A Jackson E&M: Power loss in waveguides for TE & TM modes

Hi. I'm having trouble with calculating

(8.59)
241929


from (8.58)
241930
(## \vec H_{//} ## is simply ## \vec H ## in Jackson, but that shouldn't matter.)

in Jackson. Specifically I think I'm not sure about parallel component of H field here.

For example, shouldn't I have two terms for the TE case in that

when ## \hat n = \hat x ## (for simplicity), ## \hat n \times \vec B =

- B_z \hat y + B_y \hat z

= B_z \hat y + \frac {ik} {\mu \epsilon \omega^2 - k^2} \frac { \partial B_z } {\partial y} \hat z ## (from 8.26b)

(which, by the way, does not include ## \frac {\partial B_z} {\partial x} ## as in Jackson)

And similarly, shouldn't TM mode have only one term since ## B_z## is zero in TM mode?
 

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