Implementing label, tag and eqref in KaTeX using a macro approach

Link to Eintein \eqref{eins}, and VolSolRev \eqref{volsolrev} and Vol2Curves \eqref{vol2curves} and first line \eqref{gat} or even the second line \eqref{got} and Imaginary \eqref{imag} and Calculus \eqref{calc}, Newton's Law \eqref{newtlaw}, but Newton's Second Law is not like this: \eqref{newtlaw2} and an equation with split \eqref{splitone2} and binomials \eqref{binom2}

Binomials, anyone? \begin{equation} g\left(k\right) = \binom{n}{k} p^k\left(1-p\right)^{n-k} \label{binom2} \end{equation}

A sprinkling of Euler: \begin{equation} \begin{split} e^{i\pi} + 1 &= 0 \\ e^{i\pi} &= -1 \end{split} \end{equation}

and imaginary: \begin{equation} i^2 = -1 \label{imag} \end{equation}

Einstein concluded (after considering Newton's 1st Law \eqref{newtlaw}): \begin{equation} E=mc^2.\label{eins} \end{equation}

Now, gather around, everyone, one with 2 labels: \begin{gather} \text{quantum}=b \label{gat} \\ e=\text{physics} \label{got} \end{gather}

the first line \eqref{gat} or even the second line \eqref{got} of that environment

Of course, he stood on the shoulders of giants, including Newton.

After allowing for imaginary numbers \eqref{imag}, we have an equation without labels or tags: \begin{equation} e^{i\pi} + 1 = 0 \end{equation}

Elsewhere, we'll see how to define the position, velocity and accelration vectors, at \eqref{rva_a}, but first we all need better column alignment: \begin{alignat}{2} 109&x+&3&y=2 \nonumber\\ 3&x+&125&y=4 \tag{alat} \end{alignat} and \begin{equation} \begin{split} \text{an} &= \text{equation} \\ &=\text{with}+\text{split} \label{splitone2} \end{split} \end{equation}

Newton's calculus allowed us to find the general equation for the volume of a solid of revolution for the area bounded by $y=f(x)$ and between $x=a$ and $x=b$, which is: \begin{equation} V_\text{rev} = \pi\int_{a}^b [ {f(x)} ]^2\,dx\label{volsolrev} \end{equation}

Extending volsolrev \eqref{volsolrev} to the case of rotating an area bounded by two curves, we get vol2curves:

\begin{equation} V_\text{rev} = \pi\int_{a}^b [ {y_2}^2-{y_1}^2 ]\, dx \label{vol2curves} \end{equation}

Here is an equation without labels or tags: \begin{equation} \frac{df}{dt}=\lim\limits_{h \to 0}\frac{f(t+h)-f(t)}{h} \end{equation}

Using \eqref{vol2curves}, the volume of the solid of revolution resulting from rotating the area bounded by $y_1=\sin^2(2x)$ and $y_2=\sin^2(x)$ between $x=1$ and $x=3$ is given by: \begin{align} V_\text{rev}&= \pi\int_{a}^b [ {y_2}^2-{y_1}^2 ]\,dx\nonumber\\ &= \pi\int_{1}^3 [ {\sin^2(x)}-{\sin^2(2x)} ]\,dx\nonumber\\ &= \pi(1/8 (2 \sin(2) - \sin(4) - 2 \sin(6) + \sin(12)))\nonumber\\ &\approx 1.0201 \label{calc} \end{align}

We can have a non-numbered environment with a label (which is pmat): \begin{pmatrix} a & b \\ c & d \label{pmat} \end{pmatrix} and we can have velocity \eqref{rva_v}.

We can have another non-numbered environment this time with a tag: \begin{Vmatrix} i & j \\ k & m \tag{Vmat} \end{Vmatrix}

Newton determined that :

\begin{equation} \boldsymbol{F}=m\boldsymbol{a} \label{newtlaw} \end{equation} and as an intelligent extension, we could have: \begin{equation} \boldsymbol{F_2}=m_2\boldsymbol{a_2} \label{newtlaw2} \end{equation} and further talked about position at time $t$, \eqref{rva_r}.

We don't necessarily use \eqref{pmat}, or to define the position vector $\boldsymbol{r}$, the velocity vector $\boldsymbol{v}$, and the acceleration vector $\boldsymbol{a}$ by \begin{align} \boldsymbol{r} &= x\boldsymbol{i}+y\boldsymbol{j}+z\boldsymbol{k},\label{rva_r}\\ \boldsymbol{v} &=\dot{x}\boldsymbol{i}+\dot{y}\boldsymbol{j}+\dot{z}\boldsymbol{k}, \label{rva_v}\\ \boldsymbol{a} &=\ddot{x}\boldsymbol{i}+\ddot{y}\boldsymbol{j}+\ddot{z}\boldsymbol{k}. \label{rva_a} \end{align}

But Einstein's idea in \eqref{eins} was a conjecture on acceleration \eqref{rva_a} until proof was found of time dilation effects \eqref{volsolrev}, pmatrices \eqref{pmat}, black holes \eqref{imag} and quantum physics \eqref{gat}.

Then again, most school children learn Newton's Law \eqref{newtlaw}, but Newton's Second Law is not like this: \eqref{newtlaw2}. He preferrred to work on calculus \eqref{calc}

Equations with missing items

These next two would mess up the counting if they came earlier in the page:

Here is an equation with label, but is missing a \nonumber: \begin{align} r&=\frac{a^2-c^2}{a+c\cos\theta}\\& =\frac{a^2(1-\frac{c^2}{a^2}\,)}{a(1+\frac{c}{a}\;\cos\theta)}\nonumber\\& =\frac{a(1-e^2)}{1+e\cos\theta}\nonumber\\& =\frac{k}{1+e\cos\theta}\label{ellippolar} \end{align}

This one has a tag, but is missing a \nonumber: \begin{alignat}{2} 109&x+&3&y=2 \nonumber\\ i&x+&77&j=2c \\ 3&x+&125&y=4 \tag{alat} \end{alignat}

Orphan eqref

Here's an orphan eqref for demo purposes: \eqref{wheresWally} and another \eqref{foo}.