Determine if F = (12r – 2x cos y, 2y + x² sin y) is a conservative vector field. If it is, find a potential for F.

Q: Q6: Find the outward flux of the vector field F=(x²y+e'sin(y))i+ (x²+e'cos (y)) j through the right…

A: Given vector field is F=  x2y + ex siny  i + x2 + ex cosy j Curve r2=cos2θ

Q: The vector field F = z tan-(y²) i + z³ In(x² + 1) j + (cos z + ay) k has a vector potential with…

A: The solution are next step is

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…

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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: Consider the vector field F(x, y,z)= 2.xy i+(x² +z² ) j+2yzk. c) Is F a conservative vector field?…

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Q: For which value of a the field i = (y –x)i +(2x – y)j + azk is solenoidal. Find the gradient of the…

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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: Prove that F(x, y, z) = (y² cos x + 2*)i+ (2y sin x – 4)j+ (3xz2 + 2)k - is a conservative vector…

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Q: Show that the vector field F = (x² cos y, e+y) is not a gradient vector field. %3D

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Q: Find a potential function for the vector field F(x, y) = (14xy + 7y- 8, 7z + 7z). f(x, y) =

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Q: If F (x, y) = x2 i + y2 j is a conservative vector field, find its potential function. With the…

A: The solution are next step is

Q: 3 , find a potential function for the vector field F by inspection or show that one does not exist.…

A:

Q: Show that the field F = (cos x)i + (- y sin y)j + (z3)k is conservative and Önd a potential function…

A: F = ( Cosx )i + ( -y Siny )j + ( z3 )k  F = gradient ( - Sinx - y Siny - Cosy + z4/4 ). Thus, we can…

Q: Is the vector field F = y? – 2x², 2xy + 3| conserva Yes No yz Is the vector field F y? – x2 con Vy?…

A: According to the guidelines issued by the company we are supposed to answer only first question.…

Q: Compute the curl of the vector field F = sin(x) i+ cos(y) j+ 8z3 k

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

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Q: Find a potential function for the conservative vector field. F(x, y) = xex*4Y(2yi + xj)

A: If curl f is zero then conservative field

Q: Show that the vector field F(x, y, z) = (le* cos y + yz)i+(xz - e* sin y)j+(xy + 4z) k is a…

A: Given: F(x,y,z)= (excosy+yz)i+(xz-exsiny)j+(xy+4z)k To show that F is conservative and if it is,…

Q: Is the vector field F(x, y) = (6ry cos(3x²y + e*) +e* cos(3x?y + e*), 3x² cos(3x²y + e*))…

A:

Q: Determine whether the vector field is conservative. F(x, y) = xex*sy(2yi + xj) conservative not…

A: If the vector field F is conservative (i.e, curl F =0) so that there exists a potential function…

Q: Show that the vector field given below is conservative and find its potential function. F(x, y) =…

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Q: Determine whether the vector field is conservative. F(x, у) %3D yi + xј conservative O not…

A: The given function is  fx,y=yi+xj

Q: Determine whether F(x, y) = 3xi + 7yj is a conservative vector field. find a potential function for…

A:

Q: a) Determine whether or not the vector field F 8yz, – 6 cos(y) – 8xz, – 8xy + 10z > - is…

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: Compute the flux

A: Compute the flux

Q: Find g(x, z) so that F(x, y,z) = (8x cos y sin? z + 2e2xz)î – (4x² sin y sin? z)j+ (g(x, z) cos y +…

A: Given that    Fx,y,z=8x cosy sin2z+2e2xz i^-4x2 siny sin2z j^+gx,zcosy+2e2xz k^ Now we have to find…

Q: The vector field A is given by A = 2r cos(r* + 3y")i + 6y cos(2² + 3y*)j. a) Show that A is…

A:

Q: determine if the given vector field f is conservative and hence find the potential function for the…

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

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

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Q: b) Verify whether the vector field F is conservative, if so, then find the potential function F = ,…

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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: Determine whether F(x, y) = e cos 9y i – e" sin 9yj is a conservative vector field. If so, find a…

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Q: Find the potential function for the following vector field F(x, y, z) = (2x cos y – 2z3) î + (29+…

A: F = <2xcosy-2z3, 29+2ye2-x2siny, y2e2-6xz> To check F¯ is conservative  not we have t check…

Q: 3x cmine whether F(x, y) = e³ª cos 3y i – e³ª sin 3yj is ervative vector field. If so, find a…

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

A:

Q: Determine whether F(x, y) = 6x²y i+ 4xy²j is a conservative vector field. If so, find a potential…

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Q: Show that the vector field ? = ? + sin(z)? + ycos(z)? is conservative and then find the…

A: We have to show that the vector field F=i+sinzj+ycoszk is conservative and then find the…

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: Determine whether F(x, y) = (In(y) + 2xy³) i+ (3x?y? + x/y) j is a conservative vector %3D field. If…

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Q: Evaluate ∫C F . T ds for the vector field F = x2 i - y j along the curve x = y2 from (4, 2) to (1,…

A: Given information: The value of F=x2i+yj, x=y2 and from (4, 2) to (1, -1). Calculation: Consider…

Q: Find the gradient field of the function f(x.y.z) = In 4x + 4y2 +z?. ..... The gradient field is Vf=…

A: Gradient of F is given by

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: Determine whether the vector field is conservative. 7x², F(x, y) = Yi + conservative O not…

A: If curl f is zero then vector field is conservative otherwise not

Q: Find the conservative vector field for the given potential function. f(x, y) = 9x2 + 2xy + 10y2 F(x,…

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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:

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: