2.3 WEAK FORMULATIONS 41
the description of the (physical) field as a connector between geometrical
Which somewhat devalues differential operators, too: grad, rot and
div, in this light, appear as auxiliaries in the expression of conservation
relations, as expressed by the Ostrogradskii and Stokes theorems. Their
failure to make sense locally is thus not to be taken too seriously.
Proper form is given to the foregoing ideas in
There, one forgets about the scalar or vector fields and one focuses on the
mappings they represent (and thus, to some extent, hide). Fields of
linear mappings of type
GEOMETRIC_ELEMENT --~ REAL
to 3 according to the dimension of the
geometric objects they act upon, and under regularity assumptions which
are milder than for the scalar or vector proxies, one defines a unique
operator d, the
which is realized as grad, rot, or div,
depending on the dimension.
laws of electromagnetism can be cast
in this language (including constitutive laws, which are mappings from
p-forms to (3 - p)-forms, with p = 0 to 3).
The moderate approach we now follow does not go so far, and keeps
the fields as basic objects, but stretches the meaning of the differential
operators, so that they continue to make sense for some discontinuous
fields. The main idea is borrowed from the theory of distributions: Instead
of seeing fields as collections of pointwise values, we consider how they
act on other fields, by integration. But the full power of the theory of
distributions is not required, and we may eschew most of its difficulties.
2.3 WEAK FORMULATIONS
First, some notation. Symbols
and C a for smoothness have already
been introduced, compact support 9 is usually denoted by a subscripted
0, and blackboard capitals are used in this book to stress the vector vs
scalar opposition when referring to spaces of fields. Putting all these
conventions together, we shall thus have the following list of infinite-
dimensional linear spaces:
• ck(E3 ) • The vector space of all k-smooth functions in E3,
• ok(E3 )" All k-smooth vector fields in E3,
of a function, real- or vector-valued, is the closure of the set of points
where it doesn't vanish. Cf. A.2.3.