Find the work done by the force field F =< z, x, y > in moving the particle from the point (3, 0, 0) to the point (0, T/2,3) alongover the following paths. (a) a straight line (b) the helix: æ = 3 cos t, y = t, z = 3 sin t

Q: Consider the force field F = (2xzez — 4, cos³ z, x²eª²z – 3y cos² z sin z Find the work done by F in…

A: Given: The force field is F→=2xzex2z-4,cos3z,x2ex2z-3ycos2zsinz and the given given curve is…

Q: Find the work done by the force F = 9xy i + 6yj acting along the piecewise-smooth curve consisting…

A: Given that F=9xy i+6y2 j Let r=xi+yj+zk ⇒dr=dx i+dy j+dz k Now,  F.dr=(9xy i+6y2 j+0k)(dx i+dy j+dz…

Q: Find the workdone in moving a particle in a force field given by f=3xyi-5zj+10xk along the curve…

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Q: A particle is under the influence of the force F= (-cosh (4x4) + xy)i + (e-y + x)j. The corner…

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Q: Use the Green's theorem to find the work done by a force F(x, y) = on a particle that moves…

A: find explanation below

Q: 10. Compute the work done by the force F(x, y) = (x²–y) i+xj acting on an insect as it moves along a…

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Q: Use Green's Theorem to find the work done by the force F(x, y) = x(x + y) i + xy² j in moving a…

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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: 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: Find the work done by the force field F(r, y) = x²i+ xyj on a particle that moves once around the…

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Q: 3. Use Green's theorem to find the work done by the force field F(x,y) = xyi + (;x² + yy)j on a…

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Q: A two-dimensional force field is represented by the vector field F(x, y) = (y, –x). Inside this…

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Q: 2. Let C be the curve of intersection between the cylinder r2+y = 1 and r-2y+z = 0, oriented…

A: The complete solutions are given below

Q: Find the work done under the influence of the force E = 3xyi – 5zj + 10xk along the curve x= t² + 1,…

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Q: Find the work done by the force field F(x,y) = 4yi + 2xj in moving a particle along acirclex^2 + y^2…

A: Step2

Q: consider a particle traveling within the force field F = along the parametrized curve r(t) = from…

A: Consider the given: F=−y,x,12rt=tcost,tsint,12tFrom point 0,0,0 to 2π,0,π. The work done moving the…

Q: Find the work done by the vector field ⟨4x+yx,x2+6⟩ on a particle moving along the boundary of the…

A:

Q: A particle starts at the point (-2, 0), moves along the r-axis to (2,0), and then along the…

A: Given the force field F by, F=xi+x3+3xy2j Work done by the force F in moving a particle along a path…

Q: Find the work done by the force field F(x, y)=(e* – y)i +(cos y + x) j to move a particle once…

A:

Q: Suppose a particle travels one revolution clockwise around the unit circle under the force field…

A: Step 1: Hello. Since you have posted multiple questions and not specified which question needs to be…

Q: Given the force field F, find the work required to move an object on the given oriented curve. F=…

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Q: Find the work done by the force field F(x,y) = x²i – xy²j around the circle (Clockwise direction)…

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

A: Let C covers all the line segments. Consider C1 be the line joining the points (2,0,0) and (2,2,2).…

Q: Find the work done by the force field F(x, y) = xy i + x^2j on a particle moving from (0, 0) to (1,…

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Q: A force F acts on a particle that is moving in the plane along the semi-circle parameterized by 7(t)…

A: Given that: γt=−costi+sintj; t∈0,π F=x2+y2τ, τ is the unit vector tangential to the path. To…

Q: F= (e* – y+ xcosh(x))i+ (y³/2 + x)j.

A:

Q: Compute the work done by the force along the curve (use defini- tion): 1 F = [3r – y + y] curve is…

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Q: Given the force field F, find the work required to move an object on the given oriented curve. F =…

A: Given the force field F=y, -x. Now, we need to find the work required to move an objective with a…

Q: Use Stokes' Theorem to find the work done by the force field F= on a particle that traverses counter…

A: Given data: The force field is F→=z3,12x,y The equation of circle is C:x2+y2=9 in the plane z=5.

Q: Find the work done by the force field F(x, y) = x sin y i + y j on a particle that moves along the…

A: Explanation: Given that,  Force F and a parabola from (-1,1) to (2,4)…

Q: Calculate the work done in moving a particle in the force field F = 3x2î+ (2xz-y)ĵ+ z k along a…

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

A: The work done by the path E is given by, W=∫EF.dr If the force vector is: F(x,y)=&lt;2xsiny+y2ex,…

Q: Find the work done by the force field F = in moving particle along the curve C defined by 7(t) =,0…

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Q: Can you help with 23 please?

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Q: Confirm that the force field F(x, y) = yeyi+ xey j is conservative in some open connected region…

A: We have to find the work.

Q: A particle starts at the point (-2,0), moves along the x-axis to (2,0) and then along the semicircle…

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Q: Find the work done by the gradient vector field F(x, y, z) = V (e² + y² + z²) in %3| mov ing a…

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Q: Find the work done by the force field F in moving an object from P(-9, 7) to Q(3, 3). 2.x. F (r, y)…

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Q: Use Green's Theorem to find the work done by the force  F(x, y) = x(x + y) i + xy2 j  in moving a…

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Q: Find the work done in moving a particle in the force field F = (2r + y)i + (3y – 2)j along the…

A: Work done in moving a particle in the force field along along the curve C is:  ∫CF→·dr→ here F→ is…

Q: Confirm that the force field F(c, y) = 2xy³i + 3x?y?j is conservative in some open connected region…

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Q: (b) Find the work done by force field F = 3x? î + y²j on a particle when it moves from (0, 0) to…

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Q: Use Green's theorem to calculate the work done by the force F(x, y) = xy'i + 3xyj on a particle that…

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Q: Use Stoke's Theorem to find the work done by the force field F(x, y, z) = (esin x + 2y², 6x + tan-…

A: We have to find the work done by the force field F→x,y,z=esinx+2y2,6x+tan-1y,xz in moving a particle…

Q: Exercise (3): Find the work done under the influence of the force F = 3xyi – 5zj + 10xk along the…

A: The work done by a force on displacing a object by a certain distance is given by dot product of…

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

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Q: Find the work done by the force field F in moving an object from P(-9, 5) to Q(3, 5). F (r, y)

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Q: Determine the force being experienced by a 2-microcoulomb charge at (-2,4,6) m due to another point…

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Q: Use Green Theorem to find the work done by the force field, F(x, y) = (e* + x²y,e – 2xy) in moving a…

A: According to Green's Theorem  ∫CPdx+Qdy=∬D(∂Q∂x−∂P∂y)dA where F→ = ⟨P, Q⟩

Q: Find the work done by the force field F(x, y) = (4 e* – y³ î + (2 cos(y) + x³ )j on a particle that…

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Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: