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.
    <md number="yes">

      <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>

    </md>

  </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>