The Lorentz reciprocal theorem states a relationship between two Stokes flows in the same region.  Consider  fluid filled region  bounded by surface .  Let the velocity fields  and  solve the Stokes equations in the domain , each with corresponding stress fields  and . Then the following equality holds:
Where  is the unit normal on the surface .  The Lorentz reciprocal theorem can be used to show that Stokes flow "transmits" unchanged the total force and torque from an inner closed surface to an outer enclosing surface.  The Lorentz reciprocal theorem can also be used to relate the swimming speed of a microorganism, such as cyanobacterium, to the surface velocity which is prescribed by deformations of the body shape via cilia or flagella.
The Lorentz reciprocal theorem has also been used in the context of elastohydrodynamic theory to derive the lift force exerted on a solid object moving tangent to the surface of an elastic interface at low Reynolds numbers.