4. A) Find the work done by the force F (x, y) = yx²i – y²j over the curve r (t) = t’i+t*j, 0

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: 4. Find the work done by the force field F (x,y) = (x + y)î + (xy)ĵ in moving a particle along the…

A: The force field is given by F→ = x+yi^+xyj^ The end points of the straight line along which the…

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

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

A: Given the vector field F→x,y=2xsiny+y2ex,x2cosy+2yex. The particle moves along the curve x=sinπy2…

Q: Find the tangent plane for the level surface f(x,y,z) = 5x4 + cos (4y) + In (6z) = c at the point…

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: Vy: y(0) = 1 dx dy Vy: y(0) = 0 %3D %3D dx dy - x-y; y(2) = 2 %3D dx dy VX-y; y(2) = 1 %3D dx

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: Δ denotes the Laplace operator defined by Δφ = ∂2φ/∂x2 + ∂2φ/∂y2 + ∂2φ/∂z2 . Show that if F is the…

A: A function ux,y is called harmonic if it is twice continuously differentiable and satisfies the…

Q: 8.(6 pts) Use Fundamental Theorem of Line Integrals to compute / Vf - dr where f(z, y, z) = ry? +…

A:

Q: Compute the line integral of the scalar function f(x,y,z)=2x2+8zf(x,y,z)=2x^2+8z over the curve…

A: To find the line integral, obtain dS as follows.

Q: 5. Evaluate the line integral (r+2y) dr+x² dy where C consists of line segments from (0,0) to (2, 1)…

A: In order to solve this question by the formula of line integral .

Q: 3. Evaluate the line integral Sc F. dr where F(x, y, z) = (e², xz, (x + y)) over the curve r(t) = t²…

A:

Q: 2. Calculate the directional derivative of the scalar field U(x, y,z) at the point M,, along the…

A: Given that : U=xz3y-x3y=xz3y-x32y12 The gradient will be calculated as :…

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

A:

Q: 2) Given the vector field F = (- Vx²+y² 2z2) write the integral to calculate the outward x²+y² flux…

A: We have to find the outward flux across the cylinder x2+y2=100.

Q: Verify if the vector field F = (x° cos(4y), -x sin(4y)) is conservative and evaluate the line…

A:

Q: Evaluate the line integral of the vector field (3x2 + 3y + 2xy, 3y2 + 3x + x2) along the curve y = x…

A:

Q: Consider the line integral cos(x+2y)dx+2cos(x+2y)dy where S is a curve starting at (0,0) and ending…

A: We have to show that F is conservative. When F is given as F=cosx+2yi+2cosx+2yj For the conservative…

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: Show that the isoclines of dy/dt= t are vertical lines. Sketch the slope field for −2 ≤ t ≤ 2, −2 ≤…

A:

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

A:

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

A:

Q: 2的 在年

A:

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

A: Given

Q: 6. Use the fundamental theorem of line integrals to evaluate 2x dr + 2y dy where C is the line…

A: Let's understand the Fundamental Theorem of Line Integrals:

Q: Suppose that v = (0, y, 0) is the velocity field of a fluid and let S be the surface defined by y =…

A: Given, the velocity field v→=0, y, 0and S be the surface defined by y=x2+z2, with y≤1.

Q: 4. Evaluate the line integral Vf-r'dt, where the curve C is given by the vector function r(1)…

A:

Q: In(t) i + t312j + t cos(at)k, F. dr of vector field F(x, y, z) = 3x²zi + zj + (x³ + 2yz)k along…

A:

Q: 2. Find the gradient field F = Vy for the potential function p. Sketch a few level curves of o and a…

A: Given φ(x,y)=2x2+4y2 The gradient field  F=∇·φ=∂φ∂x,∂φ∂y=4x,8y Some vectors of F are  At x=1 and y=1…

Q: Find the outward flux of the field F = 3xz2 i + y j - z3 k across the surface of the solid in the…

A:

Q: 4. Integrate f(x, y, z) = Vx² + y² over the involute curve (cos t + t sin t)i + (sin t – t cos t)j,…

A:

Q: Compute the line integral of the scalar function f(x,y,z)=2x^2+8zf(x,y,z)=2x2+8z over the curve…

A:

Q: 3. Evaluate the line integral c F. dr where F(x, y, z) = (e², xz, (x + y)) with 0 ≤ t ≤1 over the…

A: The solution is given as

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

A:

Q: b) Show that the force field F(r, y, z) = 3x zi+ 2y ln z j+ is independent of path and find its…

A: F(x,y,z)=3x2zi +2ylnzj+(y2/z +x3)k r(t) =ti +tj+t2k F(t ) =3t4i +4tlnt j +(1+t3)k dr = dt i +dt j…

Q: Consider the conservative vector field 4 8 F(x, y, z) = – 3y²cos (3nz),37y sin (3tz) – x2). 2xz, %3D…

A:

Q: 6. Show that F = [y? cos x + z3, 2y sin x – 4, 3xz2 + 2] is a conservative force field. Find the…

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Q: For the force field F = (2xe¬Y, (2y – x²e¬y)) in some open connected region containing P(1, 0) and…

A: Given that, F=2xe-y, 2y-x2e-y If F is conservative then ∂2F∂x∂y-∂2F∂y∂x=0

Q: 1. Calculate the work done when a force F = 3xyî – y?j moves a particle in the xy-plane from (0,0)…

A: Take dot product along limit of integration and integrate

Q: 12. Find the flux of the field F(x, y, z) = 4xi + 4yj + 2k outward (away from z-axis) through the…

A:

Q: 3.4.6 Evaluate the line integral I = =f( (xy dx + x dy) C where C is the closed curve consisting of…

A: We have to solve the given problem

Q: Find the work done by the force field F(x, y) = xyi+ x²j on a ,2 particle that moves along the curve…

A: Consider the given force field Fx,y=xyi+x2j Then, F⇀x,y=xy,y Write in parametric form y=t  then…

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: Q3 Show that the vector field F= 2xyi +(x -2y)j is conservative and evaluate the line integral |F•ds…

A: Solution

Q: Find the flow of the velocity field F = b. a. Na g (2,8) : 4x (0,0) X (5y² + 1)i + (10xy)j along…

A: