The free Euler equations are conservative, in the sense they are equivalent to a conservation equation: ∂ ∂ y ∂ ∂ t + ∇ ∇ ⋅ ⋅ F = 0 , or simply in Einstein notation: ∂ ∂ y j ∂ ∂ t + ∂ ∂ f i j ∂ ∂ r i = 0 i , where the conservation quantity y in this case is a vector, and F is a flux matrix. This can be simply proved.