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):