How to Read This Site - Maths: an open, verified university mathematics reference

Subjects, chapters and topics¶

The site is organised as subject → chapter → topic. A topic page is the unit you read in one sitting, typically 20 to 45 minutes. Each topic lists the topics you should know first (“Before you start”) and the topics that build on it (“Where this leads”), so you can move backwards to fill a gap or forwards to see where an idea is used.

Every topic page has the same sections, in the same order:

Why this matters: the question the topic answers, with a concrete example.
Definitions and results: the precise statements, each followed by an explanation in words and examples.
Worked examples: complete solutions with every step shown, each ending with a check.
Common mistakes: errors that students often make, why they are wrong, and how to fix them.
Rigorous track (on some pages): proofs and subtleties for readers who want full rigour.
Summary: the key points and formulas.
Exercises: tiered practice with hints, answers and full solutions.
Where this leads: the topics that build on this one.

The core text and the rigorous track¶

The core text is written for a first-year student of engineering or science. It gives the motivation, the definitions, the statements of the results, the proofs that build understanding, and plenty of worked examples.

The rigorous track is for readers who want every detail, for example mathematics majors. It appears as collapsed blocks marked “Rigorous track”. You can skip all of them on a first reading and still follow the core text. Open them when you want to see why a result is true in full detail.

Each result is handled in one of four ways:

You see	Meaning
Proof	a full proof in the core text
Proof (Rigorous track)	a full proof, collapsed; open it if you want it
Proof (Sketch)	the main idea of the proof; the text says what is left out
a link to another page	the proof needs tools from elsewhere, and the page names where it is proved

Numbers and links¶

Definitions, theorems, examples and exercises are numbered separately on each page, and the numbers start again on every page. A link to a block on another page is therefore always named (“the squeeze theorem”), never just “Theorem 2”.

Hover over a link (or tap it on a phone) to preview what it points to without leaving the page. Every block has a permanent address, so a link to a theorem keeps working even if the pages around it are reorganised.

Exercises¶

Exercises come in three tiers:

Tier	Name	What it asks
A	Check	a direct application of one definition or rule; a few minutes
B	Practice	several steps, combining two or three ideas, and choosing an approach
C	Challenge	proofs, the reason a hypothesis is needed, or a new application

Some exercises carry an extra tag. Rigorous means the exercise needs the rigorous track, so you can skip it if you are not following that track. Applied means the exercise comes from a modelling, physics or economics context.

Each exercise has, separately collapsed, one or more hints, the final answer, and a full solution. Try the exercise first. If you are stuck, open one hint at a time. Check your answer, and read the solution even when your answer is right: it may show a shorter route.

Interactive figures¶

Some figures are interactive: you can move sliders, zoom, and read off values. Each one comes with a Try this prompt saying what to change and what to notice, and a caption describing what it shows, so the content is also available without the interaction (for example when JavaScript is off). The sliders and buttons work with the keyboard: press Tab to reach one, then use the arrow keys (or Home and End) to move a slider. Here is one:

Figure 1:Graph of $y = a \sin(b(x - c)) + d$ for $-6.5 \le x \le 6.5$ , with a slider for each of $a$ , $b$ , $c$ and $d$ , and a table of values at $x = 0, 0.5, 1, 1.5, 2$ . At first $a = b = 1$ and $c = d = 0$ , which gives $y = \sin x$ . Changing $a$ stretches the graph vertically, $b$ squeezes it horizontally, $c$ shifts it to the right and $d$ shifts it up.

Try this: set $b = 2$ . How far apart are the peaks now? Then make $a$ negative: what happens to the graph, and to the values in the table?

Page status¶

Every page shows how far it has got through our quality checks:

Status	Meaning
Draft	written, but not yet checked; it may contain errors
Reviewed	the mathematics has been read and approved by the maintainer
Verified	every computation in the worked examples and every exercise answer has also been checked by a computer algebra system, and the proofs have been reviewed twice

Pages are published at every status so that they can be read and corrected early. The content status page explains the checks in more detail. If you find an error on any page, please report it.