4. (a) (b) Let F(x, y, z) = (0, y, -2) be a vector field in 3D. Draw the vector field in 3D. Consider the closed surface S formed by the two parts: • the paraboloid y = x² + 2²,0 ≤ y ≤ 1, and the disk x² + 2² <1, y = 1 Calculate (using the positive outward orientation of the surface) JI F.d.S.

Q: Find the flux of the vector field F = (y, z, x) across the - part of the plane z = 4+3x+2y above the…

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Q: Let F(x, y, z) = yi + xzj + 2xyk and G(x, y, z) = zi + xj + yk. Refine expression (FV)G (F₁0₁ + F202…

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Q: Consider the vector field: F = (ex, In(xy), e xyz) (i) Find div(F) (ii) Find curl(F) (iii) Find…

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Q: Your friends correctly calculate the gradient vector for f(x, y)=x²+y at the point (2,-3) as…

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Q: Consider the force field F(x, y) = (x, y). a) Sketch the vectors F(1,1), F(-2,1) and F(2,0) directly…

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Q: Find the curl for the vector field at the given point. F(x, y, z)= 6xyzi+6yj+6zk (6, 7, 6) O…

A: To find the curl for the vector field  at the given point.Fx, y, z=6xyzi+6yj+6zkPoint=6, 7, 6

Q: Compute the flux of the vector field F = yi + 3j – xzk through the surface S, which is the surface y…

A: We compute the flux of the vector field through the surface.

Q: .) Compute the flux of the vector field F = 4x + 4yj through the surface S, which is the part of the…

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Q: State the theorem for Surface Independence for Curl Vector Fields.

A: Stokes' Theorem Stokes’ theorem says we can calculate the flux of curl F across surface S by knowing…

Q: Determine whether →F(x,y,z)=3x^2yˆi−yzˆj+zx^2ˆk is a conservative vector field.     Component of…

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Q: point -2 has parameterization F(t) = (4(1-t), 3t) for 0 s 2

A: This is a problem of vector calculus.

Q: Find the flux of the vector field V(x, y, z) = 9xy²i + 9x²yj + z³k out of the unit sphere.

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Q: The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. (In…

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Q: a) Calculate the flux of the vector field F(x, y, z) = 275k through a sphere of radlus 4 centered at…

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Q: 4. Consider the vector function r(z, y) = (x, y, x² + 2y*). (a) Re-write this vector function as…

A: For the given vector function, PART A The surface function is given by, f(x,y)=xi^+yj^+(x2+2y2)k^…

Q: (b) In the (x, y)-plane, sketch the vector fields v = (y, 2y, 0) and w = (x−y, x+y, 0), clearly…

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Q: d) An open surface F is given by F = {(x,y, z) |z = 4 – x² – y² , z > 0} . Compute the flux of V…

A: V(x,y,z) = &lt;2x+3y,2y+3x,-4z&gt;F={ (x,y,z) : z=4-x2-y2 , z≥0}Consider, g(x,y,z) = z+x2+y2-4…

Q: Let F and G be vector fields with differentiable components. Express curl (F × G) in term of div and…

A: Let F and G be vector fields with differentiable components.Let We will compute .The cross product…

Q: Sketch (a) the level surfaces of the scalar point function f(r) = ze; (b) the field lines of the…

A: Given Information:a) Scalar point function: f(r)=ze−xyb) Vector point function: F→(r)=(−y,x,1)To…

Q: 2. Find a vector that is perpendicular to the surface x² + y² + z² = xyz + 8 at the point (1,2,3).

A: Given that x²+y²+z²=xyz+8 At (1,2,3)

Q: Calculate the flux of the vector field F(x, y, z) = (4x+8) through a disk of radius 6 centered at…

A: Given,                    To find the flux of the given vector field through a disk of radius 6…

Q: Vector Field: F(x,y) = (xv, 3x, – 2y) and Surface: z = 4 - x - 3 y? Use the normal vector formula to…

A: Given surface z=4-x-3y2

Q: Find a formula F = (F₁(x, y), F₂(x, y)) for the vector field in the plane that has the properties…

A: Answer :

Q: For the vector field (x2,0₁ - 2x² -z²), evaluate its line integral along the closed curve, made by…

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Q: Let C₁ be the line segment from the point (-4, 8) to the point (2, -4), C₂ be the arc on the…

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Q: Find the parametrization of two different curves from the point (2,4) to (3,9). Compute the work…

A: Given point (2,4)  and (3,9)now equationof line passing through two point is given by…

Q: Calculate vector field F = (2-z²+2-x²)i + (2 · y³ + 4·x)j + (2x²z³+2)k a flux from a box with the…

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Q: Compute the flux of the vector field F 3yi + 4j – 3xzk through the surface S, which is the surface y…

A: We apply formula for finding this

Q: Compute the flux of the vector field F = xi + yj through the surface S, which is the part of the…

A: Given: F→=xi→+yj→ z=36-x2+y2 ds=1+∂z∂x2+∂z∂y2dxdy       =1+4x2+4y2dxdy   and…

Q: Sketch representative vectors for the vector field by sketching it's scalar field. F(x, y) =…

A: Given: Vector field ,Fx,y=-12yi^+15xj^We need to sketch the scalar field using representative…

Q: Compute the flux of the vector field F = yi + 5j - xzk through the surface S, which is the surfac y…

A: A vector field F→=yi^+5j^-xz through a surface S, which is the y=x2+z2  , x2+z2≤1. To Find: The flux…