SA An Interesting Corollary

Section 4 An Interesting Corollary

Objectives: Fundamental Structures

This is an <objectives> element you are reading, and this is its introduction. This early section has really grown and tries to accomplish many things. Not all of them are listed here.

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Display various “blocks”, fundamental units of the flow.

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More.

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Evermore.

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This concludes the (incomplete) objectives for this section, so now we can carry-on as before.

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This is a cross-reference to one of the objectives above, forced to use the phrase-global form of the text. It should describe the objective as belonging to the section (rather than the objectives), since objectives are one-per-subdivision and are numbered based upon the containing division: Objective 1 of Section 4. For comparison this is the (forced) type-global cross-reference: Objective 4.1.

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The Fundamental Theorem comes in two flavors, where usually one is a corollary of the other.

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Subsection 4.1 Second Version of FTC

Corollary 4.1 (Leibniz, Newton).

Suppose $f(x)$ is a continuous function. Then

\begin{equation} \frac{d}{dx}\definiteintegral{a}{x}{f(t)}{t}=f(x)\text{.}\tag{4.1} \end{equation}

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Proof.

We simply take the indicated derivative, applying Theorem 2.1 at (4.2)

\begin{align} \frac{d}{dx}\definiteintegral{a}{x}{f(t)}{t}&=\frac{d}{dx}\left(F(x)-F(a)\right)\tag{4.2}\\ &=\frac{d}{dx}F(x)-\frac{d}{dx}F(a)\notag\\ &=f(x)-0 = f(x)\text{.}\tag{4.3} \end{align}

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Justification.

A justification, which is one of the variants of a proof.

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Alternate Proof.

You can have multiple proofs, and they can have titles which replace the word “Proof” as a heading. Here we just exercise displayed math with no automatic numbering, and an elective number on the middle equation.

\begin{align} \frac{d}{dx}\definiteintegral{a}{x}{f(t)}{t}&=\frac{d}{dx}\left(F(x)-F(a)\right)\notag\\ &=\frac{d}{dx}F(x)-\frac{d}{dx}F(a)\tag{4.4}\\ &=f(x)-0 = f(x)\notag \end{align}

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The alternative version of the Fundamental Theorem (FTC) in (4.1) is a compact way to express the result.

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For testing purposes, there is a simple bare Sage Cell here.

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Example 4.2. A Mysterious Derivative!

So if we define a function with its variable employed as a limit of integration, like so

\begin{equation*} K(z)=\definiteintegral{345}{z}{x^4\sin(x^2)}{x} \end{equation*}

then we get the derivative of that function so easily it seems like a mystery,

\begin{equation*} \frac{d}{dz}K(z)=z^4\sin(z^2)\text{.} \end{equation*}

That’s it.

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For testing purposes, there is a simple Sage Cell here, buried inside an example that should be a knowl (embedded in the page).

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We test a Sage cell inside a knowl, which should set the value of a variable that will be available to subsequent cells within the knowl.

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Even if you ran the cell at the top of this page, within this knowl the value of the variable c is not known, so the next cell will cause an error.

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The Sage cells on a page will “remember” results computed elsewhere on the page. If you rely on this feature, remind your readers to evaluate all the necessary cells and that they perhaps need to be evaluated in a certain order.

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There are some Sage cells in the previous (knowled) <example>. The results there are restricted to the knowl. In other words, the scope of those cells is the knowl. So if you opened the example and executed the Sage cells there, or if you skipped the example entirely, the next cell should not “know” the values of those variables and will raise an error.

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We cross-reference the example just prior, Example 4.2, to test the simple Sage cells that will now be part of a cross-reference knowl (an external file).

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Claim 4.3. An Equivalent Claim.

This claim is an equivalence: it is true if and only if it is correct.

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Proof.

Our purpose here is to show how you can structure a proof with cases, such as an equivalence structured with the arrows typically used to demonstrate the two “directions” involved in the proof, by using the @direction attribute on a <case> element.

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(⇒)

Nulla non lectus suscipit, bibendum leo quis, dignissim justo. In urna turpis, tincidunt id elementum id, faucibus ac tellus.

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(⇐)

Quisque auctor ligula turpis, ut aliquam urna consectetur hendrerit. Aenean porta dolor et justo facilisis feugiat in sed sapien. Nullam porta ex et commodo semper.

