Question

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 ????=⟨x−y,z+y+2,z2⟩v=⟨◂,▸x−y,z+y+2,z2⟩ and the net is decribed by the equation y=1−x2−z2,y=◂−▸1−x2−z2, y≥0,y≥0, and oriented in the positive y-positive y-direction.

Answer 1

To calculate the flow rate of water across the net, the surface integral of the river's velocity vector field dot product with the normal to the net's surface is needed. The net's equation is y = 1 - x² - z², and its normal vector is derived from its gradient.

Step-by-step explanation:

The flow rate of water across the net in the river can be calculated using the velocity field of the river and the surface equation of the net. The velocity vector field given is v = ⟨x - y, z + y + 2, z²⟩, and the net is defined by the equation y=1-x²-z² for y ≥ 0. The flow rate across the net (which is oriented in the positive y-direction) is determined by the surface integral of the velocity vector field dot product with the normal to the surface described by the net.

Since the flow is across a surface, we need the component of the velocity field that is perpendicular to the surface described by y = 1 - x² - z². The vector normal to this surface can be derived from the gradient of the function f(x,z) = 1 -x² - z², thus obtaining the vector n = ⟨-2x, 1, -2z⟩. The flow rate is the double integral over the region (net) of the dot product of the velocity vector and normal vector dS, where S is the surface of the net.