What does it mean to say that a vector field is conservative?

Q: Sketch the vector field F = ⟨x, y⟩ .

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Q: How do you graph the vector field F = ⟨ƒ(x, y), g(x, y)⟩?

A: Given: vector field

Q: Discuss one of the parallels between a conservative vector fieldand a source-free vector field.

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Q: State the formula for Curl of a Conservative Vector Field.

A: Given, To State the formula for Curl of a Conservative Vector Field.

Q: 4. Is this vector field F(x,y)=(ycos x)ĩ +(sin x+ y)j being the path-independent field? Please show.

A: To check if the given vector field is path-independent field or not. Given:

Q: Determine whether the vector field is conservative.

A: To determine whether the vector field is conservative.

Q: 9) Check whether the vector field F = (-cos x cos y, sin x sin y, - sec² z) is a gradient vector…

A:

Q: A vector field in space is conservative. Is the vector field irrotational? Explain.

A: Query-  A vector field in space is conservative. Is the vector field irrotational? Explain. Answer-…

Q: What does it mean if the curl of a vector field is zero throughout a region?

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Q: 4. Is the vector field F(r, y, :) = (yz,rz, ry) conservative? Justify your answer.

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Q: Can anything be said about the curl component of a conservative two-dimensional vector field? Give…

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Q: Is the following vector field conservative? F =

A: Yes the vector field is conservative

Q: Compute the outward flux of the vector field

A: Given,         F(x, y, z)=x3i+y3j+z3k And S is the surface of the solid bounded by the cone…

Q: Match the vector field with its graph. F(x, y) = yi KE y -5+

A:

Q: Consider the vector field v(x)

A: Given, the vector field        v(x)=yx2+y2i^-xx2+y2j^+6k^                               ...(1)

Q: Match the vector field with its graph. F(x, y) = yi 5 5 -5+ サ

A: To match the given vector field to its graph, where given graphs are given below.

Q: Match the vector field with its graph. F(x, y) = yi -6+

A:

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

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Q: (a) Show that the vector field F (x, y, z) = (x² + y², x, z³) is not conservative.

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Q: Find the conservative vector field for the potential function by finding its F(x, y) = x² + 5y2

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Q: Write a formula for a two-dimensional vector field which has all vectors of length 1 and is…

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Q: What does it mean if the divergence of a vector field is zerothroughout a region?

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Q: When is a vector field irrotational?

A: We have to determine when is a vector field is irrotational.

Q: 13

A: We have to find out conservative or not

Q: Give three equivalent properties of conservative vector fields.

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Q: Determine whether the vector field is conservative. F(x, y) = 6y6(yi + 7xj) conservative not…

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Q: Compute the flux of the vector field (x, (æ°, – ay*), out of the rectangle with vertices (0,0),…

A: To compute the flux of the vector field x3,-xy4, out of the rectanglewith vertices 0,0,2,0,2,4…

Q: Write a formula for a two-dimensional vector field which has all vectors of length 7 and…

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Q: The work done by a force field in a moving particle around a closed path is zero.

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Q: What are the outputs of a vector field in ℝ3?

A: What are the outputs of a vector field in ℝ3.

Q: Determine whether the vector field F(x, y) = conservative. yi + xj 1 + xy is

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Q: b) Plot the vector fields for different values of h (show all qualitative behaviors)

A: The given equation is, dxdτ=x1-x-h So, the vector field for different values of h are as follows.

Q: Why does a conservative vector field produce zero circulationaround a closed curve?

A: Let F be an arbitrary conservative field vector around a closed curve C. To prove: Let A and B be…

Q: a) What is a conservative vector field?(b) What is a potential function?

A: Conservative vector field: A conservative vector field is a vector field that is the gradient of…

Q: Compute the flux of the vector field (x3, – xy³), - out of the rectangle with vertices (0,0), (3,0),…

A: Given: F=x3 ,-xy5with rectangle with vertices (0,0) , (3,0) , (3 ,2) , (0,2) To find flux of vector…

Q: Write a formula for a two-dimensional vector field which has all vectors parallel to the y-axis and…

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Q: i) Show that F is a conservative vector field.

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Q: Explain why Green’s Theorem proves that if gx = ƒy,then the vector field B = ⟨ƒ, g⟩ is conservative.

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Q: (a) Give an example of a non-zero vector field that is not irrotational and not incompressible.

A: We will find out the required vector field.

Q: Determine whether the following vector field is conservative. If it is , find the potential…

A: F=<cos(z0, 2y-xsin9z)>=cosz i+2yi-∅sinz k i) curl F=ijk∂∂x∂∂y∂∂zcos z 2y-xsinz…

Q: Is the vector field shown in the figure conservative? Explain

A: The conservative vector fields are the path independent vector fields. A vector field, is…

Q: Show that Green's Theorem holds for the vector field F(x, y)=(x +y, 3x) at a square of side 2.

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Q: What is cural of a vector field?how can you interpret it?

A: Let F=M i+N j+P k be a vector field. Where, M, N and P are continuously differentiable functions,

Q: Show that the vector field A (3x?z + y?)a,+ 2xya, + x³a, is conservative. %3D

A: Given problem:- Show that the vector field A = (3x²z+ y²)ax + 2xyay + x³az is conservative. =

Q: Select the option that gives the vector field that could correspond to this arrow map.

A: The vector field is when a vector is assigned to each point in the given space. The cartesian form…

Q: Determine whether the vector field F(x, y) = 5y2(yi + 2xj) conservative. F is conservative Fis not…

A: Just check if curl of F Is zero or not.

Q: Compute Sc F.d3 for the vedor field in the Cylinder with out ward facing and the surface which…

A: It is given that F→=xi+yj+ezk and the surface is x2+y2=4,1≤z≤5

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: