2.3 WEAK FORMULATIONS 41
the description of the (physical) field as a connector between geometrical
entities.
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
differential geometry.
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
are called
differential forms,
of
degree 0
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
exterior differential,
which is realized as grad, rot, or div,
depending on the dimension.
All
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
C k
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,
9The
support
of a function, real- or vector-valued, is the closure of the set of points
where it doesn't vanish. Cf. A.2.3.
42 CHAPTER 2 Magnetostatics: "Scalar Potential" Approach
c0k(D),
~0 k(D)"
Same, with compact support contained in D,
being understood that domain D can be all E3, and finally,
ck(w ), ck (D),
for the vector spaces of restrictions to D (the closure
of D), of k-smooth functions or vector fields. In all
of these, k can be replaced by ~o. (In the_ inset, the
supports of a q)l E C0~(D) and a q)2 E C ~(D), which is
thus the restriction of some function defined beyond
D, whose support is sketched.)
" W
2.3.1 The "divergence" side
Now, let's establish a technical result, which generalizes integration by
parts. Let D be a regular domain (not necessarily bounded), S its
boundary, b a smooth vector field, and q0 a smooth function, both with
compact support in
E 3
(but their supports may extend beyond D).
Form u = q0 b. Ostrogradskii's theorem asserts that SD div u = Js n. u,
with n pointing outwards, as usual. On the other hand, we have this
vector analysis formula,
div(q0 b) = q~ div b + b. grad q0.
Both things together give
(9)
SD q0
div b =
-
~D b. grad q0 + Ss n. b q),
a fundamental formula.
By (9), we see that a 1-smooth divergence-free field b in D is
characterized by
(10)
S D b.
grad q) - 0 V q0 ~ C01(D),
since with q0 = 0 on the boundary, there is no boundary term in (9). But
(10) makes sense for fields b which are only
piecewise
smooth. We
now take a bold step:
Definition
2.1.
A piecewise smooth field b which satisfies
(10)
will be said to
be
divergence-free,
or solenoidal,
in the weak sense.
The q)'s in (10) are called
test functions.
1°All that is required is the integrability of b. grad q0 in (10), so 0-smoothness, that is,
continuity, of each "piece" of b is enough.

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