In general terms, Lagrange's identity for any pair of functions u and v in function space C2 (that is, twice differentiable) in n dimensions is:[1]
where:
and
The operator L and its adjoint operator L* are given by:
and
If Lagrange's identity is integrated over a bounded region, then the divergence theorem can be used to form Green's second identity in the form:
where S is the surface bounding the volume Ω and n is the unit outward normal to the surface S.
Ordinary differential equations
Any second order ordinary differential equation of the form:
can be put in the form:[2]
This general form motivates introduction of the Sturm–Liouville operator L, defined as an operation upon a function f such that:
It can be shown that for any u and v for which the various derivatives exist, Lagrange's identity for ordinary differential equations holds:[2]
For ordinary differential equations defined in the interval [0, 1], Lagrange's identity can be integrated to obtain an integral form (also known as Green's formula):[3][4][5][6]
where , , and are functions of . and having continuous second derivatives on the interval .
We have:
and
Subtracting:
The leading multiplied u and v can be moved inside the differentiation, because the extra differentiated terms in u and v are the same in the two subtracted terms and simply cancel each other. Thus,
which is Lagrange's identity. Integrating from zero to one:
as was to be shown.