Define A: C → C° by A(z,, Z2, Z3) = (a,z1, azz2, lzz), «; e C. When is A a positive operator ?

Q: Let T be function from R to R defined as T (z) = (r, x), %3D Then T is not linear operator O True…

A: FALSE  the detailed explanation is as follows below:

Q: The vector field F is shown in the xy plane. 3- 2- 2 3 4 Is div F positive, negative, or zero?…

A: Divergence is a operator which operate on vector field F to gives a scalar quantity . And divF is…

Q: 4. Is this vector field F(x,y)=(ycos x)ĩ +(sin x+ y)j being the path-independent field? Please show.

A: To check if the given vector field is path-independent field or not. Given:

Q: . Show that the operators T₁,, T4 from R2 into R2 defined by (1, 2) (1, 0) (§1, §2) (0, §₂) (§1, §2)…

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Q: When is the vector field continuous?

A: given,       Continuous vector field .

Q: Match the vector field with its graph. F(x, y) = yi -6+ 世 -5+

A:

Q: Compute the curl of the vector field F = - (x + 3y) i + (y – 3z) } – (z + 3x) k curl

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Q: Show that the operators T and T₂ from 2 R² into R² de fined by T₁: (F₁) √₂ ) → (9₂9₁) and T2=…

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Q: Match the vector field with its graph. F(x, y) = yi KE y -5+

A:

Q: . Consider the vector field F(r, y, 2) = (8r sin y, 4x² cos y – e"z, -e" – sec² z). %3D | F. dR…

A: For a vector field F→, the line integral about the path C is ∫CF→·dr. Let rt be the parametric…

Q: Prove that F(x, y, z) = (y² cos x + 2*)i+ (2y sin x – 4)j+ (3xz2 + 2)k - is a conservative vector…

A:

Q: Match the vector field with its graph. F(x, y) = yi 5 5 -5+ サ

A: To match the given vector field to its graph, where given graphs are given below.

Q: Show that F = (2xy + z³)ï + (x²)j+3xz²k is a conservative field. Find the scalar potential and the…

A: Given F = 2xy+z3 i ; x2j + 3xz2kThe objective is to show that given vector fieldis coonservative…

Q: The magnetic field along a solenoid is conservative and given by 2xy 7- (2 x ₁-2x, (x²+1)-2). ¡−2x,…

A: Given a vector field as F=2xyx2+1-2x,lnx2+1-2 We need to find the scalar potential from which this…

Q: Which are eigenvectors of the operator S:R^3->R^3 defined by S(x,y,z)=(y+z,x+z,x+y) ? Check all that…

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Q: Let UsX +Sinx Deter of X 0n [0,)where ļ Find . derivalire fa 2x+1 x²+7 te tn 6x+

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Q: Is there a vector field G on R³ such that curl G = (x sin y, cos y, z - xy)? Explain

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Q: Let u = (x²yz, ry z, ryz?). Compute curl i

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Q: Find vector x and y in R? that are orthonormal with respect to the inner product (u, v) = 4u1v1…

A: Let x=14,0, y= 12,0 Hence, by using the given inner product, we have x,y=414(0)…

Q: Is there a vector field Ğ on R³ such that curl Ġ = (xyz,–y*z³,y*z*)?

A: the given function is, G→=xyzi^-y4z3j^+y3z4k^ If there exists a vector field G then the divergence…

Q: a) Define a linear operator on R' by performing the following operations in order: i) a shear in the…

A: We are authorized to answer three subparts at a time, since you have not mentioned which part you…

Q: Prove that da(x) = a(b) a(a), %3D |

A: This is a problem from Reimann Stieltzes Integral

Q: Find the curl of the vector field F(x,y,z) = x²zi – 2xzj + yzk and the point (2, -1, 3).

A: Let the given vector field is :

Q: compute dr for the vector field where Cis the line segment from Co.o) to Cli3)

A: Given vector field is, F=1+x2,xy2  The objective is to compute ∫CF·dr, where C is the line segment…

Q: Prove that f, uv v • dỉ = – §̟ vũu • dĩ

A: Whenever the force is conservative in nature the work done over the closed loop is always zero. i.e…

Q: Is = xjx + yjy+zźz3 an inner product on R'?

A: No it is not an inner product.  Inner product looks like a dot product.

Q: Match the vector field with its graph. F(x, y) = yi -5+ -5+

A:

Q: Sketch the vector field on given byF(x, y, z )= z K

A:

Q: (b) Show that the Linear guen Operators. are not with T(f)= Lf6), f()+1].

A:

Q: Q2/ Show that {1,cosx, sinx,cos2x, sinx,...} Form an orthogonal set on [-Tt,t] *

A: A set of vector B is orthogonal if every vector in the set is orthogonal. Therefore, the dot product…

Q: Find the work F dr done by the force field F = (y, x, z) on a particle that moves along C the line…

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Q: Find the level surfaces of the scalar field U = ear, where the vectors are: a = (2, 2, 1) and r = =…

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Q: Use Cramer's rule to solve for y without solving for x, z, and w. 5 x + y + z + W = -7 5x + 3y - z +…

A: Here we have to use cramer rule for solving the system of linear equations

Q: Let Te B(H) be an invertible operator. Show that a(T-) = 0(T)-.

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Q: Find the curl of the vector field F=(2xyz, -xy,-z^2)

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Q: let c is a differentiable vector field then div(curl)=

A: This question is related to vectors.

Q: Is every liner operator T : IR2 → IR2 diagonalizable? Justify!

A: Linear transformation and diagonalization

Q: Find the curl of the vectòr field F(x, y, z) = 2xyzi + 8yzj + 5zk

A: According to our guidelines we are supposed to do only one question.  Kindly repost other question…

Q: Let F(x,y,z)=(2xz^2,−2xyz,9xy^3z) be a vector field and f(x,y,z)=x^3y^2z (∇f=(  ,  , )). (∇×F=(  ,…

A: Given:F(x,y,z)=2xz2,-2xyz,9xy3zf(x,y,z)=x3y2z

Q: 5. Is the vector field F(x, y, z) = (x(z² – y²), y(x² – z²), z(y² – x²)) incompressible? Justify…

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Q: Let C: |z-i=3 and I = §. dz 2z dz. Then %3D

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Q: Explain why Green’s Theorem proves that if gx = ƒy,then the vector field B = ⟨ƒ, g⟩ is conservative.

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Q: Define the operators Δ, (, E, μ and δ. Prove the relation μ= √(1+¼ δ²) and μ-1= 1 – δ2/8+3 δ4 /128.…

A: Defining the difference operators. Let y=f(x) be a function defined on the interval [c,d] and let…

Q: Let i = (x²yz, ry z, zyz?). Compute curl i

A: The given vector is: u→=x2yz, xy2z, xyz2

Q: Find a vector potential field for Ġ = [6xz+x³, –3r²y – y², 4.x + 2zy – 32²], i.e. find F such that V…

A: The solution is below.

Q: Sketch the vector field F(x, y) = (-x − y, x - y) at the points (x, y) where x, y = {−1,0,1}.

A: Consider the given vector field, F→x,y=-x-y,x-y Here we need to sketch the vector field at the point…

Q: Select the option that gives the vector field that could correspond to this arrow map.

A: The vector field is when a vector is assigned to each point in the given space. The cartesian form…

Q: Consider the operator T: 1 given by T(r1, 22,. ) = (0, r1, 22,. ). Is T a compact operator? Explain…

A:

Q: Let T: R² → R² be the linear operator given by 7 ] - ([ ; ) = [ - {|- } {} } -x + 2y Зх + у and let…

A: We can solve this using given information

Q: Suppose L: R → R° be defined by u, +17 Lu2 2u2 %3D Uz Uz Determine whether L is a linear operator or…

A: According to question given :L:R3→R3Lu1u2u3=u1+12u2u3We have to determine whether L is a linear…