LAGRANGIAN PERSPECTIVE (Control Mass)
Notice the red tracking box. In this mathematical framework, we lock our coordinates onto a specific "packet" of fluid and follow it as it moves through space and time. Our position coordinates (x, y) are entirely dependent on the time (t). We are sitting in a boat, floating down the river.
EULERIAN PERSPECTIVE (Control Volume)
Notice the fixed red sensor box in the center. In this mathematical framework, we do not care where individual particles go. We establish a fixed coordinate in space (a rigid grid cell) and measure the density and velocity of whatever fluid happens to pass through our sensor at time (t). We are standing on a bridge, watching the river flow underneath us.