Consider the conservative vector field 8. 4 F(x, y, 2) = (- 2xa, - 3y*cos (32)3ny³sin (3nz) – x²). - 3y²cos (3nz),3ny³sin (3rz) – x2). 2x + y 2xz, \2x +y (a) Find the potential function p of F that satisfies p(2,2,1) = 2. (b) Use the Fundamental Theorem of Line Integrals to find the work done by F along the curve with parametrization R(t) = (2 – t, -cos (tt), 3'), t e [0,1].

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

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Q: 5. Find a potential function corresponding to the vector field F(r,y, 2) = (2r, 3y, 4:).

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

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Q: Consider the conservative vector field 8 4 F(x, y, z) = - 2xz, - 3y²cos(3nz), 3ny sin(3nz) – x² ) 2х…

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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: Let the vector field F (x, y) = (3x2y-2+ 2xy-1)i + (-2x3y-3 − x2y-2)j, be a conservative field.…

A: We will use Integration and concept of total differentiation for finding scalar potential of vector…

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

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Q: ) Compute the flux of the vector field F = 5xi + 2yj through the cylindrical surface shown below, if…

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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: Find the flux of the field F = y2i + xzj - k outward (normal away from the z-axis) through the cone…

A: Definition:

Q: Find the potential function p, for the conservative vector field F(x, y, z) = (2xe² – y²sin x)î +…

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Q: 4. (a) Show that the divergence theorem holds for the vector field F = 2xî + yz} – 3z²k where o is…

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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: 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: Find the divergence of the vector field v = (x²y² - z3)i + 2xyz j + e*yzk. a) 2xy2 + 2xz + xy e*yz…

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Q: 2. Calculate the directional derivative of the scalar field U(x, y,z) at the point M,, along the…

A: Given that : U=xz3y-x3y=xz3y-x32y12 The gradient will be calculated as :…

Q: A) Consider the vector-field F(x, y) = 2xy³i+ 3y²x²j. 1. Show that F is conservative. 2. Find a…

A: The given vector field is  F⇀(x,y)=2xy3i⇀+3y2x2j→ We have to show that F→ is conservative

Q: Show that the vector field F = 2xyi + (x² – 2 y)j is conservative and evaluate the line integral…

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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: 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: Determine whether the vector field…

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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: 2. Given F(x, y) = (4x³y² - 2xy³)i + (2x¹y − 3x²y² + 4y³)j is a - conservative vector field. a. Find…

A: To find the potential function for the given conservative vector field :-

Q: Evaluate the outward flux of the vector field F = (2x? – 5e" + sin(2²), –4ry + ln(x*), 5z – 3x²y°)…

A: Given

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

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Q: Find the gradient vector field (F(r, Y, 2)) of f(x, y, z) = In(4x + 6y + 22) . F(7, y, 2) = (

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Q: 4) Consider the conservative vector field given by: F (x, y) = (y² – ye“, 2xy – e*) A potential…

A: Given vector field is F→x,y=y2−yex,2xy−ex. We have to find the potential function that generates the…

Q: Show that the following vector field is conservative and find the potential F(r, y, z) = (2ry, x² +…

A: The answer is given as follows.

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

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Q: The vector field F = (y², 2xy + e³2, 3ye³z + x) is conservative

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Q: Find the divergence of the vector field F. F(x, y, z) = In( 7x²+ 4y² )i + 28xyj + In( 4y2 + 5z2 )k

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Q: Consider the surface integral of the vector field F= [xy², 3x², x²z]over a surface S, where S…

A: From Gauss divergence theorem, we have ∬SF→⋅dS=∭VdivF→dV   The vector field is F→=xy2,3x2,x2z.…

Q: If F = (4xy – z³) î + 2x² ĵ – 3xz² k is a conservative force field, then scalar potential of F is…

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Q: F(x, y) = 6xyi + 3y²j is a conservative vector field.

A: Given :F(x,y)=6xyi+3y2j

Q: (b) Find the scalar potential of the conservative field: F = 2ry°z?i + 3x²z²y²j+ 2x?y³zk. %3D

A: Let's find.

Q: Show that the vector field F = 2.xyi +(x² – 2 y)j is conservative and evaluate the line integral…

A: Given the vector field

Q: Consider the conservative vector field 4 8 F(x, y, z) = – 3y²cos (3nz),37y sin (3tz) – x2). 2xz, %3D…

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Q: 3. Use the divergence of the vector field to compute F(x, y) = (tan-1 (2x)) ₁ + (xy-2 +²8²) J j 4 fo…

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

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Q: Find the counterclockwise circulation and outward flux of the field F = 7xyi + -5y²j C enclosed by…

A: Introduction: Let C be a piecewise smooth, simple closed curve enclosing a region R in the plane.…

Q: Compute the flux of the vector field F = 2xi + yj through the cylindrical 5 and b = 4 and it is…

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Q: Show that the vector field F(x, y)= (12x² +3y² +5y)i+(6xy– 3y² +5.x)j is a conservative vector field…

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Q: Show that the vector field F(x,y,z)=yz^2i+xz^2j+2xyzk is conservative, find a potential function f,…

A: The given vector function is F(x,y,z)=yz2i+xz2j+2xyzk The given vector field F is conservative if…

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: Compute the flux of the vector field F = 2xi + 3yj through the cylindrical surface shown below, if a…

A: We have to compute the flux of the  given vector field.