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Case 3b: The inductive step.

A case may also have a title, whose formatting and structure is entirely up to the author. This then becomes the text of a cross-reference, as well.

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Why Not Try This?

A <case> (or any other element with a default title) did not always handle title-ending punctuation correctly. So we try an example title with a question mark.

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(⇒) Necessity.

If you like, you can have both indications.

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Case.

No direction, no title, then just a generic title.

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Exciting Proof!

We test here that punctuation at the end of the title of a proof is handled correctly.

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Exact Proof.

This proof should fill exactly three lines (as of defaults in place 2018-12-31) and so the tombstone/Halmos should be on a fourth line, and then flush right. xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx.

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Claim 4.4. A List of Equivalent Statements.

The following are equivalent.

This statement is equivalent to all those below.

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This statement is equivalent to the statement above and to all those below.

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This statement is equivalent to the two statements above and to the statement below.

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This statement is equivalent to all those above.

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Proof.

Our purpose here is to show how you can structure a proof with cases to address the circular logic required to prove the equivalence of a list of statements, by using the @direction attribute on a <case> element. You should order the list of statements in the order that you would like to prove ``this statement implies the next.’’

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( ⁠(i) ⇒ (ii) ⁠)

Here we would prove that the first statement implies the second.

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( ⁠(ii) ⇒ (iii) ⁠)

Here we would prove that the second statement implies the third.

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( ⁠(iii) ⇒ (iv) ⁠) The trickiest case.

This time we include a title to describe the nature of this case. But we would still need to prove that the third statement implies the last.

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( ⁠(iv) ⇒ (i) ⁠) Wrap-around.

Finally, we would complete the cycle of logic by proving that the last statement implies the first.

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We can also use @direction set to cycle in a stand-alone proof of our TFAE claim. If we include a @ref on the <proof> that points to the original claim, then the formatting of the markers on the statement list will be honored in our cases.

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Proof. (4.4)

Once again we will prove that the four statements in 4.4 are equivalent.

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( ⁠(i) ⇒ (ii) ⁠)

Another argument that the first statement implies the second.

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( ⁠(ii) ⇒ (iii) ⁠)

And another argument that the second statement implies the third.

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( ⁠(iii) ⇒ (iv) ⁠) Not so tricky this time.

Why did we find it so difficult before to prove that the third statement implies the last?

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( ⁠(iv) ⇒ (i) ⁠) Wrap-around.

And once more we complete the cycle of logic.

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A couple more times to check that the default list markers get applied to the directional cases properly.

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Claim 4.5. Another List of Equivalent Statements.

The <ol> that creates the list below does not have @marker.

This statement is equivalent to the two below.

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This statement is equivalent to both the statement above and to the statement below.

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This statement is equivalent to the two above.

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Proof.

You know the drill by now.

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( ⁠1 ⇒ 2 ⁠)

Does the first statement imply the second?

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( ⁠2 ⇒ 3 ⁠) The “middle” case.

Does the second statement imply the third?

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( ⁠3 ⇒ 1 ⁠)

And finally, does the third imply the first?

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This proof includes a @ref to the preceding claim.

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Proof. (4.5)

( ⁠1 ⇒ 2 ⁠)

Does the first statement imply the second?

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( ⁠2 ⇒ 3 ⁠) The “middle” case.

Does the second statement imply the third?

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( ⁠3 ⇒ 1 ⁠)

And finally, does the third imply the first?

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This proof does not include a @ref, and so the direction indicators get default markers.

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Proof.

( ⁠1 ⇒ 2 ⁠)

Does the first statement imply the second?

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( ⁠2 ⇒ 1 ⁠)

And finally, does the second imply the first?

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Subsection 4.2 A Pedagogical Note about Subsection 4.1

Subsubsection 4.2.1 Symbolic and Numerical Integrals

The Fundamental Theorem explains why we use the same notation for a definite integral, which is a numerical calculation,

Which I think sometimes students lose sight of.

and an antiderivative, which is a symbolic expression.

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Checkpoint 4.6. Essay Question: Compare and Contrast.

Write a short paragraph which compares, and contrasts, the definite and indefinite integral. This is an exercise which sits in the midst of the narrative, so is formatted more like an example or a remark. It can have a hint and a solution, but this one does not. It can have a title, which this one does.

