1. Use the divergence theorem to evaluate the surface integral F · dS, S where the vector field is F = (3x + 1) i+ y (2z² + 1) j + (x³ + 2zy²) k and S is the cylindrical surface (without disks) specified by S = {(x, y, 2) | y² + 2² = 2, 0 < x < 1} , oriented outwards.

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: - Let the vector field, F = x³ i+y³ j+z³k. Verify Gauss's Theorem for surfaces S to be closed…

A: As per the question we are given a vector field F = x3i + y3j + z3k And a closed cylindrical surface…

Q: The outflow of the field F(x,y,z)=(e^y−xz^2,x−yx^2,y−zy^2), across the edge of the solid contained…

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Q: 2) If F(x, y, z) = x²i + y²j + 3zk is a vector field, use Divergence theorem to calculate the…

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Q: a) Let F. 2x³ +Y;+2 cosz+X+23³ k +2005 3 j+ -k be the vector field. Use the Divergence Theorem to…

A: The given problem is to find the flux od of vector field across the surface of solid bounded by the…

Q: Evaluate the surface integral ∬SF⋅dS∬SF⋅dS for the given vector field FF and the oriented surface…

A: Here in the given question, find the surface integral of the given vector field and the oriented…

Q: Use the divergence theorem to compute the flux integral P F . în ds where n is the outward pointing…

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Q: Attached is a multivariable calculus problem.

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Q: 3. Consider the vector field F = (y, -x, x² + y?). Find the flux of F through S if: 9– x2 – y? above…

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Q: 3. Verify Stokes' Theorem for the vector field F= 2zi +3xj+5yk, taking S to be a portion of the…

A: Given F=2zi+3xj+5yk

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

A: We will use of gauss divergence theorem

Q: Use the divergence theorem to find the outward flux F ndS of the vector field F = cos(8y + 8z) i + 4…

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Q: Use the divergence theorem to find the outward flux    ∫     ∫ SF · n dS   of the vector field F  =…

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Q: Use the extended divergence theorem to compute the total flux of the vector field F(x, y, z) = −3x^2…

A: The function given to us is F→=-3x2+2xz-y, 7y3-6y, 9x2+4z2-3x. We are asked to find the total flux…

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

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Q: (b) Let Q be the region bounded by the graphs of x² + y² = 4,z = 0 and z = 1. Let S denote the…

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Q: Use Divergence theorem to find the outward flux of F = 2xz i - 2xy j – z2 k across the boundary of…

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Q: Consider the conservative vector field 8. 4 F(x, y, 2) = (- 2xa, - 3y*cos (32)3ny³sin (3nz) – x²). -…

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Q: Use the Divergence Theorem to compute the net outward flux of the field F = (4x, y, - 2z) across the…

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Q: Let v = (5, y sin z, cos z) be the velocity field of a fluid. Compute the flux of v across the…

A: Given, v→=5,y sinz, cosz Flux=∫∫Sv⇀.ds we have divergence theorem, ∫∫Sv⇀.ds=∫∫∫Sdiv v⇀.dv div…

Q: 4. Evaluate the line integral Vf-r'dt, where the curve C is given by the vector function r(1)…

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Q: Let E be the solid region bounded by the upper half-sphere x2 + y² + z² = 4 and the plane z = 0. Use…

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Q: 5/ MATH

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Q: Q5. Consider the vector field F(x, y, z)=xi + 2yj + (x² + 4y²) k and the closed surface S made up of…

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Q: 4. (a) Show that the outward flux of the postion vector field F(x,y,z) = (x)7+ (y)] + (z) through a…

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Q: Suppose S is the surface where x² + y² + 3z² = 1. Let F be the vector field F = (xy, yz, z²).…

A: We need to evaluate integral.

Q: 2.4) Use the divergence theorem to find the outward flux of the vector field F(x, y, x) = x³i+ y³j+…

A: The divergence of the given vector field is  ∇·F=∂∂xx3+∂∂yy3+∂∂zz3=3x2+3y2+3z2 Let, S be the surface…

Q: - L ---- Let F be the vector field (r², 2r, z²). Let C be the boundary of the surface S given by 4x²…

A: Given- Let F→ be the vector field x2, 2x, z2. Let C be the boundary of the surface S given by 4x2 +…

Q: 8. Use the Divergence Theorem to compute the flux across the sphere x² + y? + z? = 1 within the…

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Q: Let S be the surface in R' that is the part of the ellipsoid 2x2 + y² + z? = 2 satisfying y > 1.…

A: In this question, we use the div.  theorem to evaluate the surface integral. given that…

Q: 4. Consider now the vector field G(xr, y, 2) = (x – 4xy? – 1, 2y – 4x*y, z). (i) Is the vector field…

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: 2. Find the surface integral SS. ř · dS for the vector field F = xyî + yzĵ + zxk where S is part of…

A: To find Surface integral ∬SF→·dS→: The Surface S is of the form z=g(x,y) and is oriented upwards.…

Q: 1) Using the Divergence (Gauss) theorem to evaluate 4-z = 3x' + 3 y z= 4 the Flux = [[F . īdS of the…

A: Given the vector field And, S is the surface bounded by the paraboloid

Q: 11. Compute the surface integral (flux) of the vector field F over parametrized surface S SSFdS…

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Q: Given a vector field Ę = 3 yz²i + x²y° į – 2xzk bounded by a solid region x² + y* =9 with height…

A: Solution:

Q: 7. Using the divergence theorem compute the outward flux of the vector field F(x, y, z) = x°i+…

A: step:-1 Divergence Theorem:-  ∬F.ndS=∭divF dV=∭divF  ρ2 sinϕ dρ dθ dϕ  Given that  F→=x3i…

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: Suppose that S is the boundary surface of the solid region B that is bounded by the paraboloid z =…

A: Given vector field F→x,y,z=2zey+xy2i→+xz2-2zexj→+x2z+2xe-yk→. To find the flux of the vector field…

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: 16. Use the Divergence Theorem to find the outward flux across the boundary of the region D where F…

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Q: 5. Use the Divergence theorem to calculate the z = 4 - x- flux of F across S where -x² + y? = 9 F =…

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Q: Consider the vector field F = 2xy i + (x² + 2y) j. Verify the Green's Theorem in the closed region…

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Q: Use the Divergence Theorem to compute the net outward flux of the field F = (4x, 3y, - 3z) across…

A: Let's find.

Q: Problem #4: Use the divergence theorem to find the outward flux F.n dS of the vector field F =…

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Q: Use Divergence theorem to find the ouward flux of F = 2xz i - 3xy j – z2 k across the boundary of…

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Q: 4- Evaluate the surface integral of the vector field F = [ax, by, cz] over the sphere r² + y² + 2² =…

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Q: 12. Find the flux of the field F(x, y, z) = 4xi + 4yj + 2k outward (away from z-axis) through the…

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Q: Verify that the Divergence Theorem is true for the vector field F on the region E. Give the flux.…

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Q: 2. Determine the outward flux of the vector field F = (xy)î + (y²)ĵ over the region R depicted %3D…

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Q: Use Divergence theorem to find the ouward flux of F = 2xz i - 3xy j – z2 k across the boundary of…

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