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

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

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Q: Determine the vector field of F. F(r, y) = yi – xj

A: The given vector function is  yi^-xj^ It is needed to plot the vector field, in order to do so…

Q: Is the vector field F(x, y) = 2xy+y 2xy+ xy+y*x x conservative? Select the correct answer below: O…

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Q: Determine whether the vector field is conservative.

A: To determine whether the vector field is conservative.

Q: Given a two-dimensional vector field F and a smooth orientedcurve C, what is the meaning of the flux…

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Q: Consider the vector field F that is depicted below. On which of the parameterized curves is the…

A: Given the vector field F in the image

Q: What is the value of a so that the vector field F(x.y)=(In y+2xy³) i+(ax²y2+ x/y) j is conservative?…

A: To find: the value of a for which given vector field is conservative.

Q: (b) Verify Green's Theorem for a vector field F(r, y) = ry²i – yx²j on a disc with center (0,0) and…

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Q: Find the gradient vector field of f(x,y) = x2 + y2 and sketch it.

A: Solution

Q: How do you determine whether a vector field in ℝ2 is conservative(has a potential function w such…

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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: Determine whether the vector field F(x, y) = 5y? (yi + 2xj)is conservative. F is conservative O F is…

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Q: (a) Is the vector field | F(x,y) = (xe, ye²) conservative? Justify your answer. If it is…

A: Let the vector field F = Pi + Qj.The above vector field is conservative if and only if dPdy=dQdx

Q: determine whether or not F is a conservative vector field. F(x,y)=(y2-2x)i + 2xyj

A: For a  conservative vector field F, curl F=∇×F=0

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: Make a sketch of the following vector field. F = (-x,2y) %3D

A: We sketch the vector field, we choose  the  points and sketch them.

Q: Determine the vector field of F. F(x, y) = yi - xj

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Q: Determine whether the vector field given by F is conservative. F(x, y) = 2xi + yj

A: Consider that any given vector field F, is said to be conservative if curl F→=0, The given function…

Q: Determine whether the following vector field is conservative on R°. If so, determine the potential…

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Q: Determine whether or not the vector field is conservative F(x,y) = 18x²y²i + 12x³yj conservative O…

A: NOTE: Refresh your page if you can't see any equations. . compare this vector field with  so here…

Q: Is the vector field F (x, y) = (2xy – y³) î + (x² – 3xy²) ĵ solenoidal, conservative, both or…

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

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Q: Consider the vector field F(2. v) = (--2) i + (8ry° tan ' y) j. i + (8ry² – tan ' y) j. a) Is F…

A: Consider the vector field

Q: Determine whether or not F(x,y) = (xy + y²,x² + 2xy) is a conservative vector field.

A: Given vector field is  Fx,y=xy+y2,x2+2xy Fx,y=F1,F2 A vector field is said to be conservative if…

Q: Consider the vector field F = (r²y³, x°y?) Is this vector field conservative? Conservative If so:…

A: Given the vector field <x2y3,x3y2>. We need to: i) Check whether the vector field is…

Q: Determine the vector field of F. F(x, y) = yi – xj

A: Given vector field is F(x, y) = yi - xj

Q: 4. Show the vector field Vu x Vv, where u and v are scalar fields, is solenoidal.

A:

Q: Determine whether the following vector field is conservative on R°. If so, determine the potential…

A: If curl f is zero then conservative vector field

Q: Show that if a vector field has the form F = ⟨ ƒ(x), g(y), h(z)⟩, then∇ x F = 0.

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Q: A constant vector field F(r, y, z) = ai + bj + ck is a gradient vector field. True

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Q: Is the vector field F = [yze"yz, xze#yz, xye"yz] conservative? %3D O Yes O No

A: The vector field is conservative if ∇×F→=0 therefore, find ∇×F→ ∇×F→=ijk∂∂x∂∂y∂∂zyzexyzxzexyzxyexyz

Q: Is for every vector field F on R3, curl(divF) = 0 ? True or False?

A: This is a problem of vector fields. General definition of the curl and divergence is,  Curl is a…

Q: let c is a differentiable vector field then div(curl)=

A: This question is related to vectors.

Q: ) Show that the vector field, F3 yz £ + zx ŷ + xy 2 c

A: Given: The vector field is, F3 = yz x^ + zx y^ + xy z^

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: Determine if the vector field F = y ln zi – x ln zj + xyk is a conservative vector field.

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Q: Test the vector field F to determine if it is conservative. F = xyi + yj + zk O Conservative Not…

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Q: F=x²y³z¹i-xyzj+(x+y+z)k

A: The vector field is F→=x2y3z4i^-xyzj^+(x+y+z)k^

Q: Verify Green 's theorem for the vector field  F (x, y) = (x^2y, -xy^2) in the triangle OAB defined…

A: The solution  is given by using the line integral of vector field and verify the greens theorem as…

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

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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: Determine if the vector field F is conservative. F=xyi + yj+zk Conservative Not conservative

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Q: Determine the vector field of F. F(x, y) yi-xj

A: see below the answer and explanation

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: Determine whether or the Vector Field is Conservative. If it is conservative, find function f such F…

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Q: Determine whether or not F is a conservative vector field. F(x, y) = (my + ²)i + (mx + 7);

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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: YouTrylt 2 Determine whether or not Fis a conservative vector field. (c) F = (x' +4xy)i+(4xy– y')j…

A: Since you have asked multiple questions in single request so we will be answering only first…

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: