Consider a body of flowing water, such as a river. We can describe the flow of river water mathematically by a vector field that describes the water velocity v at every point in the river at an instant of time. Now imagine a small, flat surface submerged in the river. (If you want to visualize something specific, imagine a square grate or a picture frame or something analogous that clearly defines an area but that water can flow through easily.) Assume that the surface is small enough that v is roughly constant over its surface, and that we describe this surface by the tile vector dA.
(a) First, imagine that the surface is perpendicular to the water flow in its vicinity, so that the flux of water through the surface is dΦv = v. d.A= +vdA. Argue that the volume of water that flows through this surface per unit time is equal to flux. (Hint: Argue that the volume of water that will flow through the surface in a time interval Δt is vΔtdA)
(b) Now imagine that the surface has an arbitrary orientation with respect to the flow. By extending your argument for part (a), argue that no matter how you orient the surface, the volume of water per unit time that flows through the surface is equal to the flux dΦv =V. dA. (Hint: What cross-sectional area does the surface present to the flow?)