Let S be the cylinder x +y = a°, 0szsh, together with its top, x² +y° sa, z=h. Let F= - 4yi + 4xj + 4x k. Use Stokes' Theorem to find the flux of V×F through S in the direction away from the interior of the cylinder. utuord through S ie

Q: Let F(x, y, z) = z tan-(y²)i + z³ In(x² + 2)j + zk. Find the flux of F across S, the part of the…

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Q: Consider the solid E determined by the inequalities x² + y² + z≤ 4, and let S denote the surface…

A: Here we will use Stokes theorem  to calculate  flux of curl (F) on surface S bounded by curve C…

Q: For F(x,y,z) = (6y-4xz/(z2+1)2)i + (-2yz-6x/(z2+1)2)j + (-3/z2+1)k,  Calculate the net flux through…

A: consider the equation

Q: Find the flux of F(x, y, z) = 4а, 32 + 2?, across the positively oriented surface 2 S given by R(u,…

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Q: Use Stokes’ Theorem to evaluate the flux integral ∫∫(∇×F)·ndS, where F =〈2y, −z, x+y −z〉, S is the…

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Q: Use the Divergence Theorem to find the flux of F(x, y, z) = (x – z)i+ (y – x)j+ (z – y)k - across…

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Q: Find the flux of the field F(x, y, z) = z2 i + x j - 3 z k outward through the surface cut from the…

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Q: Let v = (x*z, 2 – 4xyz, xz) be the velocity field of a fluid. Compute the flux of v across the…

A: Given, v→=x2z,2-4xyz,xz225x2+16z2=y-202 Then, calculating div v→ div…

Q: (b) Let V be the volume between the spherical shells of radius 1 and 2 centered at the origin.…

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Q: Find the flux of the vector field F (x, y, z) = (e-y, -y, x sin z) across the positively oriented…

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Q: Find the flux of F = 2xi + yj+ zk upward through the surface r = u?vi + uv²j+ v³k, (0 < u< 1,0 < v <…

A: We will find out the required value of flux.

Q: Use Stokes' Theorem to find the circulation of F = 6yi + 5zj + 5xk around a circle C of radius 7…

A: According to Stoke's Theorem: ∮CF→⋅dr→=∬S(∇→×F→)⋅n^dS here n^ is the vector normal to  the surface S…

Q: Let S be the cylinder x2 + y? = a?, 0szsh, together with its top, x2 + y? sa?, z=h. Let F =…

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Q: Find the net flux of the vector field F(x, y, z) = (2x, - 2y, z) over the closed surface enclosed by…

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Q: Use Stokes' theorem to evaluate frotF. ds. F(x, y, z) = xyi + e²j + xy²k; S consists of the four…

A: To find- Use Stokes' theorem to evaluate ∫SrotF·dS, Fx, y, z = xyi + e2j + xy2k; S consists of the…

Q: Let v = (4 xe cos y, y cos z, e cos y) be the velocity field of a fluid. Compute the flux of v - I…

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Q: Let S be the cylinder x2 + y2 = a2, 0&lt;=z &lt;=h, together with its top, x2 + y2 &lt;=a2, z = h.…

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Q: Find the flux of F through the surface S. F(x, y, z) = xi + yj + xzk S is the cone r(u, v) = ⟨u…

A: flux related problem.....

Q: Use the divergence theorem to compute the value of flux integral F. dS, where F(x, y, z) = 5yj - 5zk…

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Q: Consider the surfaces Si = {(2,y, 2) | 2 +y = 4 -z, 0<2< 4} and Sz = {(z,y,2) | 2²+ y² s 4, z = 0}…

A: As the upper surface is x2+y2=4-z, the limits of integration using cylindrical coordinates are…

Q: Let v (2, y sin z, cos z) be the velocity field of a fluid. Compute the flux of v across the surface…

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Q: Let F(x, y, z) = (2x, y, –2z) be the velocity field of a fluid. (b) Let o be the boundary of the…

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Q: Find the flux of F = 2xi + yj + zk upward through the surface r = u?vi + uv² j + v³k, ( 0 < u< 1,0 <…

A: To find the flux of F→ =2xi^ +yj ^ +zk^upward through the surfacer→ = u2vi^  +uv2j^ +v3k^0≤v≤1 ,  0≤…

Q: Attached picture is the question

A: Consider the provided question,

Q: Find the flux of the field F(x, y, z) = (x, y, -2z) through the surface z = ! =1– 2² – y² (0 < ¤< 1,…

