1. Consider the vector field F%3= (7.ryz, 5a z, 2²yz) . Identify the potential function for the vector field F. (a) The potential function of F is f(z, y, z) = x²y²?. (b) The potential function of F is f(2, y, z) = 2r²y"z°. (c) The potential function of F is f(z, y, z) = x²yz + 5ry?z²?. (d) No potential function exists.

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

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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: Compute the flux of the vector field F = xi + 5yj through the cylindrical surface as shown in the…

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Q: Find the work done by the force field F(x, y) =(2y'+2xy, 6xy+x +1) in moving a particle over the…

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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: 1- Prove that the force field F known as : Prove that the force field defined by F = (y*z' -6xz? )i…

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Q: 3. A flow is represented by the velocity field (y² + z²)i + xzj + 2x²y²k, check if: a) A possible…

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Q: Consider the gradient vector field V = Vg of the function g(x, y) = x² + y² + e*²-y´ cos(2xy).…

A: We will use the Divergence Theorem in 2D to solve the given flux integral.

Q: Is the two-dimensional vector field V = -2y 2x a possible velocity field x²+y² 'x² +y² for an…

A: For a two dimensional incompressible for a vector field V=ui+vj. ∂u∂x+∂v∂y=0

Q: (b) Given the force field F(x, y, z) = (z³ – 2xy)i – x²j + 3xz² k (i) Show that F is a conservative…

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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: A conservative vector field is given as F = 6xe'i + (3x²e' –10yz')j-15y 2 k. Find the potential…

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Q: Find a potential function for the vector field F by inspection. F = (4x, 4y) P(x, y) =

A: Given:

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

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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: Consider again the vector field F = (x +y)i + (x² + y²)j_what is the downward flux across the line…

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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: b) Find the flux of the vector field F(x,y,z) = 3xi – yj + 2zk; across o where o is the portion of…

A: The given question belonging to the vector calculus  in which we have to find the value flux of the…

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

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Q: Let W be the work done by the forced field F(x,y,z) =yi+zj in order to move a particle along the…

A: The formula for work done is W=∫F.dr.

Q: 4. Consider the vector field F = (y?2³ + y)i+ (2æyz³ + x – 2)j+ (3xy²z² – y + 3)E. (a) Verify the…

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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: Evaluate the outward flux of the vector field F = (2x² – 5e" + sin(z²), –4ry + In(x*), 5z – 3x²y®)…

A: We can see that the region given here is closed, so…

Q: Determine whether the vector field…

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Q: Find the work W done by the vector field ⟨2x+yx,x2+4⟩ on a particle moving along the boundary of the…

A: To find the workdone by the vector field.

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: Q4(b) Given the force field F(x, y, z) = (z^3 – 2xy)i – x^2j + 3xz^2 k (i) Show that F is a…

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

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

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Q: d) Determine whether the vector field is conservative. If it is, find a potential function for the…

A: Let a vector field is given by,                                  The vector field is conservative…

Q: A conservative vector field is given as F = 6xe'i + (3x²e" –10yz')j-15y²z°k. Find the potential…

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

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Q: Find the gradient vector fields for the following potential functions (a) f(x,y) = x²y – xy?

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Q: Show that the vector field F = i + sin(z) j + y cos(z) k is conservative and then find the…

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Q: ind the conservative vector field for the potential function by finding its gradient. f(x, y, z) =…

A: The gradient of a function is the rate of change of a function and for a function, f(x,y,z), the…

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: Find the conservative vector field for the potential function by finding its gradient. g(r, y, :) +

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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: 16. Compute the flux V· nds of the vector field V(r, y) = (x² – 2ry, y² – 2xy) across the curve C :…

A: We have to compute the flux ∫ℂV·nds of the vector field Vx,y=x2-2xy,y2-2xy across the curve…

Q: 21. Compute the flow of the vector field F = r through the surface of the ellipsoid y² + 2 = 1 in…

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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: Determine whether the vector field…

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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.