4) Consider the conservative vector field given by: F (x, y) = (y² – ye“, 2xy – e*) A potential function that generates the vector field F corresponds to: A) f(x,y) = xy² – ye" B) f(x,y) = 2xy² – 2ye" y²e" C) f(x,y) 3 + x?y – e* 2 D) f(r,y) |

Q: 5) Consider the conservative force field F(x, y) = (6x²y? + cos(x), 4x y). a) Give a general…

A: For a vector field F=P,Q that is open and simply connected in a region with continuous first order…

Q: The potential of the conservative vector field F = (y,x,1> is Of=z+xy O f=x+yz O f=y+xz Of = xyz

A:

Q: 4. Find the positively oriented flux over the surface created by the paraboloid z = 16 – x² – y2…

A:

Q: Which of the following functions is a potential function for the conservative vector field F(x, y,…

A:

Q: Compute the flux of the vector field F(x, y, z) = (2y, 3, –2xz) through the surface S, which is the…

A: Given vector field is, Fx,y,z=2y,3,-2xz. The given surface is, y=x2+z2  and  x2+z2≤1. Now for the…

Q: Consider the conservative vector field 8 4 F(x, y, z) = - 2xz, - 3y²cos(3nz), 3ny sin(3nz) – x² ) 2х…

A:

Q: Consider vector field F(x, y) = (axy - y³)i + (4y + 3x² + bxy²)j. (i) Determine the value of…

A:

Q: 9. Compute the flux of the vector field F(x, y) = (-x sin(1+xy) – 4 cos(TY), y sin(1+ xy) + 6x)…

A:

Q: Find the work done by vector field F(x, y, z) = xi + 9xyj - (x + z)k on a particle moving along a…

A:

Q: 1- Prove that the force field F known as : Prove that the force field defined by F = (y*z' -6xz? )i…

A:

Q: From the given vector field F⃗ (x, y, z) = , compute the curl of F⃗ and find the values of a and b…

A:

Q: b) Show the force field M UTM UTM F(r, y, z) = (r²y + In(yz)) i+ UTM 3 + y cos(2z) ) j+ tial…

A:

Q: Is the two-dimensional vector field V = -2y 2x a possible velocity field x²+y² 'x² +y² for an…

A: For a two dimensional incompressible for a vector field V=ui+vj. ∂u∂x+∂v∂y=0

Q: I. Use the Divergence Theorem: (² F. ds = Ifv. Fav SSS E = to calculate the flux of the field F…

A: Divergence theorem uses  Double integration converts into triple integration.

Q: Find the potential function p, for the conservative vector field F(x, y, z) = (2xe² – y²sin x)î +…

A:

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

A:

Q: A) Consider the vector-field F(x, y) = 2xy³i+ 3y²x²j. 1. Show that F is conservative. 2. Find a…

A: The given vector field is  F⇀(x,y)=2xy3i⇀+3y2x2j→ We have to show that F→ is conservative

Q: b) Show the force field (x, y, z) = (x²y + In(yz)) i+ + y cos(2z) ) j+ y? sin(2z) 3 F k is a…

A:

Q: The circuitation of the vector field F(x, y)- (3 + xy´, 4x + xy) along the boundary of the plane…

A: The solution are next step

Q: b) Show the force field F(x, y, z) = (x²y + In(yz)) i+ 3 y cos(2:) ) j+( - y° sin( sin(2z) ) k %3D…

A:

Q: What is the flux of the vector field (x2,y2,z1 ) through a cuboid with x ranging from 0 to 8.3, y…

A: use divergence theorem

Q: Given the vector field F(x,y,z) = [x, y, -2z] satisfies div(F) = 0. Find a vector field G(x,y,z)…

A:

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

A:

Q: Evaluate the outward flux of the vector field F = (2x? – 5e" + sin(2²), –4ry + ln(x*), 5z – 3x²y°)…

A: Given

Q: Compute the flux of the vector field F 3yi + 4j – 3xzk through the surface S, which is the surface y…

