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-

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,…

A:

Q: State the area of the region, determined by the lines, y = 14x, y = 4x + 2, y = 14x – 4, y = 4x + 7…

A: we know we will use jacobian to transform the change of variables here .

Q: 4. Calculate the integral (x – 3y) dA where R is the triangular region with vertices (0,0), (2, 1)…

A:

Q: Show the function f(z) = z (z) is conformal at 0 and f'(0) = 0. Does this function violate our…

A:

Q: 4. Let L: R² +R? be the linear map defined by L(x, y) = (x + y, I - y). Show that L is invertible.…

A:

Q: Bounded by the planes z= x, y =x, x+y=2 and z= 0

A:

Q: a¹y¹2'dadydz, where V is the (6) Using suitable transformations, evaluate closed region in the first…

A: Given the integral  ∫∫∫Vx4y4z4dxdydz Where V is the closed region in the first octant bounded by the…

Q: TL (7,9) dydz = T" (17,19) dzdy. For any function f(r, y) continuous on the ry-plane, )True False…

A: To find- If the given statement is true or false.

Q: Given the map f:R-R+ defined by f(x.y.z) = (0,0,x.0) then nullity of f is: O A.4 O B. 3 OCO O D-2 OE…

A:

Q: Find the Jacobian for the transformation x = 3u + v and y = u+ 3v. NEXT, use this transformation to…

A:

Q: G: Evaluate 4 px=(y/2)+1 2х - у Z. dx dy dz 3 0 Jx=y/2 by applying the transformation u = и %3D (2х…

A: Given the integral  We need to applying the transformation, u = (2x-y)/2 ,  v = y/2  and  w = z/3…

Q: and y = 2u, evaluate the integral xy³dA where c) Using the transformation x =-; 6u region R is…

A:

Q: 2. Suppose that f (x, y), g(x, y) are two continuous functions on the xy plane. Is it true that So…

A:

Q: Transform the BVP 0<x<1 u (0, t) =t u(1, t) = 1 into new one with homogenous boundary conditions, we…

A: To find -

Q: Let the function f is defined on the disk x² + y? < 1, then f has a critical point somewhere on this…

A:

Q: 1. Use an appropriate transformation of coordinates to evaluate xy dA where the region D is enclosed…

A: Special transformation is used to calculate double integral. Given integral is enclosed with the…

Q: 6. Let D be the region in the ryz-space defined by the inequalities 1 <z< 2, 0 < ry < 2, and 0 < z <…

A: Given integral, ∫∫D∫x2y+3xyzdxdydz where  1≤x≤2, 0≤xy≤2, 0≤z≤1 and transformation u=x, v=xy, w=3z…

Q: 1) Find the outward flux of F across the region D where F= (6x² + 2xy, 2y + x²z, 4x² y³ ) and the…

A: The vector field is,F¯=6x2+2xy, 2y+x2z, 4x2y3The region D is cut from the first octant by the…

Q: (6) I. adv, where E is bounded by the planes z=0, y = 0, =0, y+= =1 and +=1, as shown in the figure…

A: Given that,

Q: Let F = (-z2, 2zx, 4y - x2}, and let C be a simple closed curve in the plane x + y + z = 4 that…

A: Given F={-z2,2zx,4y-x2}. Curve in the plane (s) = x+y+z=4. Area enclosed is 16.

Q: 3) Prove that the multiplication map · : Z/nx Z/n → Z /n given by [x] · [y] = [xy] is well-defined.

A: To prove that the multiplication map · : ℤn×ℤn→ℤn given by    x·y=xy  is well-defined.

Q: Let H be the plane 2 X- 2y- z=0 in R', that is, H={(x, y, z)eR³ | 2x- 2y- z=0} and let F be the…

A:

Q: Use the transformation u = - -6x +y, to evaluate the integral [ ( (5x • y) dx dy, where R is the…

A: use the transform u=-6x+y, v=5x+ywe evaluate ∫∫R 5x+y dx dywhere R is the parallelogram bounded by…

Q: Are the following linear maps invertible? a) T: R3 → R³ defined by T(x, y, z) = (y, z, x)?

A:

Q: 2. Show that the function g defined by VVT y² + √² 2, is not continuous at (0,0). (z,y) = (0,0)…

A:

Q: Find the linearization L(x,y,z) of the f(x,y,z) = 6sqrtx^3+y^3+z^3 at the point (1,2,3).

A: The linearization of a function f(x,y,z) at a point (a,b,c) is given by the expression;…

Q: Compute fS fdS for f(z,v, z) = 2y-z and S is the part of the cone 2-2+ betweenm z 1 and z 4.

A: Integration is the process of finding the anti-derivative function of a given function. The…

Q: If the function f is defined on the disk x2 + y² < 1, then f has a critical point somewhere on this…

A:

Q: Use the transformation u = y - x, v = y, to evaluate the integrals on the parallelogram R which has…

A: We need to evaluate the integral: ∬Ry-xdA using the transformations: u=y-x, v=y. Here, R is a…

Q: 2. Use the Divergence Theorem to compute ffs F. dS for F = (xy, yz, x² z + z² ) and S the boundary…

A:

Q: Let F(x, y, z) = (x-1z, y¬lz, In(xy). (a) Verify that F = V ƒ, where ƒ(x, y, z) = z In(xy). (b)…

A: Given Fx, y, z=x-1z, y-1z, lnxy (a) Verify for F=∇f where fx, y, z=z lnxy The gradient of a function…

Q: Let z = F(x , y), where x = u ev - 4 and y = ut e -V + 3. Use the chain rule %3D az to find du at (u…

A: Concept- using the chain rule of differentiation

Q: 4. Let f(x, y) be a function defined on a disk D not containing the origin which is given by 2.x2 f…

A: as per our company guideline we are supposed to answer only one qs kindly post remaining qs in next…

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

A:

Q: Find a transformation that таps R = {z = x + iy : x > 2, y > 1} onto S = {z = x + iy : x < -1, y…

A: We need to find a transformation that maps R = { z = x+iy : x ≥ 2, y ≥ 1} onto S = { z = x+iy : x ≤…

Q: 20. Let D be the region in xyz-space defined by the inequalities 1< x< 2, 0 < xy < 2, 0 << 1.…

A:

Q: 4) Verify that Green's theorem is true for f,xy?dx – x²ydy ,where C consists of y = x² from (-1,1)…

A:

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: Find the kernel of linear map L:R R given by L(x,y,z)%3D(0,z,y)?

A: There is no such introduction for the solution.I have explained it within the solution.Please go…

Q: Suppose we have a surface f (x,y)=x²y+cos (4x) · Which of the following is the gradient of f(x,y) at…

A: First I have found the partial derivatives of f(x,y)  with respect to 'x' and 'y' at point P(π/2,…

Q: Use the transformation u = x + y and v = x- 2y to evaluate || 3xy dA for the R region R bounded by…

A:

Q: 2. Compute the line integral of E from B(0,0,0) to A(2,1,3) if E = -4xyax - (4y + 2x2)a, + 8zaz. Use…

A:

Q: Show that F(x, y, z) = (y² – e* sin z)i + (2xy + z)j + (y – e* cos z)k is conservative and hence…

A:

Q: 16. Let B be the region in the first quadrant of the xy-plane bounded by the lines x + y = 1, x +y =…

A: I have Provided this answer in step-2.

Q: What is the Jacobian of the transformation T that arises from an appropriate change of variables to…

A:

Q: Use the transformations u = xy and v=xy* to evaluate ff xyd4 where R is the √√3 1 X region bounded…

A:

Q: Let Juce) 2ntv () %3D A=O n!F (I+u+n) Bessel function Of the first Kind be a Of Order V Show that…

A:

Q: Let z = f(x, y) and r =r cos 0, y = r sin 0. Use the chain rule to express: af %3D 11. af dy Te and…

A:

Q: Let F = (r-y, z+y+3, z2) and S y-direction. Compute {(r, y, z) : y =1-r²-z², y20} oriented in the…

A: Here orientation is positive y direction, surface is open.  During the calculation we ignore many…

Q: X, Y be topological spaces and f : X Y be a continuous map. If X is completely normal then Y is also…

A: