6.Let Σ be the part of the paraboloid z = x²+y² that lies within the cylinder x² + y² = 1. Let n be its upward unit normal field. For the vector field F = (2x + e + z², In(z + 2) - y, 2), compute the flux integral .nds.

5. Let U be the region in R² defined by r ≤1+ cos in polar coordinate. Let y be the boundary of U; namely, y is given by r = 1 + cos 0 in polar coordinate. Does there exist a second continuously differentiable function f(x,y) on such that = 2x+ 3y² at every (x, y) ∈ U,and = (4x2-3y²+x,y³ –6y) atevery (x,y) ∈y? Ifyour answer is YES, construct such a function. If your answer is NO, give a proof.

6.Let Σ be the part of the paraboloid z = x²+y² that lies within the cylinder x² + y² = 1. Let n be its upward unit normal field. For the vector field

F = (2x + e + z², In(z + 2) - y, 2),

compute the flux integral .nds.

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