Consider the vector field F(x,y,z) = (3x²y)i + (x³ + 1) + (92²) and the curve C made up of the quarter circle cut from the sphere x² + y² + z² = 25 by the plane x = 3, from (3,4,0) to (3,0,4) traveling counterclockwise when looking from the positive x -axis. a. Use a line integral (as in Section 15.2) to compute the work done by the vector field in moving a particle along C. b. Show that the vector field is conservative, then compute the work done by the vector field in moving a particle along C using a potential function (as in Section 15.3) for the vector field.

Q: For the vector field G = (yey + 3 cos (3x + y))i + (xeªy + cos(3x + y)), find the line integral of G…

A: We find the required integrtion using fundamental theorem of line Integral.

Q: Express the vector field W = (x2 - y)a, + xza, in the (a) Cylinderical coordinate at P(p = 6, q =…

A: Cylindrical coordinates are ρ,ϕ,z now let us look at the general form of vector fieldW in…

Q: x-y, z + y + 3,z²) and the net is decribed by the equation y = 1- x² - z on

A:

Q: xi + yj x2 + y2 F(x, y) = k-

A: Topic- vector integration

Q: Sketch the vector field in the plane along with its horizontal and vertical components at a…

A: We shall use MATLAB to plot the given vector field on the circle.  MATLAB CODE F = @(x,y)…

Q: The surface H is defined by (x² + y²)² + z = 1; z ≥ 0. The vector field A(r) is given by A = (x, y,…

A: The surface H  is defined by    .The vector field  is given by   .We have to sketch the surface H.

Q: A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector…

A: Given that v=<x-y, z+y+9, z2> and the given curve is y=1-x2-z2,y≥0 To find: The value of ∬S…

Q: How much work is required to move an object in the vector field F= 5 y2i + 2xy J  Along the curve ,…

A:

Q: A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector…

A: Adding the plane surface y=0 to close the surface and apply the gauss divergence theorem.

Q: Find the k-component of (curl F) for the following vector field on the plane. F = (Xex)i + (8y ex)j…

A:

Q: Let F=(9xy,7y,8z)=(9xy,7y,8z). The curl of F=( Is there a function f such that F=∇f=∇?  (y/n)

A: The given function .We have to find the .

Q: Consider the vector field given by F(x, y) = (6xy – 152 y)i + (3a? + 3y – 10a°y)j find a function…

A:

Q: For any vector field F, show that div(curl F) = 0.

A: We need to prove that for any vector field F,  div(curl F) = 0.

Q: Vector Field: F(x,y) = (xv, 3x, – 2y) and Surface: z = 4 - x - 3 y? Use the normal vector formula to…

A: Given surface z=4-x-3y2

Q: Let C₁ be the line segment from the point (-4, 8) to the point (2, -4), C₂ be the arc on the…

A:

Q: Let F be the vector field (-2 sin(2x - y), sin(2x - y)). Find two non-closed curves C₁ and C₂ such…

A:

Q: Compute the curl of F = (2xy, x²y + x²).

A: Here we have to find the curl of the given vector field F.

Q: given the vector field A(r,θ,φ )= sin(θ)*r + r*θ + 6*φ, find the laplacian and evaluate it at the…

A: In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence…

Q: 6 Find the constants a and b so that F = (ax'y°z°, x°y²z°, bx°y° z²) is conservative. You may use…

A:

Q: Consider the square C oriented counter clockwise, with vertices (0, 0), (2, 0), (0, 2), (2, 2) and 7…

A:

Q: 2. A vector field is specified as G = 24xy ā, + 12(x² + 2)ā, + 18z²ā,. Given two points, P(1, 2, -1)…

A:

Q: Find a formula for the vector field V(x, y) = (vectors are shown as line segments, with a dot at the…

A: The vector field is a cluster of arrows, each with a specific magnitude and direction, that are…

Q: Show that the vector field F(x, y, z) = (-ycos(8x), 8x sin(-y), 0) is not a gradient vector field by…

A:

Q: For the vector field F = yr³i - y²j, the work on the curve y = √x from (1, 1) to (0,0) is A. B. C.…

A:

Q: The work done by the vector field F(x, y) = (-y², ²) along the arc of the parabola y = 2x² - 4x + 2…

A: According to question,Given Vector field,Fx, = -y2,x2Equation of pparabola,y = 2x2-4x+2

Q: re f an -2, 3)

A:

Q: Using arrows, clearly sketch the vector field u(r) = (y², x + y, 0) in the (x, y)-plane.

A: Given the vector field function u(r)=(y2,x+y,0). We need to plot the vector field in the…

Q: Find the curl of the vector field F = (5y cos(x), 4x sin(y)) curl F -

A:

Q: Find the curl of the vector fleld F(z, y, 2) = zi+ 8'y=²j+2rk. Enter your answer in standard unit…

A:

Q: Find curl(zi + xj+ yk).

A: Using definition of curl of vector.

Q: A child living in the ry-plane climbs a ladder from (0, 0) to (1, 1), slides down to (2,0) along the…

A:

Q: Let I be the flux of G = (4e³,9x³ex³,0) through the upper hemisphere S of the unit sphere. (a) Find…

A: