= Solution
Write the flux as
$$
A^i(x,z,\eta)=a^{ij}(x,z,\eta)\eta_j.
$$
After expanding the <divergence>, the <principal symbol of a partial differential equation> along a candidate solution $u$ is determined by
$$
A^{ik}(x,u,Du)=\frac{\partial A^i}{\partial\eta_k}
=a^{ik}+\frac{\partial a^{ij}}{\partial\eta_k}D_ju.
$$
Only the symmetric part $A_s=(A+A^T)/2$ contributes to $A^{ik}D_{ik}u$. The problem is elliptic along $u$ when $\xi^TA_s\xi\geq0$ for every $\xi$, and it is strictly elliptic where this quantity is positive for every nonzero $\xi$. It is uniformly elliptic on a set when constants $0<\lambda\leq\Lambda<\infty$, independent of the point, satisfy
$$
\lambda|\xi|^2\leq\xi^TA_s(x,u,Du)\xi\leq\Lambda|\xi|^2.
$$
These definitions separate pointwise positive definiteness from a quantitative lower and upper bound; a <degenerate elliptic operator> may lose strict ellipticity at some jets.
Solved by gpt-5.6-sol high.
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