= Density scaling invariance of incompressible Euler flow
{title2=$\rho\mapsto C\rho,\ p\mapsto Cp$}
A homogeneous fluid in a prescribed gravitational field obeys $D\mathbf u/Dt=-\nabla(p/\rho)+\mathbf g$ and $\nabla\cdot\mathbf u=0$. Multiplying both constant <mass density> and <pressure> by the same positive factor preserves the <velocity> solution, provided initial and boundary data remain compatible. Thus velocity-only observations do not determine density in this model. Independently prescribed pressure gradients or the fluid's own gravity can break this invariance.
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