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Hint.

Start writing!

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Subsubsection 4.2.2 Advice

Using an “integral sign” for an antiderivative (aka indefinite integral) would seem to make the Fundamental Theorem a fait accompli. So I would suggest not conflating the notation for two very different things until the Fundamental Theorem exposes them as being highly related.

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Example 4.7. An Example of Structure.

This is an example of an example with a bit more structure. Specifically, the example has a title, as usual, but then has a statement, which is separate from the solution. Why did we implement an example in two ways?

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Solution.

Authors asked for it and it seemed a very natural thing to do, even if we only had an unstructured version for a long time.

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Question 4.8. An Example of a Question.

Any kind of question can be marked as such with <question>. Or similarly, as a <problem>. They behave identically to examples, such as the one preceding and are numbered along with theorems, examples. etc.

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Solution 1.

You can have a solution. Or several, even if you don’t ask a question.

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Solution 2.

See?

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Checkpoint 4.9. An Inline Exercise.

There are lots of exercises in this sample article, but mostly they are in special exercise sections. Sometimes you just want to sprinkle some exercises through the narrative. We call these inline exercises, in contrast to divisional exercises. The inline exercises look a bit more like a theorem or definition, with titles and fully-qualified numbers.

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These may also have hints, answers and solutions.

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Hint.

A good hint.

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Answer.

42.

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Solution.

If your exercise feels like proving a theorem, then you might want to make some comments, but also clearly delineate which part of the solution is a the complete proof.

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Proof.

Lorem ipsum dolor sit amet, consectetur adipiscing elit. Proin lorem diam, convallis in nulla sed, accumsan fermentum urna. Pellentesque aliquet leo elit, ut consequat nunc dapibus ac. Sed lobortis leo tincidunt, vulputate nunc at, ultricies leo. Vivamus purus diam, tristique laoreet purus eget, mollis gravida sapien. Nunc vulputate nisl ac mauris hendrerit cursus. Sed vel molestie velit. Suspendisse sem sem, elementum at vehicula id, volutpat ac mi. Nullam ullamcorper fringilla purus in accumsan. Mauris at nunc accumsan orci dictum vulputate id id augue. Suspendisse at dignissim elit, non euismod nunc. Aliquam faucibus magna ac molestie semper. Aliquam hendrerit sem sit amet metus congue tempor. Donec laoreet laoreet metus, id interdum purus mattis vulputate. Proin condimentum vitae erat varius mollis. Donec venenatis libero sed turpis pretium tempor.

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Praesent rutrum scelerisque felis sit amet adipiscing. Phasellus in mollis velit. Nunc malesuada felis sit amet massa cursus, eget elementum neque viverra. Integer sagittis dictum turpis vel aliquet. Fusce ut suscipit dolor, nec tristique nisl. Aenean luctus, leo et ornare fermentum, nibh dui vulputate leo, nec tincidunt augue ipsum sed odio. Nunc non erat sollicitudin, iaculis eros consequat, dapibus eros.

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Example 4.10. An Example of with $\frac12$ math formula $\displaystyle{\int e^x \, dx}$ in the title.

Just for testing math in knowls, and also extra whitespace in a <p>.

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There are many different blocks you can employ, and they mostly behave the same way. A <project> is very similar to a <question> or <problem>

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Project 4.1. Start Exploring PreTeXt.

You could grab the minimal.xml file from the examples/minimal directory and experiment with that.

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Projects get their own independent numbering scheme, since they may be central to your textbook, workbook, or lab manual. If you process this sample article with level for project numbering set to 0 then you will get consecutive numbers from the beginning of your book, starting with 1.

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Exploration 4.2. Exploring Explorations.

This is an <exploration>. Other similar possibilities are <project>, <activity>, <task>, and <investigation>.

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Note that projects, activities, explorations, tasks and investigations share the independent numbering scheme, so it is really only intended you use one of these. If you want a variant of the name (e.g. “Directed Activity”) you can use the <rename> facility (Subsection 32.1).

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Solution.

This is a “solution” to the exploration. In practice, you might choose to not make this visible for students, but instead include it as part of some guidance you might provide to instructors (e.g. an Instructor’s Manual).

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This is quite the activity upcoming. This is a prelude authored within the activity element, but visually just prior.

