2. Compute the vector line integral: -y х — 4 Le (x – 4)² + y² dx + (x – 4)2 + y² dy where C is the circle of radius 2, centered at (4,0) with the counter- clockwise orientation.

Q: Solve the line integral with the vector field F (x, y) = (xy, seny) and the parametrization x (t) =…

A: The solution is given as

Q: Evaluate the line integral F· dr, where F(x, y, z) = 3xi – 2yj – 4zk and C is given by the vector…

A: Given that F→=3 xi→-3 yj→-4 zk→ r→=sin t i→+cos t j→+t k→ and t is varying from 0→ 3π4  then x=sin…

Q: Compute the line integral of the scalar function f(z,y, 2) = 2z² + 8z over the curve c(t) = (e*, t²,…

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Q: (a) Given the vector field F = xy + 2yzý + 3zx 2, calculate the line integral around the closed path…

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Q: Evaluate the line integral I of the one-form w = xy dx + 12y dy over the parametric curve a : [0,…

A: Given: One-form ω=xydx+12ydy and the parametric curve α:0, ln2→ℝ2 defined by α(t)=e2t, e-2t. To…

Q: Suppose we have part of a hyperboloid of two sheets -4x? -4y? + z? = 9 where z 20 (so the upper…

A: Equation of hyperboloid of two sheets is: −4x2−4y2+z2=9 Where z≥0 and z≤15

Q: Use Green's theorem to calculate the work done by force F on a particle that is moving…

A: This is a greens theorem problem

Q: Consider the vector field F = (2y +1,y, –2²) along with the paraboloid surface z = r² + y. Find the…

A: Circulation is the line integral of the vector field along the curve

Q: Find the value of the line integral ∫CF→⋅dr→if F→(x,y)=(y^3+11)i→+(3xy+11)j→is a conservative vector…

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Q: Evaluate the line integral Jc F · dr, where F(x, y, z) = xi – yj + 3zk and C is given by the vector…

A: Given F(x,y,z)=xi-yj+3zk and C is the curve given by  rt=sint,cost,t. Find ∫CF·dr.

Q: Calculate the integral of f(z,y, 2) = 6z2 + 6y² + 2³ over the curve c(t) = (cost, sin t,t) for 0<t<…

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

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Q: If a vector function v(t) = (t, Int,t-²) is given for a scalar function Ø = x In z + Jy - cos(x + 1)…

A: Given the scalar function φ=xlnz+y-cosx+1. This scalar field is to be evaluated over the path…

Q: Evaluate the line integral / V(ry + xz + yz) · dĩ over the curve F(t) = (t, 2t, 3t) for 0 <t < 4…

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Q: Evaluate the line integral F(x, y) = xyi – y²j where the curve C is repre sented by R(t) =t²i+t³j…

A: Function : Fx, y = xy i - y2 jRt = t2i + t3j from (1,1) to (4, -8)

Q: 3) Given the vector field F = circulation around the triangle formed by the intersection of the…

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Q: Evaluate the line integral F. dr, with vector field F(x, y, z) = sin xi + cos yj + xz k, and path C…

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Q: Consider the vector-field F(x, y) = y²i-xj, and let C be the arc of the parabola y = k – x² from the…

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Q: Compute (VxF). dS, where S is given by x² + y² + (z − 1)² = 2 and z ≥ 0, F(x, y, z) = (-esinzy, x +…

A: The general form of a vector-valued function is given by F1,F2,F3 where F1,F2 and F3 are the scalar…

Q: Evaluate the line integral I of the one-form w = xy dx + 12y dy over the parametric curve a : [0,…

A: Given: ω=xydx+12ydy To find: I=∫αω  where α:[0,ln2]→ℝ2 with α(t)=(e2t,e-2t) ∴x=e2t,y=e-2t Now,…

Q: Consider the force field F = (ev – e2" , (x + })ev) and the curve C = C¡U C2 where Cı is the line…

A: As per our guidelines we are allowed to do 1 question at a time. Please post other questions next…

Q: valuate the line integral Scy da +a dy where C is the parameterized path a = t2, y = t°, 1<t<4. Scy…

A: We have to find the given integral.

