Find a potential for the conservative vector field: F(x, y, z) = (2xy + 2x, x² – 2yz³, –6z – 32²y²) and use your answer in order to compute the vector line integral F. dr where C is the oriented broken line starting at (0,0,0), continuing to (1,1, 2) and ending at (2, –1,3).

Q: Find the flux of the vector field F(x, y, z) = 3xi+3yj+6zk across the portion of the surface z = 25…

A: The flux F across the surface is given by ϕ=∫σF·nds=∬RF·∇GdA

Q: The potential of the conservative vector field F = (y,x,1> is Of=z+xy O f=x+yz O f=y+xz Of = xyz

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

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

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Q: Use the divergence theorem to calculate the flux of the vector field F(x, y, z) = −1xyi – 2yzj —…

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Q: Find the outward flux of the vector field F = (x³, y³, z²) across the surface of the region that is…

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Q: Compute the flux of the vector field F(x, y, z) = (2y, 3, –2xz) through the surface S, which is the…

A: Given vector field is, Fx,y,z=2y,3,-2xz. The given surface is, y=x2+z2  and  x2+z2≤1. Now for the…

Q: Consider the vector field F = (2y +1,y, –2²) along with the paraboloid surface z = r² + y. Find the…

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Q: Find the surface integral F. dS of the vector field F(x, y, z) = (-x, -4, yz®) along the part of the…

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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: For the vector field F(x, y, z) = (15x2y + 2xz)i + (5x3 + 35y4)j + x2k, show that F is conservative…

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Q: Consider the vector field F(x, y, z) = (2ry, x², 1). (c) Calculate the line integral / F - dĩ where…

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Q: Find the potential function p, for the conservative vector field F(x, y, z) = (2xe² – y²sin x)î +…

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Q: Determine the line integral if the vector field is F(x, y) = x²yî – 2xyf On a rectilinear path that…

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Q: Compute the flux of the vector field F(x, y, z) = 2xj+ 4yk through the surface S, where S is the…

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

A: The given vector field is, F→=y, -z,  x. The given part of the plane is z=4+2x+3y And the rectangle…

Q: Find the flux of the field F = x2 j - xz k outward (normal away from the yz-plane) through the…

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Q: Find the potential function o, for the conservative vector field F (x, y, z) = (2xe² – y²sin x)î +…

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Q: b) Find the flux of the vector field F(x,y,z) = 3xi – yj + 2zk; across o where o is the portion of…

A: The given question belonging to the vector calculus  in which we have to find the value flux of the…

Q: The vector field F (x,y) = (sec²x, 3y²) is conservative. Find f(x,y) such that F = Vf. f= ytanx + xy…

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Q: Use the Divergence Theorem to find the outward flux of the vector field F(x,y, z) = xy2i + yz2j +…

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Q: Find the flux of the vector field F(x, y, 2) = yj + k where o is the portion of the cone =' = 4(x² +…

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Q: Q1 Use Stokes theorem to compute the flux of the curl of a vector field F = (2x – y)ĩ – yz? j- y²zk…

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Q: the gradient vector field of f(z, y) = In(x + 4y) %3D Work

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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: The vector field F=(x+2y)i+(2x+y)j is conservative. Find a scalar potential f and evaluate the line…

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Q: Find the work done by the force field F(x, y, z) = (x − y², y - z², z - x²) on a particle that moves…

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Q: Let ryz , (교2 + y"), 17 F = ( x2 +y? be a vector field on U = {(x, y, z) E R³ | (x, y) # (0, 0)}.…

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Q: Calculate the line integral , F · dr, where C is the straight-line path from (2,0) to (0,2),…

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Q: Find the upward flux of the vector field F(x,y,z)= (-y, x, 9) across the part of the hemisphere…

A: Consider the provided question,

Q: Consider the vector field F = (r²y³, x°y?) Is this vector field conservative? Conservative If so:…

A: Given the vector field <x2y3,x3y2>. We need to: i) Check whether the vector field is…

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: Compute the flux of the vector field F(x, y, z) = 3x + 3yk through the surface S, where S is the…

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Q: Find all potential functions of the conservative vector field F (x, y, z) = (2xz + y², 2xy,x² + %3D…

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Q: Consider the vector field F (xr, y) = (2y - 12r*y", 4ry – 9r*y?) is conservative. (a) Show that F…

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

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Q: Show that the vector field F = 2.xyi + (x – 2 y)j is conservative and evaluate the line integral %3D…

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Q: Compute the flux of the vector field F = zi + 8xk through the parameterized surface S oriented…

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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: ind the conservative vector field for the potential function by finding its gradient. f(x, y, z) =…

A: The gradient of a function is the rate of change of a function and for a function, f(x,y,z), the…

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: Compute the flux of the vector field F(x,y,z)=5xj+2yk→through the surface S, where S is the part of…

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Q: Find the flux of the vector field (y, x, z) in the negative z direction through the part of the…

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Q: Determine whether the vector field…

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Q: Let C be the helix that winds around the cylinder x2+y? = 1 (counterclockwise viewed from the…

A: here i used parametric equation of helix and done simple calculation like dot product and after that…

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

A: To evaluate ∫∫SF.dS: By the divergence theorem, ∫∫SF.dS=∫∫∫Ediv F dV where E is the solid enclosed…

Q: Find the potential function p, for the conservative vector field F(x, y, z) = (2xe? – y²sin x)î +…

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