Aligned systems

A single line is not enough. I need several equations stacked, sometimes aligned on the equals, sometimes just centred as a block.

Without an alignment mark, Typst stacks and I get a gathered block:

\begin{gathered}𝑎=𝑏+𝑐 \\ 𝑑=𝑒\end{gathered}

With an alignment mark, the equals signs line up. That is the default for a derivation I want a reader to scan vertically.

\begin{aligned}𝑎 & =𝑏+𝑐 \\ 𝑑 & =𝑒\end{aligned}

Two alignment marks, so a continuation can sit under the right-hand side instead of under the left.

\begin{aligned}𝑎 & =𝑏 & & =𝑐+𝑑 \\ 𝑒 & =𝑓 & & =𝑔\end{aligned}

Inequalities use the same columns. I want the relations stacked, not the variables.

\begin{aligned}𝑎 & <𝑏 & & <𝑐 \\ 𝑑 & \leq 𝑒 & & \leq 𝑓 \\ 𝑥 & \geq 𝑦 & & \geq 𝑧\end{aligned}

A first-order system is the same pattern with primes. I will use this for linearisation later.

\begin{aligned}{𝑥}^{\prime } & =𝑎𝑥+𝑏𝑦 \\ {𝑦}^{\prime } & =𝑐𝑥+𝑑𝑦\end{aligned}

Three unknowns, still one alignment column:

\begin{aligned}{𝑥}^{\prime } & ={𝑎}_{11}𝑥+{𝑎}_{12}𝑦+{𝑎}_{13}𝑧 \\ {𝑦}^{\prime } & ={𝑎}_{21}𝑥+{𝑎}_{22}𝑦+{𝑎}_{23}𝑧 \\ {𝑧}^{\prime } & ={𝑎}_{31}𝑥+{𝑎}_{32}𝑦+{𝑎}_{33}𝑧\end{aligned}

Initial conditions belong on the same stack:

\begin{aligned}{𝑦}^{\prime } & =𝑓(𝑥,𝑦) \\ 𝑦({𝑥}_{0}) & ={𝑦}_{0}\end{aligned}

Lorenz, because three nonlinear lines is where a converter usually gives up.

\begin{aligned}\dot{𝑥} & =\sigma (𝑦-𝑥) \\ \dot{𝑦} & =𝑥(\rho -𝑧)-𝑦 \\ \dot{𝑧} & =𝑥𝑦-\beta 𝑧\end{aligned}

A pendulum is one line of Newton and one line of the small-angle lie I will not tell yet.

\begin{aligned}{\theta }^{\prime\prime }+\left(\frac{𝑔}{𝐿}\right)\sin \theta & =0 \\ {\theta }^{\prime\prime }+\left(\frac{𝑔}{𝐿}\right)\theta & =0\end{aligned}

Van der Pol, so a first-derivative term sits next to a cubic:

{𝑥}^{\prime\prime }-\mu (1-{𝑥}^{2}){𝑥}^{\prime }+𝑥=0

A longer derivation, still one alignment column:

\begin{aligned}{(𝑔\circ 𝑓)}^{\prime }(𝑎) & ={𝑔}^{\prime }(𝑓(𝑎)){𝑓}^{\prime }(𝑎) \\ & ={𝑔}^{\prime }(𝑏){𝑓}^{\prime }(𝑎)\end{aligned}

The product rule stretched out, because I need to see every term survive a line break.

\begin{aligned}{(𝑢𝑣)}^{\prime } & ={𝑢}^{\prime }𝑣+𝑢{𝑣}^{\prime } \\ {(𝑢𝑣𝑤)}^{\prime } & ={𝑢}^{\prime }𝑣𝑤+𝑢{𝑣}^{\prime }𝑤+𝑢𝑣{𝑤}^{\prime }\end{aligned}

Integrating factor for a first-order linear equation, written the way I actually compute it:

\begin{aligned}{𝑦}^{\prime }+𝑃(𝑥)𝑦 & =𝑄(𝑥) \\ {(\mu 𝑦)}^{\prime } & =\mu 𝑄 \\ \mu (𝑥) & ={𝑒}^{\int 𝑃(𝑥)\, \mathrm{d}𝑥} \\ 𝑦 & ={\mu }^{-1}(\int \mu 𝑄\, \mathrm{d}𝑥+𝐶)\end{aligned}

Three-way alignment is just more rows. The middle line is a definition I keep rewriting until it sticks.

\begin{aligned}𝐹(𝑥) & ={\int }_{0}^{𝑥}𝑓(𝑡)\, \mathrm{d}𝑡 \\ {𝐹}^{\prime }(𝑥) & =𝑓(𝑥) \\ 𝐹(0) & =0\end{aligned}

Maxwell in SI, as a block I will need when vector calculus shows up. If the nabla and the dots fall over, the rest of the book is fiction.

\begin{aligned}\nabla \cdot 𝐸 & =\frac{\rho }{{\varepsilon }_{0}} \\ \nabla \cdot 𝐵 & =0 \\ \nabla \times 𝐸 & =-\frac{\partial 𝐵}{\partial 𝑡} \\ \nabla \times 𝐵 & ={\mu }_{0}𝐽+{\mu }_{0}{\varepsilon }_{0}\frac{\partial 𝐸}{\partial 𝑡}\end{aligned}

A trig identity I refuse to compress onto one line:

\begin{aligned}\sin (𝑎+𝑏) & =\sin 𝑎\cos 𝑏+\cos 𝑎\sin 𝑏 \\ \cos (𝑎+𝑏) & =\cos 𝑎\cos 𝑏-\sin 𝑎\sin 𝑏 \\ \tan (𝑎+𝑏) & =\frac{\tan 𝑎+\tan 𝑏}{1-\tan 𝑎\tan 𝑏}\end{aligned}