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Activity 4.3. Hints, Answers, Solutions.

Another variant of these project-like items is to possibly include a <hint> and an <answer> before the <solution>.

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Hint.

Just a little help.

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Answer.

The result, but no help in getting there.

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Solution.

Everything to get it all done, in detail.

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This was quite the activity just now. This is a postlude authored within the activity element, but visually just after.

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Note 4.11. A Note on Remarks.

<remark>, <convention>, <note>, <observation> and <warning> are designed to hold very simple contents, with no additional structure (no proofs, no solutions, etc.).

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But they do carry a title and a number, can be the target of a cross-reference, and may be optionally knowlized in HTML with the html.knowl.remark processing switch.

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And distinctly different from a <note> in a <biblio>

A gratuitous footnote to test prior bug confusing this with a <note> in a <biblio>.

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Aside: An Aside with a Formatted Title.

An <exercise> can be structured with <task>.

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Checkpoint 4.12. A very structured exercise.

This is an over-arching introduction to the whole exercise. We follow with some tasks. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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(a) A super-simple task.

This first task is very simple, just a paragraph. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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(b)

Now three paragraphs. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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(c) A title of a task that has a subtask with an `<answer>` for the Solutions.

This second task is further divided by more tasks. This is its introduction. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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(i) A task with a title and an `<answer>` for the Solutions.

A really simple subtask. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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A short paragraph, before an answer.

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Answer.

With a proof.

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Proof.

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And a bit more to say.

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(ii)

A subtask with an answer. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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Answer.

Right! In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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(iii)

Three simple sub-sub-tasks. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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(A)

First subsubtask. Short paragraph.

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(B) A second three-deep subsubtask!

Second subsubtask. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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(C)

Third subsubtask. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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In interdum suscipit ullamcorper

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The conclusion of the structured subtask. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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(iv)

A simple task as the last subtask. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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This concludes our structured second task. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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(d)

This third top-level task is intermediate in complexity, you are reading the statement, which is followed by more items. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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Hint.

One hint. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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Answer 1.

First answer. In interdum suscipit ullamcorper.

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Answer 2.

Second answer. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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Solution.

At last, the solution. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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This is a conclusion where you could summarize the exercise. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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The following <project> is nearly identical to the preceding <exercise>.

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The next block is a project, demonstrating the use of the task element to structure its parts. You are reading the prelude now. The project has lots of nonsense words, so we can test spacing the nested items. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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Project 4.4. A very structured project.

This is an over-arching introduction to the whole project. We follow with some tasks. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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A short paragraph, before an answer.

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Answer.

With a proof.

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Proof.

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And a bit more to say.

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(ii)

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Answer.

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(iii)

Two simple sub-sub-tasks. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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(A)

First subsubtask. Short paragraph.

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(B)

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(C)

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In interdum suscipit ullamcorper

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(iv)

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(d)

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Hint.

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Answer 1.

First answer. In interdum suscipit ullamcorper.

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Answer 2.

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Solution.

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This is a conclusion where you could summarize the project. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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This postlude appears visually outside the project, but is authored within, to make clear its attachment to the project. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

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The following <example>, from Elise Desgreniers, is structured with <task>.

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Example 4.13. Notation mathématique 2.

Écrivez chacun des exemples suivants avec les conditions pertinentes.

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(a)

Soit l’ensemble $A=\lbrace 1,2,3,4,5\rbrace\text{.}$

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On constate que cet ensemble contient uniquement des entiers positifs allant de 1 à 5.

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Donc, on peut écrire $A=\lbrace x\mid x\in\mathbb{N}\text{ et } 1\leq x\leq 5\rbrace\text{.}$

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(b)

$\{3,6,9,12,15,\ldots,27,30\}$

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Hint.

Ce sont des multiples de 3.

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Answer.

$\{x\mid x=3y \text{ et } 1\leq y\leq 10 \text{ et } y\in\mathbb{N}\}$

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(c)

$\{1,3,5,7,9,11,\ldots\}$

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Hint.

Ce sont des nombres impairs.

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Answer.

$\{x\mid x\mod 2=1 \text{ et } x\in\mathbb{N}\}$

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(d)

$\{2,3,5,7,11,13,17,19,23,\ldots\}$

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Hint.

Ce sont des nombres premiers.

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Answer.

