Derivatives and Integrals: An Annotated Discourse

<section>
  <title>Some Advanced Ideas</title>

  <p>
    <idx xml:id="index-finish-multicolumn" start="index-start-multicolumn" />The multi-row displayed mathematics in the proof of the Fundamental Theorem had equations aligned on the equals signs via the &amp;
    character.
    Sometimes you don't want that.
    Here is an example with some differential equations,
    with each equation centered and unnumbered,
    <md>
      <mrow>{\mathcal L}(y')(s) = s {\mathcal L}(y)(s) - y(0) = s Y(s) - y(0)</mrow>
      <mrow>{\mathcal L}(y'')(s) = s^2 {\mathcal L}(y)(s) - sy(0) - y'(0)= s^2 Y(s) - sy(0) - y'(0)</mrow>
    </md>.
    <notation>
      <usage><m>\rho</m></usage>
      <description>this symbol could be used for lots of things, but we are just trying to make a super-long description to get it to wrap within the column where it belongs, which is sometimes set to a fixed width to accomodate really complicated explanations</description>
    </notation>
        <idx>rho, a test</idx>
    Just prior to this sentence, in the middle of this paragraph, is an <tag>idx</tag> and a <tag>notation</tag>, adjacent, but separated by some whitespace in the authored source.
    That insignificant whitespace will be removed akways,
    which will be a (slightly) noticeable improvement in the <latex /> output.
    We test referencing notation here,
    placed <em>before</em> the sentence-ending period and right after some inline mathematics<mdash />for <m>\mathbb{Z}_n</m>
    <notation>
      <usage><m>\mathbb{Z}_n</m></usage>
      <description>(ring of) integers modulo <m>n</m></description>
    </notation>
    .
  </p>
  <p>
    <latex /> has a device where you can interrupt a sequence of equations with a small amout of text and preserve the equation alignment on either side.
    Here are two tests of that device,
    with aligned equations and non-aligned equations.
    Study the source to see use and differences.
    (The math does not make sense.)
  </p>
  <p>
    Aligned and numbered first.
    <mdn>
      <mrow>{\mathcal L}(y')(s)  &amp;= s {\mathcal L}(y)(s) - y(0) = s Y(s) - y(0)</mrow>
      <mrow>{\mathcal L}(y'')(s) &amp;= s^2 {\mathcal L}(y)(s) - sy(0) - y'(0)= s^2 Y(s) - sy(0) - y'(0).</mrow>
      <intertext>And so it follows that,</intertext>
      <mrow>{\mathcal L}(y')(s)^{++}  &amp;= s {\mathcal L}(y)(s) - y(0) = s Y(s) - y(0)</mrow>
      <mrow>{\mathcal L}(y'')(s)^{5} &amp;= s^2 {\mathcal L}(y)(s) - sy(0) - y'(0)= s^2 Y(s) - sy(0) - y'(0)</mrow>
    </mdn>.
  </p>
  <p>
    Now with no numbers and no alignment.
    We include two cross-references in the <c>intertext</c> portion for testing.
    <md>
      <mrow>{\mathcal L}(y')(s)  = s {\mathcal L}(y)(s) - y(0) = s Y(s) - y(0)</mrow>
      <mrow>{\mathcal L}(y'')(s) = s^2 {\mathcal L}(y)(s) - sy(0) - y'(0)= s^2 Y(s) - sy(0) - y'(0).</mrow>
      <intertext>First an external reference to <url href="http://example.com" visual="example.com" /> and internal cross-reference to <xref ref="corollary-FTC-derivative" text="type-global" />. And so it follows that,</intertext>
      <mrow>{\mathcal L}(y')(s)^{++} = s {\mathcal L}(y)(s) - y(0) = s Y(s) - y(0)</mrow>
      <mrow>{\mathcal L}(y'')(s)^{5} = s^2 {\mathcal L}(y)(s) - sy(0) - y'(0)= s^2 Y(s) - sy(0) - y'(0)</mrow>
    </md>.
  </p>
  <p>
    Tables can get quite complex.
    Simple ones are simpler,
    such as this example of numerical computations for Euler's method in just a bit.
  </p>
  <p>
    But first we make a figure with two very simple tables next to each other.
    This causes the very first instance of <tag>table</tag> to actually be a
    <q>subtable</q>, which exposes a bug provoked by Emiliano Vega and fixed around 2020-08-06. (So we have to place this early to create the same behavior that exposed the bug.)
  </p>
  <figure>
    <caption>Buggy sub-tables</caption>
    <sidebyside>
      <table>
        <title>First</title>
        <tabular>
          <row>
            <cell>One</cell>
          </row>
        </tabular>
      </table>
      <table>
        <title>Second</title>
        <tabular>
          <row>
            <cell>Two</cell>
          </row>
        </tabular>
      </table>
    </sidebyside>
  </figure>
  <table xml:id="table-euler1">
    <title>Euler's approximation for Duffing's Equation with <m>h = 0.2</m></title>
    <tabular top="major" halign="center">
      <row bottom="minor">
        <cell><m>i</m></cell>
        <cell><m>t_i</m></cell>
        <cell><m>x_i</m></cell>
        <cell><m>y_i</m></cell>
      </row>
      <row>
        <cell>0</cell>
        <cell>0.00</cell>
        <cell>0.0000</cell>
        <cell>0.5000</cell>
      </row>
      <row>
        <cell>1</cell>
        <cell>0.20</cell>
        <cell>0.1000</cell>
        <cell>0.4800</cell>
      </row>
      <row>
        <cell>2</cell>
        <cell>0.40</cell>
        <cell>0.1960</cell>
        <cell>0.4560</cell>
      </row>
      <row>
        <cell>3</cell>
        <cell>0.60</cell>
        <cell>0.2872</cell>
        <cell>0.4295</cell>
      </row>
      <row>
        <cell>4</cell>
        <cell>0.80</cell>
        <cell>0.3731</cell>
        <cell>0.4027</cell>
      </row>
      <row>
        <cell>5</cell>
        <cell>1.00</cell>
        <cell>0.4536</cell>
        <cell>0.3783</cell>
      </row>
      <row>
        <cell>6</cell>
        <cell>1.20</cell>
        <cell>0.5293</cell>
        <cell>0.3591</cell>
      </row>
      <row>
        <cell>7</cell>
        <cell>1.40</cell>
        <cell>0.6011</cell>
        <cell>0.3480</cell>
      </row>
      <row>
        <cell>8</cell>
        <cell>1.60</cell>
        <cell>0.6707</cell>
        <cell>0.3474</cell>
      </row>
      <row>
        <cell>9</cell>
        <cell>1.80</cell>
        <cell>0.7402</cell>
        <cell>0.3603</cell>
      </row>
      <row bottom="medium">
        <cell>10</cell>
        <cell>2.00</cell>
        <cell>0.8123</cell>
        <cell>0.3900</cell>
      </row>
    </tabular>
  </table>
</section>