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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