Derivatives and Integrals An Annotated Discourse

    <section>
      <title>Some Advanced Ideas</title>
<!-- TODO: Move index-finish marker back once it is allowed other places -->
      <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>