Computational Electromagnetism

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.