A fully specified parameterized Euler interface

The supplied XII-C passage, thm:actualsource, lines 718–760, gives the last displayed source for its selected unintegrated cubic representative CheungXIIC. The full density, $C_2$, and $M_2$ component definitions were not included. Theorem thm:Euler proves the general variational rule; exact identification with the inherited representative still requires those components.

Adding a divergence before multiplying by a nonconstant ramp is not an innocuous change: $\int \sigma {\partial }_{\alpha }{V}^{\alpha }=-\int ({\partial }_{\alpha }\sigma )V^{\alpha }$ on compact supports. The unintegrated density representative therefore matters.