If a vector field F (x, y, z) In (xyz), cos z), which of the following values is equal to div F (1, 2, )? (а) п (b) 1 (c) 0 (d) е — 1 (е) е? + п e

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

A:

Q: Is the vector field F(x, y) = (In y + sin(e²“), #

A:

Q: 10. For a vector field F = <2xy + y³, x² + 3xy²+ 2y7 find a function f such that F = Vr.

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: Show that the vector field F = (x² cos y, e+y) is not a gradient vector field. %3D

A:

Q: Consider the vector field F(x, y) = yî- xî+ zR and path joining (1, 0, 1) and (-1, 0, 1) as C;:…

A:

Q: b) Given the vector field SUTM F(r, y, 2) = 3z cos ri+rlnyj+e* tan rk. Determine UTM i. div F, ii.…

A:

Q: Supposing vector field G is related to the flow of fluid P is one point in the domain of G. Give the…

A: To Explain: The geometrical interpretation of G(P) where G is a vector filed and P is a point in…

Q: Determine if the vector field F(x,y)= (x+y,x−y) is conservative? If it is, find its potential.

A: Given Data The vector field is: F(x,y)=(x+y,x-y).   Write the vector field in standard form, The…

Q: Evaluate , (V. A) dV for the vector field A = x²yi +xz³j- y²z²kwithin the volume of a cube defined…

A:

Q: - Given the vector fields Å = yz az – 3x a, +x²y a, B = 3 a, – 4xy a, +y² az Petermine a unit vector…

A: Given that: A→=yzax-3xay+x2yazB→=3ax-4xyay+y2az

Q: (a) Show that the vector field F (x, y, z) = (x² + y², x, z³) is not conservative.

A:

Q: Show that the vector field F = 2xyi + (x - 2 y)j is conservative and evaluate the line integral %3D…

A:

Q: 9. Sketch the vector field F(x, y) : :(yi — х). Vx? + y?

A: Consider the given filed, Fx,y=1x2+y2yi−xj Further simplify. Fx,y=yx2+y2i-xx2+y2i=〈yx2+y2,-xx2+y2〉

Q: Write a formula for a two-dimensional vector field which has all vectors of length 1 and is…

A:

Q: Determine whether the vector field F(x, y) = 5y? (yi + 2xj)is conservative. F is conservative O F is…

A:

Q: (a) Is the vector field | F(x,y) = (xe, ye²) conservative? Justify your answer. If it is…

A: Let the vector field F = Pi + Qj.The above vector field is conservative if and only if dPdy=dQdx

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: 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: Is the vector field F(x, y) = (6ry cos(3x²y + e*) +e* cos(3x?y + e*), 3x² cos(3x²y + e*))…

A:

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

A:

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

A:

Q: Зу 1. Given the vector field F(x, y, z) = î+ Vrz j+x cos y - k, find curl F and div F.

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: a) Determine whether or not the vector field F 8yz, – 6 cos(y) – 8xz, – 8xy + 10z > - is…

A:

Q: Find the divergence of the vector field F. F(x, y, 2) = In(9x+ 8y² )i + 72xyj + In(8y² + 5z? )k #6…

A: Solution is done in step 2 and 3

Q: 4. Show, in detail, that any vector field of the form (*,J, 2) = J (*) +8(y)/+h(z)k is irrotational,…

A: To show that the vector field is irrotational.

Q: 4. Show the vector field Vu x Vv, where u and v are scalar fields, is solenoidal.

A:

Q: Show that the following vector field is conservative and find the potential F(r, y, z) = (2ry, x² +…

A: The answer is given as follows.

Q: Find a potential function for the vector field F by inspection. F = (4x, 4y) Ф(х, у) 3

A:

Q: Find the divergence of the vector field F. F(x, Y, z) = In( 9x²+ 8y² )i + 72xyj + In(8y² + 7z? )«…

A:

Q: Is the vector field F = [yze"yz, xze#yz, xye"yz] conservative? %3D O Yes O No

A: The vector field is conservative if ∇×F→=0 therefore, find ∇×F→ ∇×F→=ijk∂∂x∂∂y∂∂zyzexyzxzexyzxyexyz

Q: The divergence of the vector field F(z, y, z) = (ya°, az", zy?) is: %3D

A:

Q: Ğ(x, y, z) = (2z3 – 4y2 + Mxy, x2 - 8xy – Nz, 6xz? - My + 4) 1. Find the curl of G 2. Find M and N…

A:

Q: Find the divergence of the vector field F. F(x, y, z) = In( 7x²+ 4y² )i + 28xyj + In( 4y2 + 5z2 )k

A:

Q: Consider the vector field F(x, y) = (x², −y). If (x, y) is in the first quadrant then the vector…

A: Part 1. (x, y) is in the first quadrant.  =&gt; x &gt; 0 and y &gt; 0 =&gt; x2 &gt; 0 and - y &lt; 0…

Q: F(x, y) = 6xyi + 3y²j is a conservative vector field.

A: Given :F(x,y)=6xyi+3y2j

Q: Verify Green 's theorem for the vector field  F (x, y) = (x^2y, -xy^2) in the triangle OAB defined…

A: The solution  is given by using the line integral of vector field and verify the greens theorem as…

Q: Consider the vector field F(x,y,z)=(−3x^2,8(x+y)^2,9(x+y+z)^2). Find the divergence and curl of FF.…

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: Find the divergence of the vector field V(x, y, z) = -2e*'i – e³xyj+ 7e%yzk.

A:

Q: Show that the vector field A (3x?z + y?)a,+ 2xya, + x³a, is conservative. %3D

A: Given problem:- Show that the vector field A = (3x²z+ y²)ax + 2xyay + x³az is conservative. =

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: B) Find the work done by the conservative vector field F (x, y, z) = yzi+xzj+xyk along a smooth…

A: Since, it is given that the field is conservative.

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

A: