For the vector field F(x, y, z) = (2y — 1) 7+ (z² + 2xz)J+2xyk - the line integral over the curve C: x² + y² = 9 at z = 0 yields f F · dr = −18π. $ State Stokes' theorem and then verify Stokes' theorem in this case by calcu- lating the surface integral over the paraboloid z = 9 - (x² + y²) above the xy-plane.

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Q: The surface H is defined by (x² + y²)² + z = 1; z ≥ 0. The vector field A(r) is given by A = (x, y,…

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Q: Find the curl, the divergence and the potential (if it exists) of F = (2x + y², 2ry + 3y²z*, 4y°z³)

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Q: = (4x° – 7y)i + (9y' + 7x – 7)j %3D - ndary of the parallelogram in the following figure (note the…

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Q: Suppose F(x, y) = x² + y² and C is the line segment from point A = (1, -1) to B = (3,-5). (a) Find a…

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Q: Let C be the curve with parametrization r(t) : ,0 . Find the value of F · dr.

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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: a) Calculate the line integral of the vector field F(x, y) = yi − 5xj from the point (0, 3) to the…

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Q: For the vector field G = (yety + 4 cos(4x + y))i + (xey + cos(4x + y)) 5 I find the line integral of…

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Q: Suppose C is the curve parametrized by 7(t) = (cost, sin t, 1) and S is the portion of z = * + y…

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Q: For vector field F = (-2y, y), y > 0. Find all points P such that the amount of fluid flowing into P…

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Q: 8. Use (a) parametrization; (b) Stokes' Theorem to compute fF.dr for the vector field (x² + z)i +…

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Q: Consider the line integral (Image 1) , where C is defined by the vector function: r (t) = - t2i +…

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Q: Suppose F(x, y) = x² + y²j and C is the line segment segment from point P = (-3,-4) to Q = (-3, 1).…

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Q: ) Given vector field F (x, y) = 2y i + 3x j .Find the circulation of the vector field F over the…

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Q: r = 2 on the hemispherical surface defined by 0 ≤ Ɵ ≤ π/2 and 0 ≤ɸ ≤ 2π F = 100cosƟar Verify Stokes'…

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