92 CHAPTER 3 Solving for the Scalar Magnetic Potential
3.16. Cf. Fig. 3.8. Length b should equal a q2/3. There is numerical
evidence [MP] that such tetrahedra yield better accuracy in some
computations than the standard "cubic" grid, subdivided as Fig. 3.7
suggests. (A suitable combination of regular octahedra and tetrahedra,
by which one
can
pave [Ka], may also be interesting in this respect.)
3.17. =Label %
and 2q)2.
the solutionsl < correSPaOnding< to btl and t,t 2. Then
1/R 1 inf{~Dbt 11Vq~l q)~
}--YD~'[1 ]Vq)2] --fD~2
]Vq)a]a=1/R2 •
3.18. With respect to some origin, map D to Dx by x --+ )vx, with )~ > 0,
and assign to D~ the permeability ~ defined by btx(Kx)= bt(x). If q~ is
an admissible potential for the problem on D, then q~x, similarly defined
by q~x0~x) = q~(x), is one for the problem on Dx. Changing variables, one
sees that
bfD~
btx IVq)x 12-
ID ~ IVq) l
2, SO it all goes as if bt had been
multiplied )v (in vacuum, too!). Hence the result by Exer. 3.17.
S b
s"0
~t 0
/
s b
s"0
b
s
FIGURE 3.9. Exercise 3.19. How the presence in the domain under study of a
highly permeable part (/,t 1 >> ~), even of very small relative volume, is enough
to distort the field. (Two-dimensional drawing, for clarity. In the case of Fig.
3.1, a similar effect would be achieved by putting a high-bt thin sheet inside D.)
3.19. Since both
(4)1
and
(4)2
belong to @~, one can set qY = %
- (4)2
in both
equations (25), and subtract, which yields
Therefore,
(q)l' (4)1 -- (4)2)1 "~ ((4)2' (4)2 -- q)1)2 = O.
1~4)1--(4)21112= ID (~1--~'12 ) Vq)2" V(q)2- q)l) ~ ((1
-/.t2/~)qo 2,
(4)2--(4)1)1,
hence IIq) 1
-(p2ll I ~ C(~) [IKp2111
by Cauchy-Schwarz, where C(bt) is an
upper bound for 1 1
- ~2/~
I over D. Hence the continuity with respect
to B (a small uniform variation of bt entails a small change of the
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