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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