Find the curl of the vector field F curl F = 1.2 i + j+ 132 < 4z cos(x), 2z sin(x), 7z >

A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v=(x-y,z+y+7,z2) and the net is decribed by the equation y=1-x2-z2, y20, and oriented in the positive y- direction. (Use symbolic notation and fractions where needed.)

Find the vector parametrization r(t) of the line L that passes through the points (4, 1, 2) and (9, 3, 7). (Give your answer in the form (*, *, *). Express numbers in exact form. Use symbolic notation and fractions where needed.) r(t) =

Find the curl of the vector field F = (2y cos(x), 4x sin(y)) curl F =

F(x, y) = x²yi + 3y³x²j Find the integral curves for the vector field а. —6у + 1 = C 6xy2 ob. -6ху — 1 C 6xy2 c. -6ху + 1 = C 6x²y2 O d. -ху + 6 C x?y2

match the vector field  with one of the graphs (I)-(VI).

Find the location at t = 3 of a particle whose path satisfies dr = (181-1712-21-4) dt + r(0) = (8,7) (Use symbolic notation and fractions where needed. Give your answer in vector form.) r(3) = (54,4) Incorrect

Calculate the derivative (r × r'), where r = (5t, t²,e¹). dt (Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.) d (rxr') = (2te',-5e¹,0) dt Incorrect

Calculate the derivative (r = r'′), (r × dt ), where r = (9t, t²,e¹). (Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.) d (rx r') = (-2te¹,9,18-9e¹) dt

Find the directional derivative of the scalar field þ= x¹y + 4xz at the point (2, -2, 4) along the direction vector (2, -1, -2):