To express our problem in a Galerkin form, we begin by rewriting (1), as:
where A is the differential operator, 
.
An equivalent statement of (15) is, find 
such that
. Here, 
is an
arbitrary test function, which can be thought of physically as a
virtual potential field, and the notation,
,
denotes the inner product in 
, i.e. the space of square
integrable functions. Applying Green's theorem, we can equivalently
write,
where the notation 
denotes the inner product on the
boundary S. When the Dirichlet, 
, and Neumann,
, boundary conditions are specified on
S, we obtain the weak form of (1):
