= Symmetry of the first-order monochromatic alpha tensor
{title2=$\alpha^{(1)}_{ij}=\alpha^{(1)}_{ji}$}
The first coefficient of the <alpha tensor> is $\alpha^{(1)}_{ij}=\epsilon_{ikl}\langle u_k\partial_j u_l\rangle$. Its contraction with the <Levi-Civita symbol> is
$$
\epsilon_{pij}\alpha^{(1)}_{ij}=\langle u_j\partial_j u_p-u_p\partial_j u_j\rangle=0,
$$
by <incompressibility> and vanishing averages of total derivatives. Every <antisymmetric second-rank tensor> in three dimensions is detected by this contraction, so the coefficient is a <symmetric matrix>. The proof needs no <Fourier analysis>.
// Target: fluid-mechanics.bigb
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