Q3 Given the conservative vector field, F = (x +y²)i +(2xy +5)j , evaluate the integral F•ds where C is the path, from (1,2), to (3,4) along any curve.

Q: 5) Consider the conservative force field F(x, y) = (6x²y? + cos(x), 4x y). a) Give a general…

A: For a vector field F=P,Q that is open and simply connected in a region with continuous first order…

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: Q4: Find the curl of the vector field F(x, y,2) = xJyi +j+ zk at the point(2,4).

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: Q4: Find the curl of the vector field F(x, y.2) = x*/yi +j+ zk at the point(2,4).

A:

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: 9. Compute the flux of the vector field F(x, y) = (-x sin(1+xy) – 4 cos(TY), y sin(1+ xy) + 6x)…

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Q: Q3) Compute the work done by the force field F(x,y,z) = (3x, xz, y) on the curve C, where C is the…

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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…

A:

Q: 10. For a vector field F = <2xy + y³, x² + 3xy²+ 2y7 find a function f such that F = Vr.

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Q: 5. Find the potential function for given vector field: F = (e* In(y))i + + sin(2))j + ycos(z)k

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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. Let F(r, y) = ye²vi+ (xe"V – 1)j. (a) Show that F is a conservative vector field. (b) Find a…

A:

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: 3. Consider the vector field F = 2xy i + (x² + 2y) j. Verify the Green's Theorem in the closed…

A: Definition of green's theorem :

Q: 3. Use the fact that F = is a conservative vector field to evaluate SF. dr when C is the path r(t)…

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Q: 4. Is the vector field F(x, y, z) = (yz, xz, xy) conservative? Justify your answer.

A: Calculate divergence of given vector field. If divergence is zero then field is conservative.

Q: Q2. (a) Given a vector field, F = 2xyi + (x² + 3y²z)j+ (y³ + 2z)k. (i) Determine whether if F is…

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: Q3) Determine if the vector field F(x, y, z) = (yz,xz + 2y, xy + 1) is conservative. If it is, find…

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Q: Calculate the work done by the vector field F(x, y)= ,2xy) on a particle that moves once around the…

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

A:

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: Given that the vector field F(x, y) = (2x-3y, 2y-3x) is conservative in the xy-plane. If (x, y) is a…

A: We have to find the value

Q: Find the line integral of the vector field F= over the curve C which is the quarter of the unit…

A: We have to find line integral from point (0,1) to (1,0)

Q: Consider the vector field F (xr, y) = (2y - 12r*y", 4ry – 9r*y?) is conservative. (a) Show that F…

A:

Q: Consider the vector field F=(x^2+y^2,10xy). Compute the line integrals ∫c_1 F⋅dr and ∫c_2 F⋅dr,…

A: Given:

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: A potential function for the vector field F(x, y, z) = (yz, xz + z, xy + y+ 1) is (x. y, z) = xyz +…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: (a) Show that the vector field F(x, y, z) = (1 + 3x²y)i + x³j is conservative and find the potential…

A:

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: 6. Compute ſ, F-·dr where f(x, y) = (y² cos x, 2y sin r) and C is the straight line path starting at…

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Q: For the force field F = (2xe¬Y, (2y – x²e¬y)) in some open connected region containing P(1, 0) and…

A: Given that, F=2xe-y, 2y-x2e-y If F is conservative then ∂2F∂x∂y-∂2F∂y∂x=0

Q: Determine whether F(x, y) = e5 cos 5y i – e5" sin 5yj is a conservative vector field. If so, find a…

A: ANSWER :

Q: Which of the following functions is a potential function for the conservative vector field F (x, y,…

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Q: The vector field F(r, y) = (2xy +y) dx + (æ + 2xy) dy is conservative. %3D The function f(x, y) = x²…

A:

Q: Evaluate F•Tds for the vector field F = x²i- yj along the curve x= y? from (1,1) to (4, – 2).

A: Answer = 19.5

Q: Q2. (a) Given a vector field, F = 2xyi + (x² + 3y²z)j+ (y³ + 2z)k. (i) Determine whether if F is…

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Q: Q5: Find the divergence of the vector field y F(x, y, z) = x² √yi + =j+zk At the point (2,4,1).

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Q: 2: Compute the work done by the force field F = (y, -x) to move an object from point A = (1,0) to B…

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: 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: 25. Evaluate |F•T ds for the vector field F = x²i – yj along the curve x = y? from (4, 2) to (1,–1).

A:

Q: Find the potential function p, for the conservative vector field F(x, y, z) = (2xe? – y²sin x)î +…

A:

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: