Find the work done by the gradient vector field F(x, y, z) = V (e² + y² + z²) in %3| mov ing a particle along the circular helix r(0) = (cos 0, sin 0,0) for 0 < 0 < 2ñ.

Q: Confirm that the force field F(, y) = 2xy°i+ 3x²y² j is conservative in some open connected region…

A: Solution:- The given function is  F(x,y)=2xy3i+3x2y2jpoint P(-9,8) and Q(6,7)we have to find the…

Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y, z) =…

A:

Q: Find the work done by the force field F(x,y) = 4yi + 2xj in moving a particle along a circle x2 + y2…

A: We follow

Q: Find the gradient field F=Vo of the potential function o(x,y.z)= (3x +y +3z) -1/2 Vq = 000

A:

Q: Find the flux through through the boundary of the rectangle 0 < x < 6, 0 < y < 5 for fluid flowing…

A: Given F→=x5+y  , y cos3x flux=∫cF→.n^  ds =∫∫Ddiv F dA

Q: Find the gradient vector field of f(x, y) = ln(5x + y)

A:

Q: Compute the flux SS F.nds through the piece of the cylinder of radius 2, centered on the z-axis,…

A: The given surface is the cylinder of radius 2 centered on z-axis, where 0≤x, 0≤y and 0≤z≤4. The…

Q: Find the gradient vector field of f(r, y) = In(x + 3y)

A: Solve for the gradient vector

Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = ln(x + 5y + 6z). F(x, y, z) = =

A:

Q: Compute the work done by the force field F =< x² + 2y, 4x + y° where C is the boundary of the region…

A:

Q: The flow of a gas with a density of δ = 0.001 kg/m2 over the closed curve r(t) = (-sin t)i + (cos…

A: Given that

Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y) =…

A: It is required to find the conservative vector field for the potential function given in the…

Q: Find the flux through through the boundary of the rectangle 0 < x < 6,0 < y < 3| for fluid flowing…

A:

Q: Find the conservative vector field for the potential function by finding its gradient. 9 (r,y, 2) =…

A:

Q: Find the gradient vector field of f бу f(x, у, 2) = X COS Z (бху) (6y бу -sin бх -sin бу i cos k…

A: Given,

Q: 4. Find the flow of the velocity field F = V(xy²z³) along the line segment from (1,1,1) to (2,1,…

A:

Q: Find the conservative vector field for the potential function by finding its gradient. g (x, y, 2) =…

A: We need to find conservative vector field.

Q: Find the gradient vector field Vf of f. 1 f(r, y) = (y – 2)?

A: For a given vector fx, y the gradient vector denoted by ∇f is ∂ ∂ xf∂ ∂ yfSo,=∂ ∂ x12y-x2∂ ∂…

Q: Find the gradient vector field of f(x, y) = tan(2x − 3y).

A: Given fx,y =tan2x-3y         we find the gradient vector of fx,y

Q: Show that the vector field F = x²i + (2yz)j+ y² k is conservative and then find the corresponding…

A: A vector field F is conservative if curl(F)=0 or ∇×F=0 The potential function is given by f such…

Q: The motion of a liquid in a cylindrical container of radius 2 is described by the velocity field…

A: Given,F(x,y,z) = -19y3i + 19x3j + 7k⇒curl F = ijk∂∂x∂∂y∂∂z-19y319x37          = 0i + 0j +…

Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y) = x²…

A:

Q: Find the gradient field associated with the function φ(x, y, z) = xyz.

A: To find the gradient field associated with the function  φ(x, y, z) = xyz.

Q: Find the work done by the force field F(x,y) = (-y,x) in moving an object along the path y = V4 – x2…

A: Given force field Fx,y=−y,x and object moving along the path y=4−x2  from 2,0 to -2,0. We have to…

Q: Find a potential function for the conservative vector field given by y3/2 2T 2 2x sin z F(r, y, 2) =…

A: We will find out the required potential function.

Q: 2的 在年

A:

Q: Find the work done by the force field F(x, y) = x* +y? on a particle that moves along the quarter…

A:

Q: find the work done by the force field F(x,y) =(2y+x²,x²-2x) acting on an object as it moves along…

A: Given: The force field, Fx,y=2y+x2,x2-2x.   To find: The work done by the given force field acting…

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

A:

Q: Flow curves in the plane Let F(x, y) = ⟨ƒ(x, y), g(x, y)⟩ be defined on ℝ2.

A: The vector field Fx,y=fx,y,gx,y.

Q: Evaluate the integral curves for the vector field F(x, y) =i -i .

A:

Q: F =(-y,x,z) across the solid sphere E %3D

A: we have to find the outward flux

Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y, z) =…

A: Given: f(x, y, z) = 9x2 − xy − z2

Q: Find the conservative vector field for the potential function h(x,y,z) = 16xy ln(x+y) by finding its…

A:

Q: Find the flux of the vector field F = across the part of the plane | z = 2 + x + 3y above the…

A: Given: F¯=y,-z,x z=2+x+3y The formula to determine the flux is: ∫S∫F·nds=∫D∫(-agx-bgy+c)dA Here…

Q: Determine the gradient of the give scalar field V = e(2*+3y) cos 5z at R(-0.1, 0.2, 0.4).

A:

Q: Let W be the region between the sphere of radius 4 and the cube of side 1, both centered at the…

A:

Q: Compute the gradient field of f(x, y, z)= e^z + xy^2

A: Take gradient to solve

Q: Consider the conservative vector field 4 8 F (x, y, z) 3y cos (372), 37y sin (372) – a² > 2xz, 2x +…

A:

Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y, z) =…

A: Given : f(x,y,z)=9x2y+5z2   To Find : Conservative Vector field by finding it's gradient

Q: Find the gradient vector field of f(x, y) = x° sin(9y). F(x, y) = x°i+ sin(9y)j F(x, y) = 9x°…

A:

Q: Without using a potential function, find the total work performed by the force field F(x, y) = (2x…

A:

Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y,…

A:

Q: Find the work done by the force field F(x,y) = 4yi + 2xj is moving a particle along a circle x2 + y2…

A: Application of integral: The force acting on the particle, F⃗ =4yi+2xj  Let the particle undergo a…

Q: Find the gradient vector field (F(r, y, z)) of f(x, y, z) = x² sin(3yz) F(z, y, 2) = (

A:

Q: Find the gradient vector field of f. 5y f(x, y, z) = x cos %3D Vf(x, y, z) =

A:

Q: Find the potential function of the vector field F (x, y) = -y sin(xy)i – x sin(xy)j. Assume that the…

A:

Q: Compute the divergence and curl of the vector field cos (22 ) sin² (z² yz) 10 yz4 – xyz2 at the…

A: We have to find divergence and curl of vector field:

Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y, z) =…

A: We have to find the conservative vector field.