When can we apply the Fundamental Theorem of Line Integral to compute work done by a vector field in moving a particle along a space curve C? the vector field is not conservative the vector field is conservative can't apply to any vector field all vector fields

Q: F(x, y) = (2y³ – 4xy² – 1, 2xy² – 4x²y – 1) %3D F2(x, y) = (2y³ – 4xy – 1, 6xy² – 4x²y – 1) 1. For…

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Q: A small surface S is divided into three pieces of area 0.2. Estimate j L F · dS if F is a unit…

A: Lets consider the given surface S  is of the form with area 0.2, where F is a unit vector field…

Q: Sketch the vector field F(x, y) = xj, the line segment from (6, 1) to (6,5), and the line segment…

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Q: Consider Vector field given by A = pcos(@)a,+ a,, the line integral of Vector A over the segment 1…

A: In this question, we have to find the line integral in polar form for the given segment 1 i.e. line…

Q: A vector field in space is conservative. Is the vector field irrotational? Explain.

A: Query-  A vector field in space is conservative. Is the vector field irrotational? Explain. Answer-…

Q: True or False: Let F be a vector field defined on a region R. If the line integral of the vector…

A: Given Statement: Let F be a vector field defined on a region R. If the line integral of the vector…

Q: Use a surface integral to compute the flux of the vector field leaving a unit sphere centered at the…

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Q: What is an oriented surface? What is the surface integral of a vector field in three-dimensional…

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Q: F(x, y. =) = (y+: sin x, x+: cos y, cos x-sin y) and G(x, y, 2)={y-:sinx, x+zcos y, cos x+sin y} 1.…

A: Since you have asked multiple questions, we will solve the first question for you. If you want other…

Q: Sketch the vector field F(x, y) = xj, the line segment from (1, 4) to (7,4), and the line egment…

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Q: A particle moves along line segments from the origin to the points (1,0,0), (1, 2, 1), (0, 2, 1) and…

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Q: Compute the work done by the vector field F = in moving a particle along the circle x + y = 1 from…

A: Solution is given below

Q: Why does a two-dimensional vector field with zero divergence ona region have zero outward flux…

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Q: Assume that F is a conservative vector field. (a) What is the definition of a vector field F being…

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

A: Obtain the equation of the plane passing through the 4 points (1,0,0), (1,5,1), (0,5,1) and the…

Q: Given the vector field F = (In(x + 2z), xy, xyz). Calculate curl F(-1,1,1).

A: Solution:-

Q: Find the work done by the vector field F =( y+x 2, 2%y+z, 2yz + 2) %3D the Curve C ( all over it)…

A: The given vector field is F→=y2+xz,2xy+z2,2yz+x22 and a vector field F→ connects points 1,-1,-1 to…

Q: Check if the field is conservative, if so, calculate by the Fundamental Theorem of Calculus for Line…

A: F =2xy i +(x2+2yz) j + y2 kcurl F=ijkddxddyddz2xy(x2+2yz)y2      =(2y-2y)i-(0-0)j+(2x-2x)kcurl F…

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

A: First obtain the equation of the plane passing through the 4 points (1,0,0), (1,5,1), (0,5,1) and…

Q: Calculate the outward flux of the vector field F→ (x,y,z) = 2xi→ + yi→ +zk→ along the boundary of…

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Q: Show that the line integral is independent of path. | 2xe Ydx + (2y - x²e)dy, Cis any path from (1,…

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Q: What does it mean if the divergence of a vector field is zerothroughout a region?

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Q: Compute the curl of the vector field F = (r', y³, z*). curl(F(r, y, z)) = %3D What is the curl at…

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Q: Determine whether the line integrals of the vector fields along the oriented curves are positive or…

A: We have, the line integrals of the vector fields along the oriented curves as

Q: (1 point) Show that the vector field F-yz Ч + zz 1 J-zuz 2 k is conservative by finding a scalar…

A: To find the scalar potential function f, assume

Q: Give an example of a vector field F (x, y, z) that has value 0 on a surface and such that curl F is…

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Q: a vector field ū in spherical courdinate system Express t in the Cartesian coordinate system. U= -…

A: In spherical coordinate system, u→=urr^+uθθ^+uϕϕ^u→=-cosϕ2r+sinϕ2ϕ^ The Cartesian coordinate system,…

Q: Find work done by a force field F= in moving object along the line segment from the point (2,1,3) to…

A: Given, F=1y-x,xyz,ex2+y2+z2

Q: Find the parametrization of two different curves from the point (2,4) to (3,9). Compute the work…

A: Given point (2,4)  and (3,9)now equationof line passing through two point is given by…

Q: What does it mean to say that a vector field is conservative?

A: What does it mean to say that a vector field is conservative?

Q: Describe the different methods available for evaluating the integral of a conservative vector field…

A: To Describe: Different methods available for evaluating the integral of a conservative vecctor field…

Q: verify the Its Stoke For Upper Projection theorem Vector Field F = (2x-y)i - yzj- y²zK ² Surface the…

A: To verify Stoke's theorem: For a vector field F=(2x-y)i-yzj-y2zk over the upper half of the surface…

Q: A particle moves along line segments from the origin to the points (2, 0, 0), (2, 4, 1), (0, 4, 1),…

A: A particle moves along line segments from the origin to the points 2, 0, 0, 2, 4, 1, 0, 4, 1 and…

Q: Why does a two-dimensional vector field with zero curl on aregion have zero circulation on a closed…

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Q: 4. Verify that the two integrals in the circulation form of Green's Theorem are equal along a circle…

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Q: Explain how to compute the divergence of the vector field F = ⟨ƒ, g, h⟩ .

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Q: A particle moves along line segments from the origin to the points (2, 0, 0), (2, 4, 1), (0, 4, 1),…

A: Given Fx,y,z=z2i+5xyj+2y2k  Let r=xi+yj+zk⇒dr=dxi+dyj+dzk Now particle moves from origin O0,0,0 to…

Q: Verify Stoke's theorem for the vector field F = (2x - y) I- yz*J-yzK over the upper half surface of…

A: Stokes theorem is used

Q: Consider Vector field given by A = pcos(@)a,+ ag, the line integral of VectorA 3 over the segment 2…

A: We are given the vector field as  A=ρcosφ aρ+ρ3aφ. We need to find the line integral of vector A…

Q: A smooth vector field F has div F(5, 2, 5) = 2. Estimate the flux of F out of a small sphere of…

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Q: let c is a differentiable vector field then div(curl)=

A: This question is related to vectors.

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

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

A: Apply Stokes theorem

Q: Sketch the vector field F(x, y) = xj, the line segment from (1, 9) to (6,9), and the line segment…

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Q: Is the vector field shown in the figure conservative? O Yes O No Explain. Let C be a circle centered…

A: Graphically, a vector field is conservative if it has no tendency to swirl around. For a…

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

A: Given points, (0,0,0) (3,0,0) (3,3,1) (0,3,1) field Fx,y,z=z2i+4xyj+3y2k Now, Curl…

Q: Consider Vector field given by A = pcos(p)a,+ a,. the line integral of Vector A over the segment 3…

A: line integral of a vector field A over a path is given by ∫A.dl where . indicates the dot product of…

Q: Let C be the circle with radius 3 centered at the origin, oriented counterclockwise. Define a vector…

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Q: Give an example of a vector field F (x, y, z) that has value 0 at only one point and such that curl…

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