2.2 HONING OUR TOOLS
33
2.2
HONING OUR TOOLS
At least two things disqualify (1-3) as a proper formulation. One is the
non-uniqueness of b and h, a mild problem which we'll address later.
The other is the implicit and unwarranted assumption of
regularity,
or
smoothness, of these fields. For instance, div b = 0 makes perfect sense
if the three components
b 1, b 2,
b 3, in Cartesian coordinates, are dif-
ferentiable. Then (div b)(x) = 31bl(x)
+
32b2(x)
+
33bB(x), a well-defined
function of position x, and the statement "div b = 0" means that this
function is identically 0. No ambiguity about that. But we can't assume
such differentiability. 2 As one knows, and we'll soon reconfirm this
knowledge, the components of b are
not
differentiable, not even continu-
ous, at some material interfaces. Still, conservation of the induction flux
implies a very definite "transmission condition" on S.
2.2.1 Regularity and discontinuity of fields
Since smoothness, or lack thereof, is the issue, let's be precise, while
introducing some shorthands. D being a space domainf the set of all
functions continuous at all points of D is denoted C°(D). A function is
continuously differentiable
in D if all its partial derivatives are in C°(D),
and one denotes by C
I(D)
the set of such functions (an infinite-dimensional
linear space). Similarly, ck(D) or C~(D) denote the spaces composed of
functions which have continuous partial derivatives of all orders up to k
or of all orders without restriction, inside D. In common parlance, one
says that a function "is C k'', or "is C ~'' in some region, implying that
there is a domain D such that ck(D), or C~°(D), includes the restriction
of this function to D as a set element. "Smooth" means by default C,
but is often used noncommittally to mean "as regular as required", that
is, C k for k high enough. (I'll say "k-smooth" in the rare cases when
2This is not mere nit-picking, not one of these gratuitous "rigor" or "purity" issues.
We have here a tool, differential operators, that fails to perform in some cases. So it's not
the right tool, and a better one, custom-made if necessary, should be proposed, one which
will work also in borderline cases. Far from coming from a position of arrogance, this
admission that a mismatch exists between some mathematical concepts and the physical
reality they are supposed to model, and the commitment to correct it, are a manifestation of
modesty: When the physicist says "this tool works well almost all the time, and the
exceptions are not really a concern, so let's not bother", the mathematician, rather than
hectoring, "But you have no
right
to do what you do with it", should hone the tool in order
to make it able
also
to handle the exceptions.
3Recall the dual use of "domain", here meaning "open connected set" (cf. Appendix
A, Subsection A.2.3).

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