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 v=(x−y,z+y+3,z²) and the net is decribed by the equation y=√(1-x²-z²),y≥0, and oriented in the positive y direction.

∬v⋅dS=

Answer 1

To determine the flow rate of water across a net in a river, we need to calculate the flux of the velocity vector field through the net's surface. This can be done by evaluating the surface integral ∬v⋅dS, where v is the velocity vector field and dS is the differential surface element of the net.

Step-by-step explanation:

To determine the flow rate of water across the net, we need to calculate the flux of the velocity vector field through the net's surface. The net is described by the equation y=√(1-x²-z²), y≥0, and it is oriented in the positive y direction. The flux is given by the surface integral ∬v⋅dS, where v is the velocity vector field and dS is the differential surface element of the net.

Since the net is oriented in the positive y direction, the only component of the velocity vector field that contributes to the flow rate is the y-component. To evaluate the integral, we can substitute the equation of the net into the y-component of the velocity vector field and integrate over the surface using appropriate limits for x and z.

Performing the integration will give us the flow rate of water across the net.