Q: use a parametrization to find the flux ∬S F . n dδacross the surface in the specified direction. F =…

A:

Q: Evaluate the line integral ∫cF⋅dr, where F(x,y,z)=5xi+yj+3zk and C is given by the vector function…

A: we have given F(x,y,z)=5xi+yj+3zk r(t)=(sint,cost,t)=sint i+ cost j +t k⇒dr=(cost i -sint j…

Q: Evaluate the line integral F dr, where C is given by the vector function r(t). С F(x, y, z) =…

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Q: Sketch the vector field F(x, y) = i + 3j and calculate the line integral of F along the line segment…

A: We have to sketch the vector field.

Q: Integrate the vector field h(x,y) = y^2i + (2xy-e^2y)j over the circular arc C parametrized by r(t)…

A: The answer is given by using the formula of line integral

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: Evaluate the line integral F. dr, where F(x, y, z) = 4xi + 4yj – 5zk and C is given by the vector…

A:

Q: Calculate the flux integral f(2i + 2j + 2k) · dA where S is a disk of radius 5 in the plane æ + y +…

A: Given F=2i+2j+2k,radius=5 and x+y+z=1 plane

Q: Evaluate the line integral ſc F · n ds, where the vector field F(7,y) = (tan-" (y) +æ?, z*y) and C…

A: Given: To explain the correct option of the given statement ass follows,

Q: Evaluate the line integral ∫F⋅dr, where F(x,y,z)=5xi−yj+zk and C is given by the vector function…

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Q: Compute the line integral of the vector field F = (4y, –4r) over the circle z2 + y? = 16 oriented…

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Q: Evaluate the integral // Fn ds where F = (2r+32)i-(rz+y)3+(y² +2)& and S is the surface of the…

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Q: Calculate the line integral a) |y ds, C:x =ť², y = 2t, 0 <t < 3 C

A: Use parametric to solve

Q: Use Green's Theorem to calculate the work done by the force F on a particle that is moving…

A:

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

A:

Q: Evaluate the line integral F. dr, where F(x, Y, z) = –xi yj – 5zk and C is given by the vector…

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Q: Find the work done by F in moving a particle once counterclockwise around the given curve = (2x –…

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Q: Evaluate the line integral Sp(xi+ yyj) · dr where D is the oriented curve shown below. (1,1) y = x²…

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Q: Compute the scalar line integral xy ds where C is the part of the circle of radius 3, centered at…

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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 line integral a xy dx + yz dy + xz dz. (a) What is the vector field that is being…

A:

Q: Compute the line integral of the vector field F = (4y, –4x) over the circle x² + y? = 36 oriented…

A: Given: F= &lt;4y,-4x&gt;          x2+y2=36

Q: 2. Compute the scalar line integral xy ds where C is the part of the circle of radius 2, centered at…

A: Put x=2cost , y=2sint

Q: Compute the line integral of the scalar function f(x, y, z) = 2x² + 8z ver the curve c(t) = (et, t2…

A:

Q: Evaluate the line integral ſp(xi + Vyi) · dr where D is the oriented curve shown below. (1,1) y = x2…

A: We know that r=xi+yj As, the path followed is y=x2 So, r=xi+x2j Let x=t, then  r=ti+t2jdrdt=i+2tj…

Q: Evaluate the line integral ∫C -y dx + z dy + 2x dz, where C is the helix r(t) = (cos t)i + (sin t)j…

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Q: Compute the line integral [(yze²v² + y² + 2 − 2x) dx + (xze²v² + 2xy) dy + (xye²³² +x+32²) dz, where…

A: We have to compute the line integral ∫Cyzexyz+y2+z−2xdx+xzexyz+2xydy+xyexyz+x+3z2dz, where C is the…