Find the flux of the vector field F = across the part of the plane z=2+x+3y above the rectangle [0, 5] x [0,4] with upwards orientation.

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.)

F = (x² - y)i + + (4z)j + (x²)k Find the curl (curl calculation) of the vector field?

Find the directional derivative of the scalar field = x³y + 4xz at the point (1,-1, 2) along the direction vector (2, -1, −2):