Consider the closed cylinder x² + y² = R² for 0 ≤ z ≤ L. Find the F(r, 0, 2) = 02+z0+r²F

#2. Let flz) = (x, y) +iv(x,y) be an analytic function m a domain D. If a zeck, y) + b v(x, y) = C in D are real Constants not all 0. where a, b and c Prove that fiz) is constant in D.

Find that F = 12x²yi + (4x³ + 8yz?)j+ 8y² zk is conservative, and find a function f such that F = Vf, and use the result to evaluate , F •dr, where C is any curve from A = (0, 0,0) to B = (-3,2, 3).

Find the mass M of a fluid with a constant mass density flowing across the paraboloid z = 36 - x² - y², z ≥ 0, in a unit of time in the direction of the outer unit normal if the velocity of the fluid at any point on the paraboloid is F = F(x, y, z) = xi + yj + 13k. (Express numbers in exact form. Use symbolic notation and fractions where needed.) M = Incorrect 31740πσ > Feedback Use A Surface Integral Expressed as a Double Integral Theorem. Let S be a surface defined on a closed bounded region R with a smooth parametrization r(r, 0) = x(r, 0) i + y(r, 0) j + z(r, 0) k. Also, suppose that F is continuous on a solid containing the surface S. Then, the surface integral of F over S is given by [[F(x, y, z) ds = = D₁² Recall that the mass of fluid flowing across the surface S in a unit of time in the direction of the unit normal n is the flux of the velocity of the fluid F across S. M = F(x(r, 0), y(r, 0), z(r, 0))||1r × 1e|| dr de F.ndS X