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Q: A fluid motion is given by V=(y+z)i+(z+x)j+(x+y)k. Is this motion irrotational? If the velocity…

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Q: Use the Divergence Theorem to find the flux of F(x, y, z) = x°i + y°j+ z°k across the surface of the…

A: Gauss divergence theorem is used

Q: Let F(x, y, z) = z tan-1(y2)i + z³ In(x² + 6)j + zk. Find the flux of F across S, the part of the…

A: Using divergence theorem:  ∬SF.dS=∭VdivF.dV…

Q: Let v = (a*z, 2 – 2xyz – 3y + x*y, 3z – x*z) be the velocity field of a fluid. Compute the flux of v…

A: Given the vector field         v=x2z,2-2xyz-3y+x2y,3z-x2z

Q: Let S be the cylinder x² + y² = a², 0≤zsh, together with its top, x² + y² ≤a², z = h. Let F= -5yi…

A: By Stokes theorem  flux of ∇×F through a surface S can be given by  ∫∫S∇×FdS=∫CF.dr where C is…

Q: Verify Stokes' Theorem for F(x, y, z) = xi + yj + y²zk where S is the first octant portion of the…

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Q: Use Stokes' theorem to evaluate E (wy – y) i – In z j+ (x+x²) k) · dr, where C is the intersection…

A: The required curve C(red) is shown as follows: The intersection gives a disc D bounded by curve C,…

Q: Find the flux of F= through of the total surface of the closed cone y=(x2+z2)0.5 with 0 < y < 2…

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Q: Find the flux of F across S, where F(x,y,z) = xˆi + yˆj + zˆk and S is the part of the cone z = √x2+…

A: Surface integral problem

Q: Calculate the flux integral f(2i + 2j + 2k) · dA where S is a disk of radius 5 in the plane æ + y +…

A: Given F=2i+2j+2k,radius=5 and x+y+z=1 plane

Q: Use the Divergence Theorem to calculate the outward flux of F = (r3 + y, y + z3, z3 + 23) across S:…

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Q: .† Use Stokes' theorem to show that Sc(y+2)dx+ (z +x)dy+ (x+y)dz = 0, where C is the curve of…

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Q: Let F(x, y, z) = z tan-(y2)i + z3 In(x2 + 1)j + zk. Find the flux of F across S, the part of the…

A: Divergence Theorem states that  Where S is a closed surface, and E is the region inside that…

Q: The moment of inertia of a surface about the z axis is given by the formula I Sls x² + y² dS. Find…

A: (a) We are asked to find the moment of inertia  of the surface z=xy lying inside the cylinder…

Q: Let S be the piece of the surface z = 9– x2 that lies inside the cylinder x2 +y? = 9, and let F =…

A: According to question, Given Surface, z = 9 -x2 ,This surface lies Inside the cylinder x2 + y2 = 9,F…

Q: oriented

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Q: Let F(x, y, z) = z tan-(y2)i + z³ In(x2 + 9)j + zk. Find the flux of F across S, the part of the…

A: Get points of intersection of the paraboloid and z=3:  Thus, point of intersection is a circle in…

Q: Let S be the cylinder x+y =a,0szsh, together with its top, x +y sa, z=h Let F= -3yi + 3xj + 3xk Use…

A: Given: S is the cylinder x2+y2=a2, 0≤z≤h with its top x2+y2≤a2, z=h And, F=-yi+xj+x2k The objective…

Q: (x*z, 3 – 4xyz, xz") be the velocity field of a fluid. Compute the flux of v across the surface 16x?…

A: Use the Divergence Theorem to calculate the flux. Partial differentiate the flux and substitute the…

Q: (4) Use Stokes' Theorem to calculate the work-done by the force field F(r, y, z) = (x² + z²)î + (y²…

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Q: Consider the surface S given by y=8−x2−z2, y≥0 oriented in the positive direction of the y axis. be…

A: We will find out the required value.

Q: Let S be the surface of the solid bounded by the "fat cylinder" a* + y* = 1 and the planes z = -2…

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Q: Verify that both integrals in Stokes' Theorem are equal when F= (y – 2, z – 1, r – y), S is the cap…

A: Solve

Q: The region W lies between the spheres a? + y? + z2 = 9 and æ? + y² + z² = 16 and within the cone z =…

A: Here we can use divergence theorem to find the flux of the vector valued function. Divergence…