In linear elasticity, the equations describing the deformation of an elastic body subject only to surface forces (&/or body forces that could be expressed as potentials) on the boundary are (using index notation) the equilibrium equation:
where is the stress tensor, and the Beltrami-Michell compatibility equations:
A general solution of these equations may be expressed in terms the Beltrami stress tensor. Stress functions are derived as special cases of this Beltrami stress tensor which, although less general, sometimes will yield a more tractable method of solution for the elastic equations.
It can be shown  that a complete solution to the equilibrium equations may be written as
Using index notation:
where is an arbitrary second-rank tensor field that is continuously differentiable at least four times, and is known as the Beltrami stress tensor. Its components are known as Beltrami stress functions. is the Levi-Civita pseudotensor, with all values equal to zero except those in which the indices are not repeated. For a set of non-repeating indices the component value will be +1 for even permutations of the indices, and -1 for odd permutations. And is the Nabla operator.
The Maxwell stress functions are defined by assuming that the Beltrami stress tensor tensor is restricted to be of the form.
The stress tensor which automatically obeys the equilibrium equation may now be written as:
The solution to the elastostatic problem now consists of finding the three stress functions which give a stress tensor which obeys the Beltrami–Michell compatibility equations for stress. Substituting the expressions for the stress into the Beltrami-Michell equations yields the expression of the elastostatic problem in terms of the stress functions:
This article needs attention from an expert in equation. Please add a reason or a talk parameter to this template to explain the issue with the article. WikiProject Equation (or its Portal) may be able to help recruit an expert.(June 2010)
These must also yield a stress tensor which obeys the specified boundary conditions.
The Airy stress function is a special case of the Maxwell stress functions, in which it is assumed that A=B=0 and C is a function of x and y only. This stress function can therefore be used only for two-dimensional problems. In the elasticity literature, the stress function is usually represented by and the stresses are expressed as
Where and are values of body forces in relevant direction.
The Morera stress functions are defined by assuming that the Beltrami stress tensor tensor is restricted to be of the form 
The solution to the elastostatic problem now consists of finding the three stress functions which give a stress tensor which obeys the Beltrami-Michell compatibility equations. Substituting the expressions for the stress into the Beltrami-Michell equations yields the expression of the elastostatic problem in terms of the stress functions: