6.2. EXERCISES 187
EXERCISES
See p. 171 for Exercise 6.1, p. 174 for Exer. 6.2, p. 179 for Exer. 6.3, and p. 174
for Exers. 6.4 and 6.5.
Exercise 6.6. Let a smooth surface S be equipped with a field of normals.
Given a smooth function q0 and a smooth vector field u, we may define
the restriction of q0 to S, or
trace ¢4s,
the
tangential part
u s of u (that is,
the surface field of orthogonal projections of vector u(x) onto the tangent
plane at x, where x spans S, as in Fig. 2.5), and the
normal component
n. u of u. For smooth functions and tangential fields living on S, like
q0 s and u s, define operators grads, rots, and div s in a sensible way, and
examine their relationships, including integration-by-parts formulas.
HINTS
6.1. Notice that this approach amounts to solving (11) and (12').
6.2. Their physical
dimension
is the key. Note that components of L a
are induction fluxes, and (M a, a) has the dimension of energy.
6.3. For a tetrahedron T which contains e = {m, n}, integrate by parts
the contribution ~w h. rot w e, hence a weighted sum of the jump [n x h]
over 3T. Check that faces opposite n or m contribute nothing to this
integral. As for faces f which contain e, relate ff [n x hi . w e with the
circulation of [h] along the median. Use rot hm= 0
inside
each tetrahedron
to derive the conclusion.
6.4. Compute the divergence of u = ~
n e N ~
Wn"
6.5. Take the curls.
6.6. Obviously, grad s q0 s must be
defined as (grad q0)s, and rot s u s
as n. rot u, when q0 and u live in
3D space, for consistency. (Work
in x-y-z coordinates when S is
the plane z = 0 to plainly see that.)
Verify that these are indeed
surface
operators, that is, they only depend
on the traces on S of fields they
n
V
act on. Define div S by Ostrogradskii-Gauss (in order to have a usable
integration by parts formula on S), and observe its kinship with rot s.
You'll see that a second integration-by-parts formula is wanted. Use
notation as suggested in inset.

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