Verify the formula [[F.nds = [ div F dV in the Divergence Theorem by evaluating the surface integral and the triple integral. F(x, y, z) = 2xi - yzj + z²k where σ is the paraboloid z = x² + y² capped by the disk x² + y² ≤ 16 in the plane z = 16.

Q: Use the divergence theorem to calculate ff, FdS where F = x²yi - xy² j - xyz k. S is the surface of…

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Q: Use Stokes' theorem to evaluate 0 F. dr. F(x, y, z) = zexi + (z − 7y³)j + (x − 5z³)k, C is the…

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Q: Evaluate ſç(2x + z², − 3y + x, 6z − 4y) · ds where S is a sphere of radius R=3 and center the axes…

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Q: b) Given F(r, y, z) = (r³ + cosh z) i+ (2y3 – 3r*y)j – (2² + 4y²z) k. Use Gauss's theorem to…

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Q: Use Stokes' theorem to evaluate • Sf₂ (V x (V x F). dS where F(x, y, z) = -3yzi + 3xzj +6(x² + y²)zk…

A: Given vector field isF(x,y,z)= -3yzi+3xzj+6(x2+y2)zkAnd S is the part of paraboloid z = x2+y2  that…

Q: Let D be the unit disk D = {(x, y) | x² + y² ≤ π }. Calculate ə Ə ₁ ( [xsin (2² + y²³)] - [3² sin(x²…

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Q: Evaluate the surface integral 1 d.S where o is the portion of the plane z = 4 - 6x – 1y in the first…

A: To find  where  is the portion of the plane  in the first octant

Q: Compute the surface integral s F-ndo. F = xyi - x¹j+ (x+z)k. The surface is the top of the plane 2x…

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Q: Verify the formula $ F. T ds = || (curl F) · n dS in Stoke's Theorem by evaluating the line integral…

A: given curve Fx,y,z=9x-5y i +9y-5z j +9z-5x k verify the formula ∮C F·T ds=∫∫σcurl F· n dS in stoke's…

Q: Consider the positively oriented surface S₂ defined by the vector function R(u, v) = (u sin v-3, u…

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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: Use Stokes' theorem to calculate JJ (A) nds by A = 3yi-xzi+yz²k where S is the surface of the…

A: The given vector field is: A=3yi-xzj +yz2k The surface S is the paraboloid 2z=x2+y2 bounded by z=2.…

Q: Let Y be the portion of the sphere x² + y² + (z − 2)² = 4 which lies below the plane z = 3 with…

A: Given  ∫Cz2+3yi→+2z-2xj→+z2k→.dr→ The portion of the sphere x2+y2+z-22=4 and plane z=3

Q: Verify the formula o F T ds = || (curl F) n dS in Stoke's Theorem. by evaluating the line integral…

A: Given  Fx,y,z=5x-yi+5y-zj+5z-xk We have to verify the Stoke's theorem. x+y+z=3 is the plane.

Q: 03. a. If F = (y² + z² − x²)i + (z² + x² − y²)j + (x² + y² − z²)k, then evaluate, SSV × F·n dA…

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Q: Verify the Stokes' Theorem in evaluating F.dr if F(x, y, z) = z² i + 2xj- y³ k, where C is the…

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Q: Use Stoke's theorem to evaluate F dr where F(x,y, z)=e-7i+ +e"k and C is the boundary of the part of…

A: Let Fx,y,z be a vector field and the surface S is bounded by a simple closed curve C. The Stoke's…

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: yzi + ³j+rzk, and S is the part of the sphere Verify Stokes Theorem for fas F-dr, where F r² + y²…

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Q: and Verify Green theorem where F = 2xi-Y2j+ŻK s it represents the Surface generated by the Parabolic…

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Q: Consider the following parametric surface (u, v) E R S: r(u, v) = ui + vj + (1 + u – v)k, where Ris…

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Q: Show that the tangent plane to the quadric surface at the point (x0, y0, z0) can be written in the…

A: Given surface is x2a2+y2b2+z2c2=1 To find the equation of the plane, find the normal of the given…

Q: Use Stokes' Theorem to evaluate fF.dr if F(x,y,z) =( z-y)i +(x+z)]- (x +y)K and C is the curve of…

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Q: Let the surface S be the graph of the function z = g(x, y) = x+y over the unit disk D: x² + y² ≤1.…

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Q: For each of the following, is this a valid parametrization of the surface 2x 3y
z  4? Circle any…

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Q: Let S be the bounded portion of the paraboloid x² + y² = 8z which is cut off by the plane z = 4.…

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Q: Use Stokes' Theorem (in reverse) to evaluate (curl F) ndS where F = 6yzi + 9xj +6yzeªk,S is the…

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Q: Let S be the quadratic surface given by S = {(x, y, z) | z = 4x² - y², z ≥ 0}, oriented with the…

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