= Source-balanced Riemann invariant
{title2=$\nabla R\cdot S=0$}
For a smooth <hyperbolic system> $U_t+A(U)U_x=S(U)$, suppose $\nabla R$ is a left eigenvector of $A$ with eigenvalue $\lambda$. The chain rule gives $(\partial_t+\lambda\partial_x)R=\nabla R\cdot S$. Hence the homogeneous invariant remains an exact <Riemann invariant> when its gradient annihilates the source. For $U=(u,v)$, $A=\begin{pmatrix}v&u\\u^{-3}&v\end{pmatrix}$, $S=(-u^3,u)$ and $u>0$, the function $R=v-1/u$ has $\nabla R=(u^{-2},1)$, eigenvalue $v+1/u$, and $\nabla R\cdot S=0$. On its zero level, the other characteristic speed vanishes and $v_t=1/v$ along constant spatial positions. Source terms need not destroy every invariant.
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