In physics and engineering, mass flow rate is the mass of a substance which passes per unit of time.  Its unit is kilogram per second in SI units, and slug per second or pound per second in US customary units.  The common symbol is  (ṁ, pronounced "m-dot"), although sometimes μ (Greek lowercase mu) is used.
Sometimes, mass flow rate is termed mass flux or mass current, see for example Schaums Outline of Fluid Mechanics'. In this article, the (more intuitive) definition is used.
Mass flow rate is defined by the limit:
i.e., the flow of mass  through a surface per unit time .
The overdot on the  is Newtons notation for a time derivative. Since mass is a scalar quantity, the mass flow rate (the time derivative of mass) is also a scalar quantity. The change in mass is the amount that flows after' crossing the boundary for some time duration, not the initial amount of mass at the boundary minus the final amount at the boundary, since the change in mass flowing through the area would be zero for steady flow.