps is not linear? А. T:R²→R² defined by T(x,y) = (x²2,x+y)· В. f:R3→R2…

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Q: Let R be the relation on R given by R= {(x, y) : xy > 0}. Is R reflexive? symmetric? antisymmetric?…

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Q: Given a function f(x,y,z) = 5x² In (7y + 3z). Find the domain of the function. %3D o {(xy,z) |z >…

A: Use logarithms definition to choose the correct answer.

Q: Use the Chain Rule to find- af when r = 2, t =-1 of the function ar f(x, y, z) = e3xyz cos 2xyz ; x…

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Q: then Given the map f: R³→R4 defined by f(x,y,z) = (0,0,x,0) rank of f is: А. В. 3. Е. 2. 4-

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Q: Let f (x, y) = x² – y² . Find all all points (æ, y) such that Vf (x, y) = ±(1, 1).

A: givenf(x, y) =x2-y2∇f(x, y) =±(1, 1)

Q: let f (x, y) = x² + xy + y². Using the chain rule, calculate %3D af -(u(x, y), v(x, y)) being u = v…

A: Given that                                   f(x, y)=x2+xy+y2 And also we have…

Q: Let F(x, y, z)=z²i+ 2xj+y°k and let S be the graph of z = 4-x² -y², z 20 oriented counterclockwise.…

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Q: Find dwdt (a) by using the appropriate Chain Rule and (b) by converting w to a function of t before…

A: (a)

Q: Given the map f:R3 R4 defined by f(x,y,z) = (0,0,0,0) then rank of f is:

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Q: The domain of the function f (x, y, z) = In(3xy – 6) is %3D {(x, y) € R² : xy > 2} o {(x, y) E R? :…

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Q: (5) Determine whether Fř is independent of the path when C is any closed curve and (i) F(x, y, z) =…

A: Here we use the curl of vector function to determine its dependency on the closed curve as follows

Q: Compute f, fy, f fy, and fr of the function. f(x, y)=(2x-3y)* |

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Q: (a) express ∂w/∂u and ∂w/∂y as functions of uand y both by using the Chain Rule and by expressing w…

A: The given question should be (a) express ∂w/∂u and ∂w/∂v as functions of u and v both by using the…

Q: Let f(x, y) = x2 − 2y and g(x, y) = xey . Show that at any point (x0, y0), the level curve of f…

A: To show that at any point the level curve of f and g intersects perpendicular, it is enough to show…

Q: The domain of the function f(r, y, z) = In(6 2xy) is O {{z, y, z) E R³ ° : ay > 3} O {{z, y, 2) E R°…

A: see below the answer

Q: Consider the functions f(x, y, z, t) = (x² + 2. t²) and g(u. v) = (uy, u +v. u – v). Use the chain r

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Q: For the functions w =xy + yz + xz, x = 3u + v, y= 3u - v, and z = uv, express , and , using the…

A: Here, w=xy+yz+xzx=3u+vy=3u-vz=uv

Q: Find the domain of the piecewise function: x < 0 0<x <1 2х, y = In x, Vx-1, x 21 Select one: O a. x…

A: By finding the domain of each piece definition of functions we thereafter take the union of all to…

Q: Let f: R → R be the function which satisfies for all (z. y, z) € R° that z cos(ry) and let C C R' be…

A: Here we have to find out line integral of function f(X,Y,Z).

Q: Let z = f(x, y) = e**y while x = 2st and y = s² + t³. Use a chain rule to find ôz/8t at the points…

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Q: dw dw For the functions w= xy + yz + XZ, x = u + v, y =u - v, and z= uv, express du and using the…

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Q: Which of the following maps is not linear? O A. L-R2-R3 defined by L(x,y) = (x+y,y.x)- OB. K:R R3…

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Q: Use the chain rule to find D (f o g) (–1,2) for f(u, v, w) = (v² + w², u³ – vw, u²v+ w) and g(x, y)…

A: This question is related to derivative of composite function.

Q: ду Assume that the relation In (x°z°12) + 6(x – 1)y + 512xz + 729y°z=0 defines y implicitly as a…

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Q: 2: Let a function d: R2 x R2 R defined by d(x, y) = ((x1 – y1)² + (x2 – y2)²)* %3D = XA Vx =…

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Q: The domain of the function f(x, y, z) = In(2xy - 6) o {(x, y, z) e R': xy > 3} o {(x, y, z) e R: y…

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Q: (b) Let R= {(x, y) E R × R : 1 < x < 4 or x = 6}, and S = {(x,y) E R × R : –1< y < 1}. i. Find the…

A: Solution: The given relations are R=x,y∈ℝ×ℝ : 1≤x≤4 or x=6 and S=x,y∈ℝ×ℝ : -1&lt;y&lt;1

Q: Let RC [0, 1] ×x [0, 1] such that (x, y) E R (i.e., xRy) whenever 4x < 6y < 9x . Deter- mine if R is…

A: R is not a function, because there can be multiple ordered pairs with the same first coordinate,…

Q: For the map F: R³ → R² defined by F(x, y, z) = (x²y + e², yz − x)

A: Given,            F(x, y, z)=x2y+ez, yz-x

Q: Let F(x, y) = <3x?y3 + y", 3x°y² + y^ + 4xy³,. %3D Evaluate F. dr where C is the line segment from…

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Q: The domain of the function f (x, y, z) = In(6 – 3xy) %3D is {(x, y) E R² : xy 2} {(x, y, 2) E R³ :…

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Q: For the function P(A, B) , where both A and B are functions of t, r and w, use the chain rule to…

A: Given function is PA,B, where both A and B are the functions of t, r and w. We have to state the…

Q: Let f(x,y) = x - 5xy -y². Compute f(6,0), f(6, – 2), and f(b,c).

A: Solved below

Q: 2. Consider the function f : R²R such that f(x, y) = x²y-2xy² +6xy-3y² +18y. (a) Show that f has…

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