In Exercises 29 and 30, find the surface integral of the field F over the portion of the given surface in the specified direction. 29. F(x, y, z) = -i + 2j + 3k 0 < x < 2, S: rectangular surface z = 0, direction k 0 < y< 3, 30. F(x, y, z) = yx²i – 2j + xzk S: rectangular surface y = 0, -1 < x < 2, 2

Q: Compute the upward flux of the vector field Ĥ(x, y, z) = (x, 0, 2z) across the portion of the…

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Q: Please solve this question in 30 Minutes Question 4 Let Y define the surface Y: z=(x^2-y^2)/2…

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Q: Confirm that the force field F(, y) = 2xy°i+ 3x²y² j is conservative in some open connected region…

A: Solution:- The given function is  F(x,y)=2xy3i+3x2y2jpoint P(-9,8) and Q(6,7)we have to find the…

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Q: Evaluate the surface integral ∬SF⋅dS for the given vector field F and the oriented surface S. In…

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Q: Use the Divergence Theorem to compute the net outward flux of the field F=(2x, 3y, -z)  across the…

A: To calculate the net outward flux of the given vector field across the given surface

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

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Q: Confirm that the force field F(x, y) = 2xy³i + 3x²y²j is conservative in some open connected region…

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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: 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: Consider the conservative vector field 8 4 F(x, y, z) = - 2xz, - 3y²cos(3nz), 3ny sin(3nz) – x² ) 2х…

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Q: Compute the flux SS F.nds through the piece of the cylinder of radius 2, centered on the z-axis,…

A: The given surface is the cylinder of radius 2 centered on z-axis, where 0≤x, 0≤y and 0≤z≤4. The…

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: Compute the flux of the integral of the vector field F(z,y, 2) = (2,y, 2) through the half cylinder…

A: Given vector field is, Fx,y,z=x,y,z. It can be written as,  Fx,y,z=xi^+yj^+zk^. Flux of vector field…

Q: Although it is not defined on all of space R°, the field associated with the line integral below is…

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Q: Evaluate the scalar line integral of the vector held F(z. y. 2) = zi + yj - zk around the closed…

A: Let F(x,y,z)=-yi+xj+zk  around a closed circular path c of radius combined at the point(-2,0,3)…

Q: 10.) Use the Fundamental Theorem for Line Integrals to find the work done by the conservative vector…

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Q: Use the Fundamental Theorem for Line Integrals to find the work done by the conservative vector…

A: If F(x,y,z) is a conservative field then there exist a scalar function f, such that ∇f = F. By…

Q: Evaluate the line integral in a vector field F.dr C if the vector field is F(x, y, z) = (x² – y, 4z,…

A: Here, we have given that the vector field Fx,y,z=x2-y,4z,x2 and C is the curve defined as where the…

Q: Define the force field ₹(x, y, z) =. Let S be the portion of the plane z = 2 - 2x in the first…

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Q: 4. Verify that the two integrals in the circulation form of Green's Theorem are equal along a circle…

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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: Consider the vector field F (x, y, z) = (2e2" cos(3y), –3e2" sin(3y) +VV?) Y y' 2 COS a. Find…

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Q: Confirm that the force field F(x, y) = yeyi+ xey j is conservative in some open connected region…

A: We have to find the work.

Q: Verify Stoke's theorem for the vector field F = (2x - y) I- yz*J-yzK over the upper half surface of…

A: Stokes theorem is used

Q: . Verify Stoke's theorem for the vector field F = (2x - y) I– yz²J -y z K over the upper half…

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Q: Use Green's Theorem to find the work done by the force field F(x,y)= (x3-x2y)i+xy2j   on a…

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Q: 3. Find a potential for the conservative vector field: F(x, y, z) = (2xy + 2x, x² – 2yz³, -6z –…

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Q: Let W be the region between the sphere of radius 4 and the cube of side 1, both centered at the…

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Q: Consider the conservative vector field 4 8 F (x, y, z) 3y cos (372), 37y sin (372) – a² > 2xz, 2x +…

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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: If F = xy i+ (x² + 4yz)j+ 2y²z k. Evaluate the line integral between A (0, 2, 0) and B (3, 6, 1).…

A: Given, F= xy i +(x²+4yz)j +2y²zk r0= (0,2,0) r1= (3,6,1)

Q: Confirm that the force field F = xy'i+ x²yj is conservative in some open connected region containing…

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Q: Use the Divergence Theorem to compute the flux of the field 1 cos (y)) out of the region E that lies…

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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: 6) Given the vector field F = (-x,-y,-z) '(x²+y2+z²)3/2! write the integral to calculate the outward…

A: First we parameterize field F then we calculate the Flux across the sphere using following formula:

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. Verify that the two integrals in the circulation form of Green's Theorem are equal along a circle…

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

A: According to the divergence theorem:

Q: Show that the force field F(x, y, z) = (81xz)i + (1 −6yz³)j + (42²-9y²22) k is conservative and find…

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Q: Compute the outward flux of the vector field F = (5xy, 5y²) for the region enclosed by the curves y…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

Q: A particle at (5, 0) traverses the upper semicircle x² + y² = 25 and then closes the loop along the…

A: Formula: The work done by the force F→ is given by, ∮CF· dr, where C is the path along the region.…

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: Use the Divergence Theorem to evaluate the flux of the field F(x, y, z) = (e,3y+ sin (x'z), 8z + Vx2…

A: Given: Fx,y,z=ez2,3y+sinx2z,8z+x2+9y2 The surface S is bounded by the paraboloids x2+y2=z and…

Q: • (2,1,1) 20. (2x In y – yz) dx + хz ) dy — ху аdz - у (1,2,1)

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Q: Although it is not defined on all of space R³, the field associated with the line integral below is…

A: The solution are next step

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: 3.7 If the integral| F dl is regarded as the work done in moving a particle from A to B, 'A find the…

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Q: Consider the 3-dimensional solid E in octant one bounded by = 2- y, y = 1, and x y.S is the surface…

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Q: Q3 Show that the vector field F= 2xyi +(x -2y)j is conservative and evaluate the line integral |F•ds…

A: Solution