Let Q be a solid bounded by the cylinder x² + y² = 4, the plane x + z = 6, and the xy-plane. Use the Divergence Theorem to evaluate [/F S where S is the surface of Q and F(x, y, z) = (x² + sin z) i + (xy + cos z) j + e³ k. F. ds

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Q: 3) Let S be the unit sphere x² + y² + z² = 1 and F(x, y, z) = (2x, y²,z²). Use the divergence…

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Q: Find a parametrization of the line in which the planes x- 2y - 2z = 3 and -3x + 3y + 3z = -6…

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Q: Compute the flux of F = xi+yj+zk through the curved surface of the cylinder x² + y² = 1 bounded…

A: The given vector field is F→=xi+yj+zk. The goal is to evaluate the flux or surface integral of the…

Q: Let r(t) be a positively oriented rectangle with corners (0,0), (0,1), (2,1) and (2,0) and F(r, y)…

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Q: Evaluate the line integral Jc F · dr, where F(x, y, z) = xi – yj + 3zk and C is given by the vector…

A: Given F(x,y,z)=xi-yj+3zk and C is the curve given by  rt=sint,cost,t. Find ∫CF·dr.

Q: Let S {(x1, x2, x3) € R³ | x3 = 5-x²-x², x3 ≥ 1}, a portion of a circular paraboloid. Endow S with…

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Q: b) Given F(.r, y. :) = (r + cosh 2)i+ (2y-3r*y)j-(r + 4y*:) k. Use Gauss's theorem to calculate F.n…

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Q: Suppose F(x, y) = x*i + yj and C is the line segment segment from point P = (-5, 5) to Q = (-8, 9).…

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Q: Let z = g(x, y) = f(3 cos(xy), y + ev) provided that f(3,6)= 7, f₁(3,6) = 2, f₂(3,6)= 5. i) Find g₁…

A: note :Since you have posted multiple questions, we will provide the solution only to the first…

Q: Compute the flux of F = xi+yj+zk through the curved surface of the cylinder x² + y² = 1 bounded…

A: The vector field is F=xi^+yj^+zk^. The curves surface is x2+y2=1 bounded below by the plane x+y+z=2…

Q: Suppose f(x, y) satisfies the basic existence and uniqueness theorem in some rectangular region Rof…

A: Consider the given: Let the function f(x,y) satisfies the basic existence and uniqueness theorem in…

Q: b) Use Gauss theorem to evaluate F.ndS, where F(r, y, z) = (2x²y² + lIn 2) i – ry° j – tanh(x² + y³)…

A: Given function  F(x, y, z) = (2x2y2+lnz)i -xy3j -tanh(x2+y2)k we have to find ∬σF.nds by using guass…

Q: Graph the solid that lies between the surface z= 2xy/( x2 + 1) and the plane z =x + 2y and is…

A: Given surfaces are z=2xyx2+1, z=x+2y  and are bounded by planes x=0 , x=2 , x=-1 , y=0 , y=4. To…

Q: Evaluate the line integral faF · dr, where F(x, y, z) = -2xi – 5yj + 5zk and C is given by the…

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Q: Evaluate 7. dr

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Q: A) Show that the function u(x; y) = 5x4y - 10 x²y³ + y5 is harmonic and determine the harmonic…

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Q: Use Stokes' Theorem to evaluate of intersection of the plane x + 3y +z = 12 with the coordinate…

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Q: b) Given F(x, y, z) = (x³ + cosh z) i+(2y³ – 3x²y) j – (x² + 4y²z) k. Use Gauss's theorem to…

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Q: Integrate g(x, y, z) = x + y + z over the portion of the plan 2x + 2y + z = 2 that lies in the first…

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Q: Let F = –9zi+ (xe"² – 2xe³) }+ 12 . Find ſg F ·dÃ, and let S be the portion of the plane 2x + 3z = 6…

A: The vector field is F→=-9zi→+xexz-2xex2j→+12k→ The equation of the plane is 2x+3z=6⇒ gx, y=6-2x3.…

Q: Use divergence theorem to evaluate where F(x, y, z) = 3xi + xyj + 2xzk and s is the surface of the…

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Q: 7. Let F(x, y,z)=-2yi +2xj. Use Stokes’ Theorem to find F dr where C is a circle (a) Parallel to the…

A: By stoke's theorem

Q: In V = R³. Let W₁ be the xy - plane and let W₂ be the z zxis: 2 and Show - W₁ = {(x, y, 0) : x, y ≤…

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