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:φ<sub>''v''</sub>(L''u'')=φ<sub>''v''</sub>(''g'')
for all ''v'' in V. From now on, we will use ∫<sub>T</sub> for the double integral ∫<sub>0</sub><sup>1</sup>∫<sub>0</sub><sup>1</sup>. One can see, via [[integration by parts]], and noting that because of the periodic boundary condition the non-integral term
the ''fg'' term
<math>
\int_0^1 \left[ v_x u_x \right]_0^1 dy +\int_0^1 \left[ v_y u_y \right]_0^1 dx
</math>
vanishes, that
<math>
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