A: We apply formula for finding this

Q: Find the work done by the force field F(x, y, z) = (x − y², y - z², z - x²) on a particle that moves…

A:

Q: Find the gradient vector field (F(r, Y, 2)) of f(x, y, z) = In(4x + 6y + 22) . F(7, y, 2) = (

A:

Q: Consider the vector field F (x, y, z) = (2e2" cos(3y), –3e2" sin(3y) +VV?) Y y' 2 COS a. Find…

A:

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: Find the flux of the field F up through the part of the 1+ xy paraboloid z = 1 – x² – y? that…

A: FORMULA: Flux=∬F.n^ dS According to Stokes theorem, ∬F.n^ dS =∬∇×F . n^ dS According to the…

Q: Compute the flux of the vector field F(x, y, z) = 3x + 3yk through the surface S, where S is the…

A:

Q: Find g(x, z) so that F(x, y,z) = (8x cos y sin? z + 2e2xz)î – (4x² sin y sin? z)j+ (g(x, z) cos y +…

A: Given that    Fx,y,z=8x cosy sin2z+2e2xz i^-4x2 siny sin2z j^+gx,zcosy+2e2xz k^ Now we have to find…

Q: 4. (a) Show that the outward flux of the postion vector field F(x,y,z) = (x)7+ (y)] + (z) through a…

A:

Q: 2.4) Use the divergence theorem to find the outward flux of the vector field F(x, y, x) = x³i+ y³j+…

A: The divergence of the given vector field is  ∇·F=∂∂xx3+∂∂yy3+∂∂zz3=3x2+3y2+3z2 Let, S be the surface…

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

A:

Q: Consider the surface integral of the vector field F= [xy², 3x², x²z]over a surface S, where S…

A: From Gauss divergence theorem, we have ∬SF→⋅dS=∭VdivF→dV   The vector field is F→=xy2,3x2,x2z.…

Q: If F = xy i+ (x² + 4yz)j+ 2y²z k. Evaluate the line integral between A (0, 2, 0) and B (3, 6, 1).…

A: Given, F= xy i +(x²+4yz)j +2y²zk r0= (0,2,0) r1= (3,6,1)

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

A: Given :F(x,y)=6xyi+3y2j

Q: Find the divergence of the vector field F. F(x, y, z) = In 5x²+ + 10xyj + In( 2y2 + 722)k div F(x,…

A:

Q: Show that the vector field F = 2.xyi +(x² – 2 y)j is conservative and evaluate the line integral…

A: Given the vector field

Q: Show that the vector field F(x, y, 2) = (2x cos y – 3)i – (x² sin y + 2²)j – (2yz – 2)kis a…

A:

Q: 1. Is the vector field V (x, y, z) = (2ry, x² – 2², –2yz) conservative?

A: Since you have asked multiple question,as per our guidelines we are supposed to answer only one…

Q: 6) Given the vector field F = (-x,-y,-z) '(x²+y2+z²)3/2! write the integral to calculate the outward…

A: First we parameterize field F then we calculate the Flux across the sphere using following formula:

Q: Why does a two-dimensional vector field with zero divergence on a region have zero outward flux…

A:

Q: Show that the vector field F(x, y)= (12x² +3y² +5y)i+(6xy– 3y² +5.x)j is a conservative vector field…

A:

Q: 2: Compute the work done by the force field F = (y, -x) to move an object from point A = (1,0) to B…

A:

Q: Show that the vector field F(x,y,z)=yz^2i+xz^2j+2xyzk is conservative, find a potential function f,…

A: The given vector function is F(x,y,z)=yz2i+xz2j+2xyzk The given vector field F is conservative if…

Q: Determine whether the vector field…

A:

Q: a) What is the flux of the constant vector field F = (a, b, c) through any closed surface? b) Is the…

A: Given That : A constant  vector field F=a,b,c and F=x3,y3,z3 To Find : i) The Flux through of…