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is

E_x = E_1 cos(n_1 x \pi/L) sin(n_2 y \pi/L) sin(n_3 z \pi/L) sin(\omega t)

E_y = E_2 sin(n_1 x \pi/L) cos(n_2 y \pi/L) sin(n_3 z \pi/L) sin(\omega t)

E_z = E_3 sin(n_1 x \pi/L) sin(n_2 y \pi/L) cos(n_3 z \pi/L) sin(\omega t)

with E_1 n_1 + E_2 n_2 + E_3 n_3 = 0.

In counting the number of modes, the counting is restricted to non-negative

values of n_1, n_2 and n_3. Is there a simple way to show that

a) any mode in which one or more of the n_1, n_2 and n_3 are negative, can

be written as a linear combination of the modes that are included in the

counting and

b) the modes that are included in the counting are all independent?

Also, is the perpendicular component of the magnetic field on the surface

of a plane conductor required to be zero? The vanishing electric field in the

conductor only implies that the the time derivative of the perpendicular

component of the magnetic field vanishes.

Thanks!