5. Use Stokes' Theorem (and only Stokes' Theorem) to evaluate ffs curlF - dS for the vector field F(1, y, 2) = (2²,2x, –-y³) and the surface S that is the upper half of the unit sphere r² + y³ + z² = 1. (so just to be clear, if you want to evaluate this and use Stokes' Theorem then you must calculate F - dr). You must show all your work.

Q: Consider once again the notion of the rotation of a vector field. If a vector field F(x, y, z) has…

A: Question 39 - F(x, y, z) = -sinx, 3y3, 4z+12Curl of F = ijk∂∂x∂∂y∂∂zFxFyFzCurl of F =…

Q: Show that the vector field F(x, y, z) = xz i + xyz j - y 2 k can’t be written as the curl of another…

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Q: Compute the circulation of the vector field F=(4cos(y))i+(3x^2sin(y))j around C. Where C is the…

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Q: 4. Find the positively oriented flux over the surface created by the paraboloid z = 16 – x² – y2…

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Q: 5) By using Stokes' Theorem, find the work performed by the vector field Ex² + 4xy³i+ y²xk,on a…

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Q: 4. Consider the vector field given in Cartesian coordinates by A(r) = (x² + y³, 3xy², z²). (a)…

A: Given : Ar=x2+y2,3xy2,z2

Q: Find all points (a, b, c) on the sphere x2 + y2 + z2 = R2 where the vector field F = yz2i + xz2j +…

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Q: Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F and…

A: To use Green's theorem to find the counterclockwise circulation and outward flux for the field F and…

Q: From the given vector field F⃗ (x, y, z) = , compute the curl of F⃗ and find the values of a and b…

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Q: b) Show the force field M UTM UTM F(r, y, z) = (r²y + In(yz)) i+ UTM 3 + y cos(2z) ) j+ tial…

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Q: Show that Gauss's Theorem holds for the vector field F(x, y)=(xy2, -yx2) in a side square 3.

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Q: Consider the vector field F(x, y) = yî- xî+ zR and path joining (1, 0, 1) and (-1, 0, 1) as C;:…

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Q: Supposing vector field G is related to the flow of fluid P is one point in the domain of G. Give the…

A: To Explain: The geometrical interpretation of G(P) where G is a vector filed and P is a point in…

Q: ind the flux of the vector field F = yzi + 7yj + yk over the sphere x2 + y2 + z2 = 4.

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Q: Consider the vector field F (x, y, z) = (y - zsinx, x, 2z + cosx). The work that performs the F…

A: Given that, F(x,y,z)=y-zsinx, x, 2z+cosxA body is displaced from point A(3π,-1,1) to point Bπ,2,0

Q: I. Use the Divergence Theorem: (² F. ds = Ifv. Fav SSS E = to calculate the flux of the field F…

A: Divergence theorem uses  Double integration converts into triple integration.

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

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Q: 2. Use Gauss' Theorem to compute the outward flux of the vector field F(r) = (x, y, z²) through the…

A: Outward flux by using gauss divergence theorem

Q: 5. Find the outward flux of the vector field F(x, y,z) = x²i + y°j+z²k through the first octant…

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Q: 8. Show that the vector field, -y a² + y?' x² + yj²' ) is not conservative in R³, despite the fact…

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Q: Determine the value of k for which (u, v, w) is an orthogonal coordinate system if x = −(u2 + kv2),…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F = (4x…

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Q: 5) Find the work done by the force field F(r, y, z) = ri-zj+y k in moving an object along the…

A: Work = Force X Displacement

Q: The circuitation of the vector field F(x, y)- (3 + xy´, 4x + xy) along the boundary of the plane…

A: The solution are next step

Q: Compute the curl of the vector field F = (xy+ z² ,x², xz – 2). curl(F(x, y, z)) =

A: Please refer the attached image for complete solution.

Q: b) Show the force field F(x, y, z) = (x²y + In(yz)) i+ 3 y cos(2:) ) j+( - y° sin( sin(2z) ) k %3D…

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Q: What is the flux of the vector field (x2,y2,z1 ) through a cuboid with x ranging from 0 to 8.3, y…

A: use divergence theorem

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: Given the vector field F(x,y,z) = [x, y, -2z] satisfies div(F) = 0. Find a vector field G(x,y,z)…

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Q: Compute the flux ∫F · n ds of F across the curve C for the given vector field and curve using the…

A: Solution; Consider the given function: F(x, y) = xy, x-y = P, Q P = x y Q = x -y and the boundaries…

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: Find the vector field flow F(x,y,z)=(x2,y2,z2)  through S, where S is a flat surface x+y=2, bounded…

A: The Divergence theorem, Let S be a piecewise, smooth closed surface that encloses solid E in space.…

Q: 5. Find the curl of the following vector field in both Cartesian and cylindrical coordinates. F=…

A: Curl of a vector field represents the circular nature of vector field. Curl of vector field F=a,b,c…

Q: Let C be any smooth simple closed curve with positive orientation that lies entirely in a plane with…

A: Step:-1 Stock's Theorem:- ∫CF.dr=∬Scurl(F). ds=∬Scurl(F). n dA Part (a):- Given that F→=9y, 3x, 6z…

Q: Show that the vector field F(x, y, z) = (2y cos(-3x), –3x sin(2y), 0) is not a gradient vector field…

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Q: b) Find the curl of the vector field, CurlF, if F = Ax²z²i– By:'j+Cz*k, where the coefficients A,…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Find the level surfaces of the scalar field U = ear, where the vectors are: a = (2, 2, 1) and r = =…

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Q: Verify Stoke's Theorem for the vector field F = (2y + 2) i + (x –2)ĵ+(y – x) . over the portion of…

A: The formula for stokes theorem is, ∫Pdx+Qdy+Rdz=∬∇×F.ndS Given that, F=(2y+z)i+(x-z)j+(y-x)k Then,…

Q: 4. Show, in detail, that any vector field of the form (*,J, 2) = J (*) +8(y)/+h(z)k is irrotational,…

A: To show that the vector field is irrotational.

Q: 3. Find the work of the vector field F(2, y, ) = yi along the curve which is obtained as the…

A: To find the work of the vector field Fx,y,z=yi along the curve obtained by the intersection of…

Q: (b) Verify Green's Theorem for a vector field F(x,y) = xy²i – yx²j on a disc with center (0,0) and…

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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: Let F be the vector field (x², 2r, 2²). Let C be the boundary of Lv **---- the surface S given by…

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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: 2) Given that the total flux across the surface, S of the tetrahedron bounded by the planes x = 0, y…

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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: 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: 4- Evaluate the surface integral of the vector field F = = [ax, by, cz] over the sphere a2 + y² + z²…

A: From the given problem:   F→=axi+byj+czk   Sphere:  x2+y2+z2=36.   From the divergence theorem:…

Q: 2: Compute the work done by the force field F = (y, -x) to move an object from point A = (1,0) to B…

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Q: a) What is the flux of the constant vector field F = (a, b, c) through any closed surface? b) Is the…

A: Given That : A constant  vector field F=a,b,c and F=x3,y3,z3 To Find : i) The Flux through of…