Sample 02 — Mathematics and Scientific Notes

A scientific note becomes easier to follow when its assumptions, equations, examples, and sources have distinct roles. Here is a small worked example: estimating the derivative of using a forward difference.

The numbers are calculated examples, not experimental measurements. Each source block can be copied into another article; the rendered material below it shows the result.

Choose the notation packages

Core MathJax notation supports fractions, sums, matrices, and document-local macros. This article also enables optional STEM packages in its frontmatter:

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math:
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  packages: [physics, mathtools, units, mhchem, cancel]

In Studio, the same choices live in article Settings → Mathematics. Enable only what the article uses. Physics changes some commands: \div means divergence, while \divsymbol supplies the familiar division sign.

State the definition before using it

A derivative describes a local rate of change; a difference quotient estimates that rate using a nonzero step. The limiting definition and its geometric interpretation are introduced in [1]. Our example below studies the finite-step error directly.

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:::definition[Forward difference]{#def-forward}
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For $h>0$, define $D_h f(x) = \frac{f(x+h)-f(x)}{h}$.
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:::

The label def-forward is a stable reference target. Its displayed number is assigned automatically, so adding an earlier statement does not require rewriting every reference.

Build a short argument

The statement components share a document-local counter. Choose the name that matches the role of the statement; numbering alone does not establish that a claim has been proved.

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:::lemma[Increment of a square]{#lem-increment}
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For real $x,h$, $(x+h)^2-x^2=2xh+h^2$.
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:::
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:::proof[Proof of the increment identity]
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Expand $(x+h)^2=x^2+2xh+h^2$ and subtract $x^2$.
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:::
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:::theorem[Exact forward-difference error]{#thm-error}
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For $f(x)=x^2$ and $h>0$, $D_h f(x)-f'(x)=h$.
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:::

Other statement names express different parts of the same argument:

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:::proposition[Smaller positive steps]{#prop-smaller}
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If $0<h_1<h_2$, the absolute error for $h_1$ is smaller than for $h_2$.
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:::
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:::corollary[Halving the step]{#cor-half}
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Replacing $h$ with $h/2$ halves the error for this function in exact arithmetic.
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:::
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:::example[A concrete evaluation]{#ex-quarter}
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At $x=1$ and $h=1/4$, $D_h f(1)=2.25$, whereas $f'(1)=2$.
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:::

Both Proposition 4 and Corollary 5 follow immediately from Theorem 3. This is a claim about a particular polynomial and exact arithmetic, not a general promise that arbitrarily small floating-point steps always improve a numerical derivative.¹

Number a derivation as one equation

An equation container holds exactly one display equation. Put several related lines inside aligned to keep them together.

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:::equation[Forward difference of a square]{#eq-forward}
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$$
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\begin{aligned}
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D_h f(x)
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  &= \frac{(x+h)^2-x^2}{h} \\
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  &= \frac{2xh+h^2}{h} \\
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  &= 2x+h.
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\end{aligned}
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$$
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:::
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The finite-step term is visible in :ref[eq-forward].

The finite-step term is visible in Equation 1. Notice that the reference stays readable without manually inserting an equation number.

Compare exact values in a table

Use a captioned table when the values need a label, a note, or a cross-reference. Alignment comes from the ordinary GFM delimiter row.

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:::table[Forward differences at x = 1]{#tab-forward}
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| Step $h$ | Estimate $D_h f(1)$ | Error |
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| -------: | -----------------: | ----: |
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| $1/2$    | 2.500              | 0.500          |
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| $1/4$    | 2.250              | 0.250          |
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| $1/8$    | 2.125              | 0.125          |
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These values use exact arithmetic; the decimal expansions terminate.
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:::

Table 1 illustrates Corollary 5. Keep units, precision, and missing values explicit in a real dataset. The booktabs presentation uses horizontal rules and numeric alignment rather than a full grid.

Define a document-local macro

Keep a definition in a math expression before its first use. The definition itself produces no visible symbol.

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$\newcommand{\sampleD}{D_h}$
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Our local operator is $\sampleD f(x)$.

Our local operator is . Its definition belongs to this article and does not affect another article. Source completion can offer document-defined macros as well as built-in commands.

Lay out matrices and piecewise functions

These common structures work with core notation. Use & to separate columns and \\ to separate rows.

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\begin{bmatrix} 1 & h \\ 0 & 1 \end{bmatrix}
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\begin{bmatrix} x \\ v \end{bmatrix}
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= \begin{bmatrix} x+hv \\ v \end{bmatrix}

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g(x)=\begin{cases}
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0, & x<0, \\
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x^2, & x\ge 0.
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\end{cases}

Add the STEM notation you need

Physics adds convenient derivatives and vector notation:

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\dv{x}{t}=v,\qquad
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\dv[2]{x}{t}=a,\qquad
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\pdv{u}{x},\qquad
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\vb{F}=m\vb{a}.

Units, chemical formulas, and cancellation use their own enabled packages:

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v=\units[3]{m/s},\qquad
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\ce{H2O},\qquad
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\frac{\cancel{x}y}{\cancel{x}}=y\quad(x\ne0).

These are notation examples rather than a single physical model. Prefer separate display equations if a combined line becomes too wide. On a phone, a wide equation scrolls within its own region.

Cite a real, traceable source

Bibliography entries are written explicitly. :cite[key] creates a numbered citation; :ref[label] points to a component in this document. They serve different purposes.

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The derivative is introduced in :cite[openstax-derivative].
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::::references[References]
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:::reference[OpenStax — Defining the Derivative]{#openstax-derivative}
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OpenStax. *Calculus Volume 1*, Section 3.1.
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[Read the chapter](https://openstax.org/books/calculus-volume-1/pages/3-1-defining-the-derivative).
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:::
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::::

Before adapting this note, check each assumption, retain stable labels, enable the required packages, and review the rendered equations as well as the source.