Find the flux through through the boundary of the rectangle 0 < x < 6, 0 < y < 5 for fluid flowing along the vector field (x + 5, y cos(3x)). Flux =

Q: A quarter cylinder of radius 2 and height 5 exists as shown in the figure below. Surface integral of…

A: Given that the quarter cylinder is of radius 2 and height 5. This means that the radical distance…

Q: Find the total work done in moving particle in a free field given by F = 3xyi – 5zj + 10xk along the…

A: The total work done by a vector field, F→ moving a particle along the curve r→(t) is calculated…

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: Let v = (x*z, 2 – 4xyz, xz) be the velocity field of a fluid. Compute the flux of v across the…

A: Given, v→=x2z,2-4xyz,xz225x2+16z2=y-202 Then, calculating div v→ div…

Q: Compute the flux of the integral of the vector field F(z,y, 2) = (2,y, 2) through the half cylinder…

A: Given vector field is, Fx,y,z=x,y,z. It can be written as,  Fx,y,z=xi^+yj^+zk^. Flux of vector field…

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 work done by the vector field F = in moving a particle around the circle x2 +y2 = 4 from…

A: The given vector field is F=x-y, xy. The particle is moving around the circle x2+y2=4 from 2,0 to…

Q: Let = (xz, 2 4xyz, rz) be the velocity field of a fluid. Compute the flux of i across the surface…

A: Flux of velocity about a surface is the measure of amount of velocity of fluid hitting the surface.…

Q: The continuity equation for incompressible fluids states that the divergence of the velocity vector…

A: The given statement is The continuity equation for incompressible fluids states that the divergence…

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: Find the flux of the vector field F = (x, z, - y) through the helicoid with parameterization r(u, v)…

A: Given :-  The flux of the vector field  F→ =  x3 , z , -y through the helicoid with the…

Q: Evaluate line integral of vector field F(z,9) = i+ 1- x? y2 2x along the curve x = cos(t), y =…

A:

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: Geometrically represent the velocity field of a fluid given by i(x, y) = yi – xj, , note that (xi +…

A:

Q: Find the work done by the force field F(x, y, z) = on particle that moves along the line segment…

A:

Q: Find the flow of the velocity field F =(x + y)i - (x2 + y2)j along each of the following paths…

A: The velocity field F=(x+y) i -x2+y2j along each of the following paths from 1,0 to -1,0 in the xy…

Q: A particle moves in a velocity field V(x, y) = (x2, x + y2). If it is at position (x, y) = (7, 2) at…

A:

Q: y' = 2 – 3xy; {-1<x<4, -4 < y s 4}

A: To plot direction field we plot small lines with slope calculated form differential equation after…

Q: A particle moves along line segments from the origin to the points (3, 0, 0), (3, 5, 1), (0, 5, 1),…

A: Given:Fx, y, z=z2i^+2xyj^+5y2k^Particle moves from line segments from the origin to the points 3, 0,…

Q: Compute the work done by the force field F(r, y) = (2, 3r) acting on an object as it moves along the…

A:

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: Integrate f(x, y, z) = x2 -y + z over the line segments from (0, 0, 0) to (1, 1, 0)  and then to the…

A:

Q: 1. Verify the Divergence Theorem in the calculation of the flux of the vector field: through the…

A:

Q: Find the work done in moving a particle in the force field F : 3x²i + (2xz - y)j + zk along the…

A:

Q: Find the work done by the force field F(x, y) = (x^2-lny)i + (e^x^2+y^2)j on a particle that travels…

A:

Q: The work done by a force field in a moving particle around a closed path is zero.

A:

Q: 2的 在年

A:

Q: Compute the flux of the vector fields F(x,y, z) = (, y, z) across the 1 and z = 4 whose vertical…

A:

Q: Evaluate line integral of vector field F(x, y, z) = y sin(x²) i+ xyz j+ (x +y + z) k along line…

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: A particle starts at the point (-1, 0), moves along the x-axis to (1, 0), and then along the…

A:

Q: 4) Write the integral to calculate the outward flux through F = (-yx2, xy,-z) across the sphere with…

A:

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: A particle moves along line segments from the origin to the points (3, 0, 0), (3, 3, 1), (0, 3, 1),…

A: From stokes theorem ∮F.dr=∯∇×FdS.  Thus, calculate the curl of F=z2i+3xyj+4y2k:…

Q: Calculate the line integral of the vector field F = -2i+ 4jalong the line from the point (3,0) to…

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: Express the field D = (x2 + y)-(xax + yay) in cylindrical components

A:

Q: A particle moves in the positive direction once around the square with vertices (0, 0), (0, 2),(2,…

A: Given, the force field Fx,y=x-sinxi+y2+xj. We have to find the work done.

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 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: The outward flux of the vector field F=(x,0, z) across the cylinder x + y = 1, for 0<z<1, is %3D…

A:

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

A:

Q: Let - (xz, 5 - 4xyz, xz) be the velocity field of a fluid. Compute the flux of i across the surface…

A:

Q: Find the flux of F = (3z + 1)k upward across the hemisphere x2 + y2 + z2 = a2, z&gt;= 0 (a) with the…

A:

Q: The necessary and sufficient condition that a velocity potential exists is that curl of V is…

A: We have to solve given problem:

Q: A particle at (5, 0) traverses the upper semicircle x² + y² = 25 and then closes the loop along the…

A: Formula: The work done by the force F→ is given by, ∮CF· dr, where C is the path along the region.…

Q: 6. Show that the system with vector field -x + sin(y) F(x, y) : -y+ cos(x), admits no periodic…

A:

Q: Calculate the line integral of the vector field F -3 i+ 2j along the line from the point (4, 0) to…

A:

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.