$\{x\mid x\text{ est un nombre premier positif}\}$

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(e)

$\{1,4,9,16,25,36,\ldots,961\}$

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Hint.

Ce sont des carrés parfaits.

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Answer.

$\{x\mid x=y^2 \text{ et } 1\leq y\leq 31\text{ et } y\in\mathbb{N}\}$

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(f)

$\{1,8,27,64,125,\ldots\}$

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Hint.

Ce sont des cubes parfaits.

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Answer.

$\{x\mid x=y^3 \text{ et } y\in\mathbb{N^*}\}$

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Notes or examples related to computation or technology can go in blocks of the same name.

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Technology 4.14. Sample Use of Sage.

This would be a good place to talk about Sage, including a cell or two.

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But you might want to describe how to use some other calculator, or maybe some numerical method.

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A `<paragraphs>` with a `<project>` with an `<answer>`.

The solutions to a project (and similar) once did not migrate to the automatically-generated solutions.

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Project 4.5.

A simple project, no tasks, just an answer.

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Answer.

Here’s the answer we are looking for.

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Exercises 4.2.3 Exercises

1.

This is an exercise in an “Exercises” subdivision at the level of a subsubsection. There is no question other than if the numbering is appropriate. Here is a self-referential link: Exercise 4.2.3.1.

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The subsubsection has no title in the source, so one is provided automatically, and will adjust according to the language of the document.

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Solution.

This solution will migrate to a list of solutions in the backmatter. We include a sidebyside as a test.

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This is a skinny paragraph which should be just 30% of the width.

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And another skinny paragraph which should also be just 30% of the width.

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2.

An <exercise> can be structured with parts, called <task>. This is the <introduction>.

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(a)

Do this.

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(b)

And the other thing.

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3. STACK integration question.

A test question. (2025-10-17: you will need to ask for a “new” question to make this render.)

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Reading Questions 4.2.4 Reading Questions

A set of reading questions may have an <introduction>, perhaps for preparatory explanation.

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If a student has logged in to the HTML version, then they can answer the reading questions directly in the book. Inline math LaTeX can be entered using $...$ or $...$ delimiters, and inline AsciiMath using backticks `...` as delimiters. Here are some `gratuitous backticks` to check that AsciiMath is only active in the answers to reading questions.

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1.

This is a reading question that you might have a student answer prior to a class session, based on reading part of the book. A quick glance before class can help you tailor class time to the specific needs of your students. The perfect reading question will reveal whether the student has read and understood the material, and will be difficult to answer if they have not. What do you think of that?

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2.

And a second one, with a cross-reference to the first, as a check on numbering: Reading Question 4.2.4.1. Reading questions are allowed to have answers, but providing answers misses the point of a reading question, and the answer knowl interacts poorly with the mechanism used to allow students to answer directly in the book. Do you think the schema should ban answers to reading questions?

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And for symmetry, a <conclusion>.

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Glossary 4.2.5 Glossary

A glossary may have a <headnote>, perhaps with some explanation. This glossary is a specialized division within a section. Placement in the back matter is another option, see the Glossary.

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bar. 🔗: A part of foobar. See foobar.
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foobar. 🔗: A synonym for the acronym FUBAR.
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Solutions 4.2.6 Solutions for This Subsection

This is an introduction, where you might explain that this division of this subsection contains various hints, answers, solutions of inline exercises, divisional exercises, and/or project-like blocks. See the source to see just how this solutions division was built.

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4.2.1 Symbolic and Numerical Integrals

Checkpoint 4.6. Essay Question: Compare and Contrast.

Hint.

Start writing!

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4.2.2 Advice

Checkpoint 4.9. An Inline Exercise.

Hint.

A good hint.

🔗

Answer.

42.

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Solution.

If your exercise feels like proving a theorem, then you might want to make some comments, but also clearly delineate which part of the solution is a the complete proof.

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Proof.

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Exploration 4.2. Exploring Explorations.

Solution.

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Activity 4.3. Hints, Answers, Solutions.

Hint.

Just a little help.

🔗

Answer.

The result, but no help in getting there.

🔗

Solution.

Everything to get it all done, in detail.

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Checkpoint 4.12. A very structured exercise.

4.12.c A title of a task that has a subtask with an `<answer>` for the Solutions.

4.12.c.i A task with a title and an `<answer>` for the Solutions.

Answer.

With a proof.

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Proof.

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And a bit more to say.

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4.12.c.ii

Answer.

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4.12.d

Hint.

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Answer 1.

First answer. In interdum suscipit ullamcorper.

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Answer 2.

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Solution.

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Project 4.4. A very structured project.

4.4.c

4.4.c.i

Answer.

With a proof.

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Proof.

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And a bit more to say.

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4.4.c.ii

Answer.

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4.4.d

Hint.

🔗

Answer 1.

First answer. In interdum suscipit ullamcorper.

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Answer 2.

🔗

Solution.

🔗

Project 4.5.

Answer.

Here’s the answer we are looking for.

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4.2.3 Exercises

4.2.3.1.

Solution.

This solution will migrate to a list of solutions in the backmatter. We include a sidebyside as a test.

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This is a skinny paragraph which should be just 30% of the width.

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And another skinny paragraph which should also be just 30% of the width.

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And a conclusion to this solutions division, which may not be readily apparent as distinct from the final division’s worth of solutions, but since it is not prefixed with a number, it may be different enough.

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Subsection 4.3 Theorem-Like Environments

There are a variety of pre-defined environments in PreTeXt. All take a title, and must have a statement. Some have proofs (theorems, corollaries, etc.), while some do not have proofs (conjectures, axioms, principles).

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Principle 4.15. The Title Principle.

It is a fundamental principle that many elements can have a title. Try it and see. If you get better formatting, then it is being recognized. If it looks very plain, check the documentation and perhaps make a feature request.

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More precisely, <theorem>, <corollary>, <lemma>, <algorithm>, <proposition>, <claim>, <fact>, and <identity>, all behave exactly the same, requiring a statement (as a sequence of paragraphs) followed by an optional proof, and may have an optional title. The elements <axiom>, <conjecture>, <principle>, <heuristic>, <hypothesis>, and <assumption> are functionally the same, barring a proof (since they would never have one!). Definitions are an exception, as it is natural to place <notation> within—see the source for Definition 2.2 for an example.

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Subsection 4.4 Linking Sage Cells

Sage cells share their results on a per-webpage basis, or a per-knowl basis, so if you move to a new chapter, section, or subsection that happens to be on another webpage, your Sage computations are gone and you start fresh. But maybe you need some results from elsewhere. As an author, you can make an exact copy of a cell in another location by placing the code in an external file, which is pure text, freed from any need to format for XML processing. So, in particular, there is no need to escape ampersands and angle brackets, nor is there employment of the CDATA mechanism. But the real value is that there is just one version to edit, and any changes will be reflected in both copies. We demonstrate this in the sample book, since it has the xinclude mechanism in place. In the chapter on groups, find the section on Sage and then find the discussion of subgroups, and you will find an example of two identical Sage cells produced from one source file.

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You can also specify certain cells to be auto-evaluated, by setting the @auto-evaluate attribute to yes. The resulting cell will not have a button for evaluation (so editing it would be pointless). See the source of this sample article for the two examples below.

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2023-08-17: support just now is for the use case of a small portion of code, not a huge library of helper routines.

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Two cells with the default language sage.

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Two cells with language python.

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Subsection 4.5 Hierarchy

Structure.

This section of this article has subsections and subsubsections. In a book you can have chapters enclosing multiple sections. There is one finer subdivision, it is achieved with the paragraphs element.

🔗

It is basically a sequence of paragraphs, where the first one gets an inline title. You are reading the second, and final, paragraph of one right now. It is useful for organizing very short documents, where numbered subdivisions might be overkill.

🔗

A Second Paragraphs.

This is a second consecutive paragraphs element, so should seem related to its title, but distinct from the two paragraphs in the grouping with the title “Structure” immediately prior.

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Assemblages: Collections and Summaries.

An <assemblage> is a collection, or summary, that does not have much structure to it. So you are limited to paragraphs and friends (p, blockquote, pre) and side-by-sides that do not contain captioned items (sidebyside, sbsgroup). The intent is that contents are not numbered, so cannot be cross-referenced individually, and so also do not become knowls. You may place <image>, <tabular>, and <program> inside a <sidebyside>, in addition to other objects that do not have captions. Note that p may by extension contain lists (ol, ul, dl). Despite limited structure, the presentation should draw attention to it, because the contents should be seen as more important in some way. It should be “highlighted” in some manner. If you need to connect the entire assemblage with material elsewhere, you can do that with the usual xref/xml:id mechanism.

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What have we seen so far in this (disorganized) sample?

Theorems, definitions and corollaries. (Section 2)
🔗

🔗
Sage cells, including with R. (Section 3)
🔗

🔗
Lots of document structure, like introductions and conclusions (next). (Section 4)
🔗

🔗

🔗

A sample table, as a tabular inside a sidebyside with no caption, follows.

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A	B	C
Uno	Dos	Tres

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This is a small assemblage with no title, simply to make sure the surrounding box behaves properly, especially for LaTeX output.

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Assemblages containing $\mu \forall \tau \mathbb{H} = \emptyset \kappa$.

It is acceptable for an assemblage to contain mathematical content, even in its title.

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Subsection 4.6 Introductions and Conclusions

An Introductory Introduction.

Any subdivision may have a sequence of paragraphs within an <introduction> that precedes subsequent further subdivisions. You are reading one now. They are always leaves of the document structure, so are rendered on some pages that reference the following subdivisions.

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An introduction or conclusion is an extremely restrictive container with simple presentation. A title is optional (and probably not advisable). Content is meant to be short and unstructured, in particular, nothing that can be numbered is allowed. If this feels too restrictive, then place your content in an initial numbered subdivision and perhaps title it “Introduction”. Or make your entire subdivion unstructured and place whatever you want into it.

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This ends this introduction to introductions.

🔗

Subsubsection 4.6.1 Test One

An intervening subsubsection just after an introduction.

🔗

Subsubsection 4.6.2 Test Two

An intervening subsection section which contains an <exercises> division which must be at the level of a Subsubsubsection.

🔗

Checkpoint 4.16.

An inline exercise to examine any clash with divisional exercises below.

🔗

Answer.

An answer so there is something to appear in a <solutions>.

🔗

Reading Questions What Did You Learn?

1.

A mock exercise to appease validation.

🔗

Answer.

An answer so there is something to appear in a <solutions>.

🔗

2.

And a second to help with formatting the division heading.

🔗

Exercises Exercises

1.

A mock exercise to appease validation.

🔗

Answer.

An answer so there is something to appear in a <solutions>.

🔗

2.

And a second to help with formatting the division heading.

🔗

Subsubsection 4.6.3 Test Three

An intervening subsubsection just before a conclusion.

🔗

Entirely analogous to introductions are conclusions. Any subdivision may have a sequence of paragraphs within a <conclusion> that follows previous further subdivisions. You are reading one now. They are always leaves of the document structure, so are rendered on some pages that reference the preceding subdivisions.

🔗

This concludes this conclusion (and this subsection and this section).

🔗

Subsection 4.7 Some Paragraph-Level Markup

Text within a paragraph may be emphasized with <em> or if you want to take it to the next level you can identify the text as an alert with <alert>.

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Similarly, within a paragraph, you can identify edits between versions as inserted text that has been added with <insert> or as ~~deleted text that has been removed~~ with <delete>. Note that these identified edits are slightly different than ~~stale text that you want to retain, but which is no longer relevant~~, which is accomplished with <stale>. The original request for stale text came from an instructor with an online list of student topics for presentations, and as students claimed topics they were marked as no longer available for other students.

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If you need a “fill-in blank”, like this , it can be obtained with an empty <fillin> element that defaults to roughly a 10-character width. You can use the @characters attribute to make the rule longer or shorter, such as a 40-character blank: . The character count is approximate, based on typical character widths within a proportional font carrying English language text. Adjust to suit, or request a language-specific adjustment if it is critical.

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This paragraph is intended to make a <fillin> appear right at the start of the second line in print and then the next paragraph has nothing but a <fillin>. Both are for testing purposes.

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The following are <fillin> with @rows and/or @cols attributes (at least one of which is greater than 1): (2 × 1 array), (1 × 3 array), (2 × 3 array).

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Long after we started this mess, we added PreTeXt tags to mark up tags and attributes. The elements are: <tag>, <tage>, <attr>. Examples of how these render are (respectively): <section>, <hash/>, @width. Perhaps this document will make greater use of these tags.

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We supply two provisional cross-references for testing purposes only: [provisional cross-reference: a first incomplete cross-reference], [provisional cross-reference: a second incomplete cross-reference].

🔗

A conclusion here, which we fill with some numbering tests.

🔗

This is a cross-reference to one of the outcomes, forced to use the type-global form of the text. It should describe the outcome as belonging to the section (rather than the outcomes), since outcomes are one-per-subdivision and are numbered based upon the containing division: Outcome 2 of Section 4. For comparison this is the (forced) type-global cross-reference: Outcome 4.2.

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Outcomes: Fundamental Structures, Revisited

This is a <outcomes> element you are reading, and this is its introduction. This early section has really grown and we have tried to accomplish many things. Not all of them are listed here.

🔗

Display various “blocks”, fundamental units of the flow.

🔗
More, and this is what the cross-references above are pointing to.

🔗
Evermore.

🔗

This concludes the (incomplete) outcomes for this section, so now we can carry-on to the next section.

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Prev Top Next

Section 4 An Interesting Corollary

Objectives: Fundamental Structures

Subsection 4.1 Second Version of FTC

Corollary 4.1 (Leibniz, Newton).

Proof.

Justification.

Alternate Proof.

Example 4.2. A Mysterious Derivative!

Claim 4.3. An Equivalent Claim.

Proof.

(⇒)

(⇐)

Case 3b: The inductive step.

Why Not Try This?

(⇒) Necessity.

Case.

Exciting Proof!

Exact Proof.

Claim 4.4. A List of Equivalent Statements.

Proof.

( ⁠(i) ⇒ (ii) ⁠)

( ⁠(ii) ⇒ (iii) ⁠)

( ⁠(iii) ⇒ (iv) ⁠) The trickiest case.

( ⁠(iv) ⇒ (i) ⁠) Wrap-around.

Proof. (4.4)

( ⁠(i) ⇒ (ii) ⁠)

( ⁠(ii) ⇒ (iii) ⁠)

( ⁠(iii) ⇒ (iv) ⁠) Not so tricky this time.

( ⁠(iv) ⇒ (i) ⁠) Wrap-around.

Claim 4.5. Another List of Equivalent Statements.

Proof.

( ⁠1 ⇒ 2 ⁠)

( ⁠2 ⇒ 3 ⁠) The “middle” case.

( ⁠3 ⇒ 1 ⁠)

Proof. (4.5)

( ⁠1 ⇒ 2 ⁠)

( ⁠2 ⇒ 3 ⁠) The “middle” case.

( ⁠3 ⇒ 1 ⁠)

Proof.

( ⁠1 ⇒ 2 ⁠)

( ⁠2 ⇒ 1 ⁠)

Subsection 4.2 A Pedagogical Note about Subsection 4.1

Subsubsection 4.2.1 Symbolic and Numerical Integrals

Checkpoint 4.6. Essay Question: Compare and Contrast.

Subsubsection 4.2.2 Advice

Example 4.7. An Example of Structure.

Solution.

Question 4.8. An Example of a Question.

Solution 1.

Solution 2.

Checkpoint 4.9. An Inline Exercise.

Proof.

Example 4.10. An Example of with \(\frac12\) math formula \(\displaystyle{\int e^x \, dx}\) in the title.

Project 4.1. Start Exploring PreTeXt.

Exploration 4.2. Exploring Explorations.

Activity 4.3. Hints, Answers, Solutions.

Note 4.11. A Note on Remarks.

Aside: An Aside with a Formatted Title.

Checkpoint 4.12. A very structured exercise.

(a) A super-simple task.

(b)

(c) A title of a task that has a subtask with an <answer> for the Solutions.

(i) A task with a title and an <answer> for the Solutions.

Proof.

(ii)

(iii)

(A)

(B) A second three-deep subsubtask!

(C)

(iv)

(d)

Project 4.4. A very structured project.

(a)

(b)

(c)

(i)

Proof.

(ii)

(iii)

(A)

(c) A title of a task that has a subtask with an `<answer>` for the Solutions.

(i) A task with a title and an `<answer>` for the Solutions.

A `<paragraphs>` with a `<project>` with an `<answer>`.

4.12.c A title of a task that has a subtask with an `<answer>` for the Solutions.

4.12.c.i A task with a title and an `<answer>` for the Solutions.