Authoring

Reference for writing content on this site. This page is also the live test of every custom shortcode — if something here renders wrong, the shortcode is broken.

Front matter

Pages use TOML front matter delimited by +++.

+++
title = 'Limits and Continuity'
description = 'One sentence, shown in section listings and search results.'
weight = 30
chapter = 2
+++

Key	Effect
`title`	Page title, sidebar entry, browser tab
`description`	Shown in parent-section listings and search results
`weight`	Sidebar ordering within its section; lower sorts first
`chapter`	Prefixes statement numbers on the page, e.g. Theorem 2.4
`type = 'chapter'`	Renders as a section landing page rather than an article
`draft = true`	Excluded from builds unless `hugo -D`

Start new pages from an archetype rather than by hand:

hugo new content books/calculus/limits.md

hugo new content books/calculus/_index.md --kind chapter

Mathematics

Write LaTeX directly. $...$ is inline, $$...$$ is displayed, and MathJax loads on every page.

The Euler identity $e^{i\pi} + 1 = 0$ sits inline in the sentence. A displayed equation gets its own line:

$$\int_a^b f'(x)\,dx = f(b) - f(a)$$

Long displayed equations scroll within their own box rather than forcing the whole page sideways on a phone.

Because Goldmark passes $...$ through untouched, backslashes and underscores inside mathematics need no escaping: $a_1 + a_2$ gives $a_1 + a_2$, not italics.

Statements

statement renders a numbered definition, theorem, example, or proof. The body is parsed as Markdown, so mathematics and emphasis work inside it:

{{< statement kind="theorem" name="Fundamental Theorem of Calculus" >}}
If $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$, then
$$\int_a^b f(x)\,dx = F(b) - F(a).$$
{{< /statement >}}

Which renders as:

Definition 4 (Continuity at a point)

A function $f$ is continuous at $c$ if $\lim_{x \to c} f(x) = f(c)$ — which requires that $f(c)$ is defined, that the limit exists, and that the two agree.

Theorem 5 (Fundamental Theorem of Calculus)

If $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$, then

$$\int_a^b f(x)\,dx = F(b) - F(a).$$

Proof.

Define $G(x) = \int_a^x f(t)\,dt$. By the first part of the theorem $G' = f$, so $G$ and $F$ are antiderivatives of the same function and differ by a constant. Then $F(b) - F(a) = G(b) - G(a) = \int_a^b f(x)\,dx$.

Example 6

The function $f(x) = 1/x$ is continuous at every $c \neq 0$, and is not continuous at $0$ for the simplest possible reason: $f(0)$ is undefined.

Parameters

Parameter	Purpose
`kind`	`definition`, `theorem`, `lemma`, `corollary`, `proposition`, `example`, `exercise`, `notation`, `remark`, `proof`
`name`	Optional italicised name shown in parentheses after the number
`number`	Explicit number; omit to take the next number on the page
`id`	Explicit HTML id for cross-references; defaults to `kind-number`

Numbering

Every numbered kind shares one counter per page, so a page reads Definition 1, Theorem 2, Example 3. That is the usual textbook convention, and it keeps a cross-reference unambiguous without having to name its kind.

Set the page’s chapter front matter to prefix numbers with the chapter — chapter = 3 gives Theorem 3.2.

Proofs and remarks are never numbered; a proof belongs to the statement above it.

Figures

Figures are pre-rendered SVGs committed to the repository. The site build never runs Asymptote, so deploying needs neither Asymptote nor LaTeX installed.

Keep the .asy source and its .svg output together in the page bundle:

content/books/calculus/riemann-sums/
├── index.md
├── left-sum.asy
└── left-sum.svg

Then reference the SVG by name:

{{< figure src="left-sum.svg"
           title="Figure 1:"
           caption="A left Riemann sum for $f(x) = x^2$."
           source="left-sum.asy" >}}

Parameter	Purpose
`src`	SVG filename; a page resource first, then a site asset
`title`	Bold caption lead-in, e.g. `"Figure 1:"`
`caption`	Caption body; LaTeX between $...$ is rendered
`source`	`.asy` filename to show in a collapsible block
`alt`	Accessible description; falls back to caption, then title
`width`	CSS width, e.g. `"60%"`

Passing source puts an Asymptote source disclosure under the figure — worth doing on any figure a reader might want to adapt. There is a live example on the Asymptote Library page.

A missing src or source fails the build rather than rendering a broken image.

Re-rendering

After editing any .asy under content/:

scripts/render-figures.sh

That renders only sources whose SVG is missing or older than the source. Pass --force to re-render everything, which you want after changing the library’s theme file.

Notices

The theme’s notice shortcode covers asides:

A tip

Use style="tip", "note", "info", "warning", or "primary".

A warning

Reserve warnings for things that will actually cost the reader time.