Let v = (x*z, 2 – 4xyz, xz) be the velocity field of a fluid. Compute the flux of v across the surface 25x? + 16z2 = (y – 20) where 0 < y < 20 and the surface is oriented away from the origin.

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

Calculate the gradient of the function f(T, y) = cos (4x - 4y) O a. -8x sin (4x2 4y)i + 4 sin (4r2 - 4y)j O b. 8r sin (4x² – 4y)i – 4 sin (4x² – 4y)j O c.- sin (4x2 4y)i + sin (4x² 4y)j O d. sin (4x - 4y)i sin (4x² – 4y)j

Suppose that over a certain region of space the electrical potential V is given by the following equation. V(x, y, z) = 5x² - 3xy + xyz (a) Find the rate of change of the potential at P(5, 2, 5) in the direction of the vector v = i + j - k. (b) In which direction does V change most rapidly at P? 73°F Mostly cloudy (c) What is the maximum rate of change at P? Need Help? Read It Show My Work (Optional)? Watch It Q

Calculate the gradient, Vf, for the following scalar functions (a) ƒ = xyz. (b) ƒ = x² + y² + z². (c) f = xy +yz+xz. (d) ƒ = 3x² – 42². - (e) ƒ = e¯# sin y.