4. Show, in detail, that any vector field of the form (*,J, 2) = J (*) +8(y)/+h(z)k is irrotational, F. g, and h are differentiable, Give an example

Q: Show that the vector field F(x, y, z) = xz i + xyz j - y 2 k can’t be written as the curl of another…

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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 if a vector field has the form F = ⟨ƒ(y, z), g(x, z), h(x, y)⟩, thendiv F = 0.

A:

Q: 8. Compute the flux of the vector field. F(x,y,z) = (2x -z,-2y, 1-2x) through S= {(x,y,z) € R³: z =…

A:

Q: 4. Consider the vector field given in Cartesian coordinates by A(r) = (x² + y³, 3xy², z²). (a)…

A: Given : Ar=x2+y2,3xy2,z2

Q: In problems 11 and 12, do the computation two ways: i) by using the definition of either flux of a…

A: Using i) the definition of the flux across the surface is  ∬SF.dS.

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: 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: 41.) Use given grid to sketch the vector field F = (y, -x). Include vectors at all integers points…

A: Given vector field is F=12y,-x

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

A:

Q: Let the vector field F (x, y) = (3x2y-2+ 2xy-1)i + (-2x3y-3 − x2y-2)j, be a conservative field.…

A: We will use Integration and concept of total differentiation for finding scalar potential of vector…

Q: What is the divergence of the vector field f = x²i+ xyzj – zk at the point (1, 2, 3). Select one: а.…

A: f=x2i+xyz j-zK at (1,2,3)  div f=i ∂f∂x+j.∂f∂y+K.∂f∂z

Q: 15) Consider the vector field F (x, y, 2) = (In y, x, z) and the oriented curve C:r (t) = (t, e', 2…

A:

Q: 5. Let F be the gradient vector field of the function f (x, y) = 4xy + 3y". Evaluate JaF. dr, where…

A:

Q: 7) Consider the vector field F (x, y, z) = (xZ + cos y, XZ - x sin y, Xx). The work that the F field…

A: A) 10,57 J

Q: Determine and state the work done by the vector field over the given path in each situation. F…

A:

Q: 4. (a) Show that the divergence theorem holds for the vector field F = 2xî + yz} – 3z²k where o is…

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Q: 5. Compute div F and curl F if F = (2x – cos y)i – x²e32j + (z² – 7x)k. 6. Determine if the vector…

A:

Q: 2. Show that any vector field of the form F(r, y, z) = f(y, z)i + g(x, z)j+ h(x, y)k is…

A:

Q: 8. Show that the vector field, -y a² + y?' x² + yj²' ) is not conservative in R³, despite the fact…

A:

Q: 10. Evaluate the divergence of the following vector field at the point (0,1,2): F = (xyz, x? + y? +…

A: Given: F=xyz,x2+y2+z2, xy+yz+xz       The point 0,1,2 To find the divergence

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: 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: 5. Suppose that the vector field F = is conservative. Find a potential function of the vector field…

A:

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: Determine and state the work done by the vector field over the given path in situation. F (x,y) = (x…

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: Show that F = (3, 1, 2) is conservative. Then prove more generally that any constant vector field F…

A: Given: F = (3, 1, 2) To show: F is conservative. Also to show any constant vector field F = (a, b,…

Q: 1. Determine whether F(x, y) et sin yi + e" cos syj is a conservative vector field.

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: 2. Which of the following statements are true for all vector fields, and which are true only for…

A: according to our guidelines we can answer only three subparts and rest can be reposted.

Q: Find the work done by the three-dimensional inverse-square field F(r) : 1 r on a particle that moves…

A:

Q: 1 Find the divergence of the vector field F -(xi + yj+ zk) Vx? +y? +Z 2 x² + y² + z? 1 x² + y? + z?…

A: If the vector field is given by F=F1i+F2j+F3k then the divergence of F is ∇·F=∂F1∂x+∂F2∂y+∂F3∂z.

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

A:

Q: How much work is required to move an object in the vector field F = 5 y² ỉ + 2xy] Along the curve C,…

A: Given: F=5y2i+2xyj  C1 is the line y=0 C2 is the line x=1 C3 is the line segment joining points 1,1…

Q: Show that if a vector field has the form F = ⟨ ƒ(x), g(y), h(z)⟩, then∇ x F = 0.

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.

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Q: 2. Sketch the vector field F(x, y, z) = xi + yj + zk.

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: 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: 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: Determine and state the work done by the vector field over the given path in situation. a. i. F (x,…

A:

Q: 2. State whether the divergence of the vector field shown below is on average positive, negative or…

A: If the divergence is positive the rate of change is outward like a heated gas that is expanding. If…

Q: 7. Find the outward flux of the field F(x, y) = -yi + x²j across the closed counter-clockwise curve…

A: Consider the given field: F→(x,y)=-yi^+x2j^ and

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: 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