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For the barotropic equation of state $P=w\rho$, the continuity equation separates asSince $a(t_0)=1$,which is the constant-equation-of-state density scaling.

When $Q=0$, the expanding branch satisfieswhere $H_0=\dot{a}(t_0)>0$. Because $w>-1$, integration givesand hence the flat constant-equation-of-state scale factorIt vanishes atas $|w|<1$ ensures $1+w>0$.

Put $K=8\pi G/(3c^2)$. Differentiating $Q=K\rho a^2-{\dot{a}}^2$ givesThe continuity equation and the Friedmann acceleration equation implyandThese terms cancel, soThis is the First integral of the Friedmann acceleration equation.

For a comoving region, $E=\rho V$ and $V=a^3{V}_0$. The work law givesso differentiation with respect to time yieldsSince $\dot{V}/V=3\dot{a}/a$, division by $V$ gives the cosmological perfect-fluid continuity equation

Let $\hat{b}=B/|B|$. For a constant uniform magnetic field, choose the symmetric gaugewhich has $\nabla \times A=B$ and $A\cdot \hat{b}=0$. With $E=0$, one may take $\varphi =0$. The coordinate $r_{\parallel }=r\cdot \hat{b}$ is then absent from the Lagrangian, so it is an ignorable coordinate. Its generalized momentumis conserved. Hencewhich is the canonical momentum parallel to a uniform magnetic field in this gauge.

In Cartesian components,andThe Euler-Lagrange equation therefore givesUsingand $[\dot{r}\times B{]}_i={\dot{r}}_j({\partial }_i{A}_j-{\partial }_j{A}_i)$, this becomes the Lorentz force

Under the stated gauge transformation, the charged-particle electromagnetic Lagrangian becomesThusThe action changes only by $q[f(t_2)-f(t_1)]$, whose variation vanishes for fixed endpoints. By total-time-derivative invariance of a Lagrangian, the Euler-Lagrange equation is unchanged.

One parse tree can be recorded unambiguously by the nested expressionEvery internal node uses one of the given productions, and the leaves read from left to right areEquivalently, the corresponding leftmost grammar derivation isHence the required word is generated by the grammar.

The length isIndeed, the parse tree has $n$ terminal-production nodes and, as a full binary tree with $n$ leaves, $n-1$ binary-production nodes. This is the Chomsky-normal-form derivation length.

A context-free grammar is in Chomsky normal form when every production is of one of the formswhere $A,B,C$ are nonterminal symbols and $a$ is a terminal symbol. The conventional exceptional production $S\to ϵ$ may be allowed when the language contains the empty word, with $S$ excluded from right-hand sides.

The first two modified stages giveThe correction stage isUsing $A_x{A}_y=A_y{A}_x$,Therefore

On the common eigenvector $v^{(p,q)}$, the eigenvalue of $D$ isSet $a=-\mu {\lambda }_p>0$ and $b=-\mu {\lambda }_q>0$. ThenBoth numerator and denominator are positive, andHencefor every mode and every $\mu >0$. Orthogonal diagonalization now gives $\Vert D{\Vert }_2<1$, so the method is stable without a time-step restriction. This is the unconditionally stable corrected directional diffusion splitting.

At an interior grid point, the five-point finite difference method givesLetwhere the zero Dirichlet boundary values are already incorporated. Stack the unknowns with the $y$ index varying fastest. Using the Kronecker product, defineThen the semidiscrete equation isMoreover,Thus the directional matrices commute, as in the five-point Dirichlet Laplacian as a Kronecker sum.

Let ${\rho }^{\prime }$ and $p^{\prime }$ be the density and pressure perturbations and let $\mathbf{u}$ be the velocity perturbation. The linear acoustics equations about a uniform quiescent state areThe last relation is the homentropic pressure perturbation law.

Taking the scalar product of the momentum equation with $\mathbf{u}$ givesThe continuity and pressure relations implyTherefore the acoustic energy conservation law iswith