For the case of zero body forces, the Galerkin vector is biharmonic. However, it was pointed out that in curvilinear coordinate systems, the individual Galerkin vector components might not necessarily be biharmonic. Consider the cylindrical coordinate case where V = V_rer + V_θeθ + V_zez. Using the results of Section 1.9, first show that the Laplacian operator on each term will give rise to the following relations:

Data from section 1.9

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(Ve) = (vv - 1/2) er + (Ver) = Ve x2 (Voco) = (vV - 12) eo Ve 2 av, eo 30 2 ave er 2 0 V (Vez) = VVzez Using these results, show that the biharmonic components are given by 2 4 8v, av, - [ (v-2) v + 82] + [ (-2) dov ** Vr er eg 30 [** - * _ ( - )] + [ ( - )f-]-(~AA = AAA = (3A) zAzA and thus only the component V will satisfy the scalar biharmonic equation.