Reynolds transport theorem

An equation relating the time rate of change of an extensive property in a moving control volume to its local change and flux across the boundary.

Material derivative: The time rate of change of a quantity following a moving fluid particle. The theorem converts this particle-following rate into control-volume storage and boundary flux.
Continuity equation: A local conservation equation stating that changes in density are balanced by mass flow. Applying the theorem to mass yields the integral mass balance and its local form.
System (thermodynamics): A specified quantity of matter or region selected for thermodynamic analysis. The theorem connects changes in this material system to a control-volume description.
Lagrangian description: A description of motion that follows individual material particles through time. The theorem translates this system-following viewpoint into a control-volume balance.
Control volume: A chosen region in space across whose boundary mass, momentum, or energy may flow. The theorem expresses system changes using balances over this spatial region.
Momentum equation: A conservation equation relating a system’s momentum change to the forces acting on it. Applying the theorem to linear momentum gives control-volume momentum balances.
System boundary: The real or imagined surface separating a thermodynamic system from its surroundings. Boundary motion and transport across it determine the theorem’s flux contribution.
Eulerian description: A description of flow properties at fixed positions in space. Control-volume balances use this spatial viewpoint, unlike particle-following system descriptions.
Extensive property: A property of a system whose value scales with the amount of matter, such as mass or energy. The theorem applies to extensive quantities and their associated densities.
Angular momentum equation: A balance relating angular momentum change to the net external torque. The theorem converts system angular-momentum change into control-volume storage and flux.

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