70 CHAPTER 3 Solving for the Scalar Magnetic Potential
A little more abstractly, suppose a problem has been cast in the form
Ax = b, where b symbolizes the data, x the solution, and A some
mapping of type
SOLUTION --~ DATA.
Solving the problem means, at a
high enough level of abstraction, finding 7 the inverse A -1, which may
not be defined for some values of b (those corresponding to sharp
corners, let's say, for illustration). But if there is a solution x n for each
of some sequence that converges toward b, it's legitimate to
define the limit x = lim n _, ~ x n as the solution, if there is such a limit, and
if there isn't, to invent one.
That's the essence of completion. Moreover,
attributing to Ax the value b, whereas A did not make sense, a priori,
x, constitutes a prolongation of A beyond its
initial domain, a thing which goes along with completion (cf. A.4.1).
Physicists made much mileage out of this idea of a generalized solution,
as the eventual limit of a parameterized family, before the concepts of
modern functional analysis (complete spaces, distributions, etc.) were
elaborated in order to give it status.
Summing up: We now attribute the symbol O* to the completion of
the space of piecewise smooth functions in D, null on sh0 and equal to
Shl , with respect to the norm IIq011,
[~D ~ I grad
q) i2] 1/2.
Same renaming for
(which is now the closure of the previous one in
O*). Equation (1), or Problem (3), has now a (unique) solution. The next
item in order s is to
But what do we mean by that? Solving an equation means being able to
answer specific questions about its solution with controllable accuracy,
whichever way. A century ago, or even more recently in the pre-computer
era, the only way was to represent the solution "in closed form", or as
the sum of a series, thus allowing a numerical evaluation with help of
formulas and tables. Computers changed this: They forced us to work
from the outset with
representations. Eligible fields and solutions
7An unpleasantly imprecise word. What is required, actually, is some
the inverse, by a formula, a series, an algorithm.., anything that can give
SWhether Problem (3) is well posed (cf. Note 1.16) raises other issues, which we
temporarily bypass, as to the continuous dependence of q~ on data: on I (Prop. 3.2 gave
the answer), on ~ (cf. Exers. 3.17 and 3.19), on the dimensions and shape of the domain
(Exers. 3.18 and 3.20).