13 Given the conservative vector field, F = (x +3y?)i +(6.xy +5)j , evaluate the 120) integral F.ds where C is the path from (1,1), to (4,4).

Q: The potential of the conservative vector field F = (y,x,1> is Of=z+xy O f=x+yz O f=y+xz Of = xyz

A:

Q: 2. Verify that the vector field F(1, y, 2) = sec? y+ -32 2.r tan y + e csc r, 3e -32 cot x + sec z…

A:

Q: 6. Let F = (x³ sin y, a* cos y). COS (a) Is F a conservative vector field? (b) If your answer to…

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Q: The vector field F = ( , ,2²) has a scalar potential in the region R= {(r, y, z) E R³ : x> 0}

A: ANSWER: TRUE

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: 2. Find the curl of the vector field. F = z°e* sin(2 y)i – e' cos yj+ xyzk to

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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: 8. The component of the field F= (1,2,3) normal to the plane z = -3x + 3y -1 is (a) 0. (b) –2//19.…

A: If we are given a plane equation in the form Ax+By+Cz=D, then the normal vector to the plane will…

Q: 13 Is this vector field F(x, y)=(ycosx)i +(sin x+ y)j being conservative? Please show.

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Q: 1. Consider the vector field F(x, y, z) = (2e2" cos(3y), –3e²" sin(3y) + Y a. Find divF(x, y, z) and…

A:

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

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

Q: 5. Is the following vector field F(r, y) = (2x – y, -x + 3y?) conservative? If so, find a potential…

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Q: Q2: Find the work done S. F.dr in the force field F = yzî + xzĵ + xyk in moving from (1,1,1) to…

A: The equation of a line joining by the points and is given as the following.

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: Compute the line integral of the vector field F = (3zy-¹, 4x, -y) over the path c(t) = (e¹, e¹, t)…

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Q: Consider again the vector field F = (x +y)i + (x² + y²)j_what is the downward flux across the line…

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Q: 2) Given the vector field F = (- Vx²+y² 2z2) write the integral to calculate the outward x²+y² flux…

A: We have to find the outward flux across the cylinder x2+y2=100.

Q: 8. Show that the vector field, -y a² + y?' x² + yj²' ) is not conservative in R³, despite the fact…

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Q: Sketch the vector field F(x, y) = xi + yj and calculate the line integral of F along the line…

A: The solution are next step

Q: 5) Find the work done by the force field F(r, y, z) = ri-zj+y k in moving an object along the…

A: Work = Force X Displacement

Q: Compute the line integral of the vector field F = (3zy1, 4z, -y) over the path c(t) = (e*, e', t)…

A:

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…

A:

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: Sketch the vector field F(x, y) = xi + yj and calculate the line integral of F along the line…

A:

Q: 5. Show that F =(y cos(x),sin(x)+ y) is a conservative vector field and use this to calculate the…

A: Given: F → = y cos(x), sin(x) + y. A vector field F→ = P, Q is said to be conservative if partial…

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: 6. Use the fundamental theorem of line integrals to evaluate 2x dr + 2y dy where C is the line…

A: Let's understand the Fundamental Theorem of Line Integrals:

Q: Зу 1. Given the vector field F(x, y, z) = î+ Vrz j+x cos y - k, find curl F and div F.

A:

Q: 4. Show, in detail, that any vector field of the form (*,J, 2) = J (*) +8(y)/+h(z)k is irrotational,…

A: To show that the vector field is irrotational.

Q: 4. Evaluate the line integral Vf-r'dt, where the curve C is given by the vector function r(1)…

A:

Q: Consider the following vector field: F = (y* + 1, 3ry + 1). (8.1) Is ſ. F · dr path-independent?…

A: The given vector field is F=y3+1, 3xy2+1. (8.1) For a vector field F=P,Q, the line integral ∫CF·dr…

Q: Q4 Given the conservative vector field, F = (sinx +2y²)i + (4 xy +cost)j, evaluate the integral…

A: A detailed solution is given below.

Q: 1.) Given the conservative vector field F(x,y,z) = ( y In z,x In z, xy find the work required to…

A: The conservative vector field Fx,y,z=y ln z, x ln z, xyz. The parametric equation of line segment…

Q: 2. Find the gradient field F = Vy for the potential function p. Sketch a few level curves of o and a…

A: Given φ(x,y)=2x2+4y2 The gradient field  F=∇·φ=∂φ∂x,∂φ∂y=4x,8y Some vectors of F are  At x=1 and y=1…

Q: Q2: Find the work done S. F.dr in the force field F = yzî + xzj + xyk in moving from || (1,1,1) to…

A: The given question is related with work done. We have to find the work done ∫CF→ . dr  in the force…

Q: (10) Given that the vector field F= 2z i- 3 x j + 4 y k Evaluate f F. dr where S is the portion of…

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Q: If F = xy i+ (x² + 4yz)j+ 2y²z k. Evaluate the line integral between A (0, 2, 0) and B (3, 6, 1).…

A: Given, F= xy i +(x²+4yz)j +2y²zk r0= (0,2,0) r1= (3,6,1)

Q: 9. (~Ch16 Review #13) Find the work done by the conservative vector field F(x, y) = along the curve…

A: given a conservative vector field  Fx,y=4x3y2-2xy3,2x4y-3x2y2+4y3 and a curve  rt=t+sinπt,2t+cosπt…

Q: o) Find the flux of the vector field F (x,y,z) = 3xi – yj + 2zk; across o where o is

A: The given question belonging to the vector calculus  in which we have to find the value flux of the…

Q: 1. Is the vector field V (x, y, z) = (2ry, x² – 2², –2yz) conservative?

A: Since you have asked multiple question,as per our guidelines we are supposed to answer only one…

Q: F(x, y, 2) = (y² cos r + 2*)i+ (2y sin æ – 4)j+ (3xz² + 2)k Cos z

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Q: 5) Integrate the vector field h a,y)= y2i+ (2xy-e*)J C parumetrizal by t€ £0, 5 3. over the circular…

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: Consider the one-form w = x' cos(y) dx + xe13% dy+4z² dz on R3. The associated vector field is: F(x,…

A: For the solution follow the next step.

Q: B) Find the work done by the conservative vector field F (x, y, z) = yzi+xzj+xyk along a smooth…

A: Since, it is given that the field is conservative.

Q: 3. Determine if the vector field F(x, y, z) = e" sin yzi + ze² cos yzj + ye² cos yzk is conservative…

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Q: ermine whether the vector field F(x, y, z) = (x³, 3zy², y³) is conservative on R³. If so, find a…

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Q: 4. Given the vector field Č = p² sin o a, + (z² cos – 1)as + (p² – z)a, Determine the x-component of…

A: Given the vector field

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: