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:f = rac{d F}{d V}\,

Where F is force and V is volume. Examples of common body forces include:

In general, any Acceleration that a body undergoes will cause a body force given by:

:f = ho a\,

Where ho is the Density of the substance at a given point and (lowercase) f is the body force. This result may derived by applying Newton's Law to a small Volume and then taking the appropriate Limit :

:\Delta F = \Delta m a \quad \Rightarrow \quad rac{\Delta F}{\Delta V} = rac{\Delta m}{\Delta V} a
\quad \Rightarrow \quad
\lim_{\Delta V o 0}\left( rac{\Delta F}{\Delta V} ight) = \lim_{\Delta V o 0}\left( rac{\Delta m}{\Delta V} a ight)
\quad \Rightarrow \quad rac{d F}{d V} = rac{d m}{d V} a\,

By definition, the derivative of mass with respect to volume is Density ; from this f = ho a follows. A body force doesn't necessarily involve acceleration. A prime example of such a body force is the action of gravity on an immobile object, other examples include various stresses in solids and pressure gradient of a steady fluid flow.

The Navier-Stokes Equations are written in terms of body forces.