Suppose K is a region in the x,y plane and P(a, b, c) a point above K. Let S be the 'conical surface' formed by drawing lines from P to the boundary of K. If D is the ‘conical solid' enclosed by S and K, prove that Volume(D) P(a,b,c) S X D K Z = Area(K). y Hint: Apply the divergence theorem to the vector field F = (x – a)i + (y − b)j + (z – c)k.

Q: Given the vector field F(x, y, z) = sin zi + cos yj + zzk, evaluate the line integral JF-dr where…

A: If curl F =0 It implies that F is conservative.

Q: Consider the solid D given by O= {(r,y, 2) € R°; 7+ỷ <=<2- VF +y}. Let F(1, y, 2) = (2rz + e )ï+…

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Q: Use the divergence theorem to find the outward flux (F.n) ds of the given vector field F. F- xy'i +…

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Q: a field of forces defined by DE IR le closed curve C inside D,it is sa S. I F.dr = 0 A) F(x, y) = (y…

A: We have to choose the correct answer from the given option according to the given condition

Q: Consider vector field: F = e i-cos y j+ sin² z k. Obtain divergence and curl for F at points (0,,).

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Q: ) Use Green's Theorem to evaluate the line integral along the given positively oriented curve: F. dr…

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Q: Let the surface S be part of the sphere r +y² +z² = 4 (oriented away from the origin) that lies…

A: Given: Let the surface S be part of the sphere x2+y2+z2=4x (oriented away from the origin) that lies…

Q: Consider the conservative vector field F = ⟨yz + 1, xz + ez cos y, xy + ez sin y⟩ a) State why it…

A: Consider the conservative vector fieldF = ⟨yz + 1, xz + ez cos y, xy + ez sin y⟩a) State why it is…

Q: let F be the vector field Cx, 2x,3). let c be the of the Surface S gien by 4xt y1855 it. 871.Given c…

A: We will use reverse of Stokes theorem for vector  field F to convert given integral into line…

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: By using the divergence theorem I need to find the outward flux of the vector field    F(x, y, x) =…

A: Using divergence theorem  the outward flux of the vector field: ∯SF.dS=∰V∇·F.dV…

Q: For closed surface bounded by the coordinate planes, x + 2z = 6 and y = 3, verify that the surface…

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Q: 5) Let F(x, y, z) = x²i+ y²j+ z2k be a vector field defined on the surface S which is the portion of…

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Q: i F - y,-1 is vector field and R = {(r, v): 45 2 + y s 9, y S -z} is region in zy- plane. R is the…

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Q: Let F be the vector field defined as follows: F = (r°, y°, 2x²z + 2y²z). (i) Compute V. F in…

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Q: Consider the vector field F(r, y, z) = (x, y, 6z). Evaluate the surface integral I = [[; F · dS…

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Q: Consider the field of vectors F(x, y, z) = (x − y, x + y, z). Calculate the flow of F through…

A: The field of vectors is Fx,y,z=x-y,x+y,z. The flow of F through the surface S has to be determined.…

Q: Use the Divergence Theorem to find the outward flux of the vector field F(x, y, z) = 7x3i +…

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Q: Consider the vector field F = (x²,xy, tan(x²y)) and the surface of the tetrahedron with vertices…

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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: Let S be the portion of the ellipsoid x² + 4y² + z² = 4 where z ≥ √3x. Suppose that S is oriented so…

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Q: Use the divergence theorem to evaluate the surface integral F · dS, S where the vector field is F =…

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Q: Also, with the picture, what would be the flux going through the surface of just the cone?

A: Consider the provided question, (a)

Q: Let F¯be the vector field given by F = (x, y, z) = [xz, – yz, xyz] Let T be the area bounded by the…

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Q: Given a vector field F(x, y, z) = z²î+x²ĵ_y²k and S is the surface of portion of the cylinder x² +…

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Q: A potential function for the vector field F(x, y, z) = (yz, xz + z, xy + y+ 1) is (x. y, z) = xyz +…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Evaluate the surface integral F. ds for the given vector field F and the oriented surface S. In…

A: The S is composed of the surface in the paraboloid in the form y=g(x,z) f(x,y,z)=y-g(x,z)=y-x2-y2…

Q: Ben can wash a car in 2 hours while his younger brother James can do it in 3 hours. How long will it…

A: In 1 hour Ben can wash 12 part of car. in 1 hour his younger brother can wash 13 part of car.

Q: Use the Divergence Theorem to compute the net outward flux of the vector field F across the boundary…

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Q: Use the divergence theorem to calculate the flux of the vector field F(x, y, z) = 4xyi + 5yzj + 2xzk…

A: The given problem is to find the flux of the given vector field through the sphere of radius 4…

Q: Consider r = |r| and r = (X, Y, z). Find the curl Find the divergence of .

A: Given, r=x,y,z=xi+yj+zkr=r The main objective is to find the curl rr3 and the divergence of rr3.

Q: Evaluate the surface integral F. ds for the given vector field F and the oriented surface S. In…

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