Work done by a force field F=[1,2x] along a closed path C: x²+y = 16 is given by: Select one: O a. 4TT b. 16T с. 32п d. none of these

Q: Please compute the work done by the vector field F(r, y) = (sin r+ry?)i+(ev+ a)j in R?, where C is…

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Q: Given the vector field E = 4zy² cos 2xa, +2zy sin 2xay + y² sin 2xa₂ for the region [x], [y], and…

A: The general form of a vector-valued function in three dimensions is given by F=F1ax+F2ay+F3az, where…

Q: Evaluate the line integral fe F n ds, where the vector field F(x, y) = (tan-'(y) + x², x?y) and C is…

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Q: Given the vector field and oriented curve C, evaluate F.Tds. C (x.y) on the line segment r(t) = (t,…

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Q: Given two points A(3, 0) and B(), find the work done by the following force field in moving a…

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Q: Show that work performed by a constant force field F over any path C from P to Q is equal to F - PÓ.

A: A vector has 2 components, magnitude and direction which is represented by cos and sin angles. A…

Q: Given the following vector field and oriented curve C, evaluate SF.T ds. C F= (-3y,3x) on the…

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Q: Given two points A(3, 0) and B(), find the work done by the following force field in moving a…

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Q: Given the following vector field and orientated curve C, evaluate JF•Tds. (x.y) on the curve r(t) =…

A: Solution:

Q: Find the work done by the force field F(x,y) = x² i– 2xy j in moving a particle along the quarter…

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Q: 13 Given the conservative vector field, F = (x +3y?)i +(6.xy +5)j , evaluate the 120) integral F.ds…

A: Here we use the way of line integral. By finding the parametric path between two points.

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

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Q: Find the work done by the force field F on a particle moving along the given path. F(x, y) = xi +…

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Q: Find the potential function for the following vector field F(x, y, z) = (2xy – 29) î + (x² + z²)j+…

A: To find the potential function, f, integrate each partial derivative. Integrate ∫fxdx=∫(2xy-29)dx,…

Q: 20

A: I have stated the used formulas in the solution

Q: Q3 Given the conservative vector field, F =(x +y²)i +(2xy +5)j , evaluate the integral F•ds where C…

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Q: Q3 Given the conservative vector field, F = (x +y²)i +(2xy +5)j , evaluate the integral F•ds where C…

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Q: The work done by the force field F = grad(x+y+z), along the curve C:r()-[1,1,1], from a point A:…

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Q: Given the vector field F(x,y,z)= (z, x, y). Evaluate the line integral S F.dr, along the curve C: C…

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Q: Consider the force field F (ev – e2", (x + )e") and the curve C = C1 U C2 where C1 is the line…

A: Given : force field F→=ey-e2x, x+12ey and the curve C=C1∪C2 where C1 is the line segment from 0, 0…

Q: Given that F(x, y, z) = (ey, xey + ez, ye²) is a conservative vector field. a. Find a potential…

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Q: Given the following vector field and orientated curve C, evaluate F•Tds. (x,y) F = on the curve r(t)…

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Q: Consider the vector field F = p? + yzi+ + In(s* + :² + 1)k „j + In(=* + :² + 1)k 12 + y²" defined on…

A: We will use the basic knowledge of vector calculus and integral calculus to answer this question.

Q: 8. Given the vector field G(x, y, z) = e>+2z (1 + x j + 2xk) : a) Show G(x, y, z) is conservative…

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Q: Given the force field F (x, y, z) = 1.) Show that SF · dR is independent of path without computing…

A: We know that the line integral of a vector field is path independent if the vector field is…

Q: Find the work done by the force field F in moving an object from A to B. F(x, y) = 6y3/2 i + 9xyj…

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Q: Evaluate the integrals

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Q: 问题的回答系 极性问题是预期答案等 于 是和否的问题(因此, Given the vector field F(r. y, z) = 3: cos.ri+a lny j + e° tanr k.…

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Q: 2. Given that F(x, y, z) = (e, xey + e², ye²) is a conservative vector field. a. Find a potential…

A: We are authorized to answer one question at a time, since you have not mentioned which question you…

Q: Given the following vector field and oriented curve C, evaluate F•T ds. F = (y,x) on the line…

A: Given: F=<y,x> C be the line segment from (3,1) to (7,10)

Q: Given the conservative vector field, F=(x +y³)i +(2xy +5)j , evaluate th integral F•ds where C is…

A: We evaluate the given integral by using line integral. We parameterized the curve between two…

Q: Consider the semicircle r(t) = a cos(t)i + a sin(t)j with 0 0. For a given vector field F, the flux…

A: Given

Q: Find the work donc By the Force Field E în movirg an objcet From Pto ♡ 444 53 F(X,y)= 15X 'y %3D -…

A: The work done is given by  dot product of force and displacement vector

Q: Find the work done by force field F on an object moving along the indicated path. F(x, y, z) = xi +…

A: To solve the following problem

Q: Identify the work done, J. F.dr by an electric field, E = (x + xy)î + (y + 5)f – 5zk V/m to move an…

A: Work done along the curve C by vector field F . Where C is given by parametric equation r(t)…

Q: The work done in moving a particle in the force field F= x- i+ 3xy j from (0,0) to (1,2) along the…

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Q: Given the following vector field and orientated curve C, evaluate F•Tds. (x.y) F = on the curve r(t)…

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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: Given the following vector field and oriented curve C, evaluate F•Tds. C F= (x,y) on the parabola…

A: Given,F = <x,y>and r(t) = <14t, 3t2> , for 0≤t≤1. now,

Q: Find the work done by the force field F on a particle moving along the given path. F(x, y) = xi +…

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Q: Show that the vector field ? = ? + sin(z)? + ycos(z)? is conservative and then find the…

A: We have to show that the vector field F=i+sinzj+ycoszk is conservative and then find the…

Q: ind the work done by the force field F on a particle moving along the given path. F(x, y, z) = yzi +…

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

A: Let F=Pi^+Qj^ ∂Q∂x=∂P∂y=1 The force field is conservative.

Q: Given two points A(3, 0) and B( ), find the work done by the following force field in moving a…

A:

Q: Given the conservative vector field, F=(x +y³)i +(2xy +5)j , evaluate th Q3 integral | F•ds where C…

A: Given a conservative vector field, F=x+y2i+2xy+5j. Here, M=x+y2 and N=2xy+5. Since, F is…

Q: Find the work done by the force field F in moving an object on the line segment from A to B, where…

A: Given, Fx,y=2y32 i +3xy j To find the work done by the force field F in moving an object on the line…

Q: Given vector field F(x, y) = (4x³y? – 2xy³,2x*y – 3x?y² +6y³) and curve C defined by C: 7(t) = (t+…

A: Here we will find out the required value.

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: