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**Unformatted text preview: **to �xy + ik, where �xy is just the x and y derivatives
and k = k z , so that
ˆ
�
��
�
�
�
b [�xy + ik] ·
b�xy ·
bk·
ˆk =
D
=
+i
.
a [�xy + ik]
a�xy
ak
´ �
1
ˆ
¯
+ a v · v� , we obtain two terms from �w, Dk w� �,
ˆ
ˆ
from the two terms above. The ﬁrst, from the �xy terms, works exactly like D (since we analyzed D in any
number of dimensions, and this is just the 2d case), and so is anti-Hermitian as before. The second, from the k
terms, is:
�
�
ˆ
bk·
¯
�w, i
w� � =
(¯ik · v + v · iku)
u
ak
Ωxy
ˆ
�
�
=−
iku · v + ik · vu
Plugging this into the 2d inner product �w, w� � = ¯�
b uu �1 Ωxy Ωxy �
= �−i
ˆ
ˆ∗
and so this part is anti-Hermitian too. Hence Dk = −Dk .
1 bk·
ak � w, w� �, (c) Here, we just apply the ordinary quotient/product rules of diﬀerentiation:
�
�
ˆ
D
ˆ
ˆw
ˆ
ˆ
� ∂ wk , iDk wk � +...

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