1. Given vector field F=(-y,x). By using Green's theorem, evaluate dA where A R is the area of a circle with radius, r equal to 3.

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

A: Given vector field : F=x,yx2+y2 As rt=t,7t; 1≤t≤6 Thus, the parametric representation of x and y…

Q: Given vector field A = 5p sinø a, - pz cosø a, + 2p a̟. At point P(1, T, -1), the vector component…

A: A= 5ρsinϕaρ-ρzcosϕaϕ+2ρaz  at point  P(1,π,-1) =0aρ-1aϕ+2az A that is parallel to  ϕ =π is…

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: Show that for any scalar, vector or tensor field y and any material volume V: d. prpdV = dt v Dt AP-

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Q: A vector field v is expressed in spherical coordinates as v(r, 0, o) = 2r cos 0 e, - rsin o eg.…

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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: Is the vector field F(x, y) = (In y + sin(e²“), #

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Q: Suppose S is the surface where x? + y? + 3z² = 1. Let F be the vector field F = (xy, yz, z?).…

A: suppose S is the surface where x2+y2+3z2=1 let F be the vector field Fxy, yz , z2 claim- evaluate…

Q: Find the flux out of the unit shpere for the vector field F(2,4, ;) = 1ā+3+K). %3D r answer =

A: Flux of F through the closed surface S is ∫∫sF→·n→dsS is the unit sphere x2+y2+z2=1Letϕ:…

Q: Compute the curl of the following vector field. (x, y, z) 5/2 F= (* +y? +z?)* The curl of F is (Di+

A: On solving this problem we will get

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: 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: A vector field v is expressed in spherical coordinates as v(r, 0, 0) = 2r sin cos o e, +r cos o cos…

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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: Is the vector field F(x, y) = (sin(x)ely=3 + 2r cos(r>)y°, -2e)y-3 + 6 sin(x³)»*) conservative?…

A: Consider the given vector field.

Q: If a vector field F (x, y, z) In (xyz), cos z), which of the following values is equal to div F (1,…

A: Finding divF

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: A vector field v is expressed in spherical coordinates as v(r, 0, 6) = 2r cos 0 e, – r sin 0 eg.…

A: For the solution follow the next step.

Q: Compute the line integral of the vector field F=⟨3zy−1,4x,−y⟩ over the path c(t)=(et,et,t)for −4≤t≤4…

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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: At how many points is the vector field F(x, y, z) = (z– ,r – 2², y – x²) irrotational? Select one:…

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Q: given the vector field A(r,θ,φ )= sin(θ)*r + r*θ + 6*φ, find the laplacian and evaluate it at the…

A: In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence…

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

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Q: he vector field at the given point F (x, y, z) = e* sin yi – e

A: Here we have to find the curl F for the vector field at the given point (0,0,1)   we have,…

Q: Given the following vector field and oriented curve C, evaluate T ds. F = (x.y) on the parabola r(1)…

A: In this question we will find the line integral of F over c.

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

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Q: Consider the vector field F = (a*y°, æ°y*) Is this vector field conservative? Select an answer If…

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Q: Is the vector field F(x, y) = (sin(x) cos(y), cos(x) sin(y)) conservative? Select the correct answer…

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

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

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Q: F=cos(x) î+sin(y) ĵ+cos (z) k

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Q: A vector field is given by F = (2y + 3)i + xz j + ( yz -x)k, evaluate ∫ ?. ?? c

A: Given: F→=(2y+3)i⏞+xzj⏞+(yz-x)k⏞ C is the curve given by r→(t)=2ti⏞+tj⏞+t3k⏞ where t varies from a=0…

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: Given the vector field H = pz cos o a, + sin ap +p²az and a point Q 1,,0), solve the following and…

A: Part (a) The given equation is, H→=ρzcosφap+sinφ2aφ+ρ2az The vector component of H→ normal to the…

Q: Consider the vector field F = (x*y, x*y? Is this vector field conservative? Select an answer If so:…

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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: For what values of a & b such that the vector field f= (ay'e" + xz)i +(ye" +z')}+| byz' + is…

A: f is given. We find a,b so that f is conservative (irrotational).  f is conservative (irrotational)…

Q: Given the following curve: 7: [0, 1] → R? kt² tHy(t) = 3t2 +2 and the following vector field: R? →…

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Q: 5. If F = A × OP, where A is a constant vector field, show that V x F = 2A 6. If R = OP,r = ||R||,…

A: The solution are next step

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: Given the vector field F(x,y) = [x/((x^+y^2)^(3/2)), y/((x^+y^2)^(3/2))], Find a fucntion f(x,y)…

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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: 9. Given the vector field F =, use Stokes' Theorem to compute SS. (V× F) · ñ dS where S is that…

A: Solution is given below

Q: Given the following vector field and oriented curve C, evaluate F = (x,y) on the parabola r(t) =…

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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: