5. If F = A × OP, where A is a constant vector field, show that V x F = 2A 6. If R = OP,r = ||R||, find V × R and if F = p(r)R, where o is differentiable, demonstrate that

Q: If the vector field v = [mz+x, -y, x]is irrotational, the mis equal to O A. 2 O B. 0 O C. 1 O D.…

A: Curl of v is zero.

Q: Show that work performed by a constant force field F over any path C from P to Q is equal to F - PÓ.

A: A vector has 2 components, magnitude and direction which is represented by cos and sin angles. A…

Q: 6. Let F = (x³ sin y, a* cos y). COS (a) Is F a conservative vector field? (b) If your answer to…

A:

Q: 4. Is the vector field F(r, y, :) = (yz,rz, ry) conservative? Justify your answer.

A:

Q: Suppose the field lines of a gradient field F: R2 + R² are hyperbolas and the length of each vector…

A: To find- Suppose the field lines of a gradient field F : ℝ2 → ℝ2 are hyperbolas and the length of…

Q: (b) Suppose C is a curve of length N, and ||F|| < M, where M is some positive number. Prove that ·…

A:

Q: Match each three-dimensional vector field with the corresponding plot. レ 4レ/ / トー / tト44レ O (x, y,…

A:

Q: What is the value of a so that the vector field F(x.y)=(In y+2xy³) i+(ax²y2+ x/y) j is conservative?…

A: To find: the value of a for which given vector field is conservative.

Q: A vector field F = x(x² – y²) + ŷ(2xy + y?) a) f d.č =? c on the closed curve in the figure b)…

A: a) Consider the figure:

Q: 7. Determine if the given vector field F is conservative on some domain and, if it is, find the…

A:

Q: Given: r = |F| ř = (x, y, z) And letting F be the vector field F = f/rP. Please solve in general the…

A:

Q: → If I is a path-independent vector field, with 00 F. dr = 4.1 and ((11) F · ďr = 3.3 and (0) F · dr…

A:

Q: Suppose C is a curve of length N, and ||F|| < M, where M is some positive number. Prove that F. dr <…

A:

Q: Find the divergence of the field. F= (4x + 5y + 7z)i + (8x - 8y + 3z)j + (2x - 7y + 4z)k div F =

A:

Q: (2) For a differentiable vector field f, which of the following is (are) incorrect? (a) ▼ × (▼ × ƒ)…

A: Since we know Formula a→×(b→×c→)=(a→·c→)b→-(a→·b→)c→

Q: For a force field F, if curl F= 0, then the field is nonconservative field. True False

A: False

Q: 3. Use the fact that F = is a conservative vector field to evaluate SF. dr when C is the path r(t)…

A:

Q: Given: r = F| i = (x, y, 2) And letting F be the vector field F = f'/rP. Please solve in general the…

A:

Q: 11. LetF(x, y) = (ye™ – 6x))i + (xe™v + 4)7 (a) Show that F~ is a conservative vector field. (b)…

A:

Q: 4. Is the vector field F(x, y, z) = (yz, xz, xy) conservative? Justify your answer.

A: Calculate divergence of given vector field. If divergence is zero then field is conservative.

Q: If a vector field F (x, y, z) In (xyz), cos z), which of the following values is equal to div F (1,…

A: Finding divF

Q: 2. Given that F(x, y, z) = (e, xey + e², ye²) is a conservative vector field. a. Find a potential…

A: We are authorized to answer one question at a time, since you have not mentioned which question you…

Q: 14. Find the curl of the vector field F = x²i – xevj+ 2xyzk

A:

Q: 3 2) Integrarte the vector field hlxry,2)= xyi + ya J+xa k over the twisted cubic PIt)=ti+t?g + t3 k…

A: Given that, the vector field        h(x, y, z) = xy i + yz j + xz k        r(t) = t i + t2 j + t3 k…

Q: 3. Let F be the vector field (ze²z + yz², xz², xe*z + 2xyz)

A:

Q: 5. Show that F =(y cos(x),sin(x)+ y) is a conservative vector field and use this to calculate the…

A: Given: F → = y cos(x), sin(x) + y. A vector field F→ = P, Q is said to be conservative if partial…

Q: At how many points is the vector field F(x, y, z) = (z– ,r – 2², y – x²) irrotational? Select one:…

A:

Q: Q5) If the vector field T = (axy + Bz³)a, + (3x² – yz)a, + (3xz² – y)a, is irrotational, determine…

A:

Q: Given the vector field: F = (ry², y² + za", sin(x2)) (a) Find the curl of the field. (b) Find the…

A:

Q: Let Σ be the sphere x2`y2`z2 “ 25 with outward orientation and let F~ be a vector field with divF~ “…

A:

Q: Find div ccurl F) = v(♡XFJ For the V cetor Field given By F(AoyoZ)= X=i -6xzj + yz K

A:

Q: 2. Letr =√x² + y² + z². (a) For n ≥ 1 and r> 0, express V(r-n) in terms of r and the radial vector…

A: Given  solution(a): Radial vector field  solution(b):

Q: Consider the following vector field: F = (y* + 1, 3ry + 1). (8.1) Is ſ. F · dr path-independent?…

A: The given vector field is F=y3+1, 3xy2+1. (8.1) For a vector field F=P,Q, the line integral ∫CF·dr…

Q: 4. Determine whether or not the vector field F(x, Y, z) = (-xye®, -xe + e° + 2y, 2z) is…

A:

Q: Q4. A vector field F is defined as F = psin pa,+ 2p² zas + zcos oa,. The V × F at point P(1, 7/2, 2)…

A:

Q: 1).Evaluate ∮CF•dr, whereF=⟨e^x−y^3,cosy+x^3⟩, and C is a circle of radius 2, centered at the…

A:

Q: 2 1 Given the vector field F=rcos a.+ =sing a tra evaluate the curl of the vector field.

A:

Q: Which of the following is the divergence of the vector field F=x,y,z)? none of these 3. -3

A: The correct option is 3rd option .i.e. 3 .

Q: 6. Compute ſ, F-·dr where f(x, y) = (y² cos x, 2y sin r) and C is the straight line path starting at…

A:

Q: 7. Choose an option in which the vector field is conservative: None of them OF = (xy, x+ y) O 2,…

A: A vector field F=∇f is conservative if and only if ∂M∂y=∂N∂x

Q: Find the divergence of the field. F= 5z e 8xyz i+ 5x e 8xyzj + 5x e 8xyZk div F=O

A:

Q: 5. Show that there is no vector field G such that: V x G = 2xi + 3yzj – xz?k.

A: Given that ∇×G→=2xi^+3yzj^−xz2k^. We have to show that there is no vector field G→ such that…

Q: Given an n xn-matriz A over a field K ry column vector b, there is a unique column v lution to Ar=0…

A:

Q: For what values of a & b such that the vector field f= (ay'e" + xz)i +(ye" +z')}+| byz' + is…

A: f is given. We find a,b so that f is conservative (irrotational).  f is conservative (irrotational)…

Q: If the scalar field is given by f = 2p², the result of V × Vƒ at P(4, ,4) 3

A: The expression for the gradient in the cylindrical coordinates  is given as . Following are the dot…

Q: Determine whether the following vector field is conservative on R. F- (x.4y)

A:

Q: Given the vector field F(x,y) = [x/((x^+y^2)^(3/2)), y/((x^+y^2)^(3/2))], Find a fucntion f(x,y)…

A:

Q: ermine whether the vector field F(x, y, z) = (x³, 3zy², y³) is conservative on R³. If so, find a…

A:

Q: 4. Given the vector field Č = p² sin o a, + (z² cos – 1)as + (p² – z)a, Determine the x-component of…

A: Given the vector field