undergradmath.typ

// Meta data
#set document(title: "Typst Math for Undergrads", author: "johanvx")

// headcolor
#let headcolor = rgb("004225")

// Margin and footer
#set page(
  margin: 0.5in,
  footer: context {
    if counter(page).display() == "2" {
      grid(
        columns: (1fr, 1fr),
        [],
        block(
          inset: 4pt,
          stroke: (top: headcolor),
          text(headcolor)[johanvx (https://github.com/johanvx) #h(1fr) #datetime.today().display()]
        )
      )
    } else {
      []
    }
  }
)

// Font size
#let scriptsize = 7pt
#let normalsize = 10pt
#let large = 12pt
#set text(size: normalsize, lang: "en")

// Some horizontal spacing
#let kern(length) = h(length, weak: true)
#let enspace = kern(0.5em)

// For table/grid, something like "lhs \enspace rhs"
#let cell(lhs, rhs) = box(lhs + enspace + rhs)
// Grid for code blocks
#set grid(columns: (2em, auto))
// Table for math-code listing
#set table(stroke: none, align: horizon + left, inset: 0pt, row-gutter: 0.45em)

// LaTeX and TeX logos
#let TeX = style(styles => {
  let e = measure(text(normalsize, "E"), styles)
  let T = "T"
  let E = text(normalsize, baseline: e.height / 2, "E")
  let X = "X"
  box(T + kern(-0.1667em) + E + kern(-0.125em) + X)
})
#let LaTeX = style(styles => {
  let l = measure(text(10pt, "L"), styles)
  let a = measure(text(7pt, "A"), styles)
  let L = "L"
  let A = text(7pt, baseline: a.height - l.height, "A")
  box(L + kern(-0.36em) + A + kern(-0.15em) + TeX)
})

// Unavailable (last check version)
#show "??": box(text(red, [v#sys.version #emoji.crossmark]))
// Tricky
#show "!!": box(text(blue, emoji.drops))
// No idea
#show "?!": box(text(orange, [No idea #emoji.face.unhappy]))
// Tricky figure numbering
#set figure(numbering: n => {
  ([??], [!!], [?!]).at(n)
})
// No prefix
#set ref(supplement: "")

// Justified paragraphs
#set par(justify: true)

// Run-in sections, like LaTeX \paragraph
#show heading.where(
  level: 1
): it => text(
  size: normalsize,
  weight: "bold",
  fill: headcolor,
  it.body + h(0.67em)
)

// Black raw code
#show raw.where(block: false): it => { it.text }

// Two-column layout
#show: rest => columns(2, rest)

// Title
#align(center, link("https://github.com/johanvx/typst-undergradmath")[
  #text(large, headcolor)[*Typst Math for Undergrads*]
])

// Put this here to avoid affecting the title
#show link: underline

This is a Typst port with typst #sys.version of _#LaTeX Math for Undergrads_ by Jim Hefferon.
The original version is available at #link("https://gitlab.com/jim.hefferon/undergradmath").

= Meaning of annotations
// #figure(
//  table(
//    columns: (1fr, 2fr),
//    [??], [Unavailable until typst #sys.version.],
//  )
// ) <unavailable>
#figure(
  table(
    columns: (1fr, 2fr),
    [!!], [Tricky. A simpler method is needed.],
  )
) <tricky>
// #figure(
//   table(
//     columns: (1fr, 2fr),
//     [?!], [Don't know how to get this.],
//   )
// ) <noidea>

= Rule One
Any mathematics at all, even a single character, gets a mathematical setting.
Thus, for "the value of $x$ is $7$" enter `the value of $x$ is $7$`.

= Template
Your document should contain at least this.

#grid(
  "",
  ```
  -- document body here --
  ```
)

= Common constructs
#align(center, table(
  columns: 4,
  align: (right, left, right, left),
  column-gutter: (1em, 1.5em, 1em),
  [$x^2$], [`x^2`],
  [$sqrt(2)$, $root(n, 3)$], [`sqrt(2)`, `root(n, 3)`],
  [$x_(i, j)$], [`x_(i, j)`],
  [$2 / 3$, $2 \/ 3$], [`2 / 3`, `2 \/ 3` or `2 slash 3`], // Maybe use `slash`?
))

= Calligraphic letters
Use as in `$cal(A)$`.

$ cal(A B C D E F G H I J K L M N O P Q R S T U V W X Y Z) $

#show math.equation: set text(stylistic-set: 1)
Get script letters, such as $cal(P)$ from `$cal(P)$`, by changing the `stylistic-set` parameter of `text()` to the corresponding set.

= Greek
#align(center, table(
  columns: 4,
  align: (right, left, right, left),
  column-gutter: (1em, 1.5em, 1em),
  [$alpha$], [`alpha`], [$xi$, $Xi$], [`xi`, `Xi`],
  [$beta$], [`beta`], [$omicron$], [`omicron`],
  [$gamma$, $Gamma$], [`gamma`, `Gamma`], [$pi$, $Pi$], [`pi`, `Pi`],
  [$delta$, $Delta$], [`delta`, `Delta`], [$pi.alt$], [`pi.alt`],
  [$epsilon.alt$], [`epsilon.alt`], [$rho$], [`rho`],
  [$epsilon$], [`epsilon`], [$rho.alt$], [`rho.alt`],
  [$zeta$], [`zeta`], [$sigma$, $Sigma$], [`sigma`, `Sigma`],
  [$eta$], [`eta`], [$sigma.alt$], [`sigma.alt`],
  [$theta$, $Theta$], [`theta`, `Theta`], [$tau$], [`tau`],
  [$theta.alt$], [`theta.alt`], [$upsilon$, $Upsilon$], [`upsilon`, `Upsilon`],
  [$iota$], [`iota`], [$phi.alt$, $Phi$], [`phi.alt`, `Phi`],
  [$kappa$], [`kappa`], [$phi$], [`phi`],
  [$lambda$, $Lambda$], [`lambda`, `Lambda`], [$chi$], [`chi`],
  [$mu$], [`mu`], [$psi$, $Psi$], [`psi`, `Psi`],
  [$nu$], [`nu`], [$omega$, $Omega$], [`omega`, `Omega`],
))

= Sets and logic
#align(center, table(
  columns: 6,
  align: (right, left, right, left, right, left),
  column-gutter: (1em, 1.5em, 1em, 1.5em, 1em),
  [$union$], [`union`], [$RR$], [`RR`, `bb(R)`], [$forall$], [`forall`],
  [$sect$], [`sect`], [$bb(Z)$], [`ZZ`, `bb(Z)`], [$exists$], [`exists`],
  [$subset$], [`subset`], [$bb(Q)$], [`QQ`, `bb(Q)`], [$not$], [`not`],
  [$subset.eq$], [`subset.eq`], [$bb(N)$], [`NN`, `bb(N)`], [$or$], [`or`],
  [$supset$], [`supset`], [$bb(C)$], [`CC`, `bb(C)`], [$and$], [`and`],
  [$supset.eq$], [`supset.eq`], [$diameter$], [`diameter`], [$tack.r$], [`tack.r`],
  [$in$], [`in`], [$nothing$], [`nothing`], [$models$], [`models`],
  [$in.not$], [`in.not`], [$alef$], [`alef`], [$without$], [`without`],
))

Negate an operator, as in $subset.not$, with `subset.not`.
Get the set complement $A^(sans(c))$ with `A^(sans(c))` (or $A^(complement)$ with `A^(complement)`, or $overline(A)$ with `overline(A)`).

// https://www.ctan.org/tex-archive/fonts/newcomputermodern
//
// README
//
//     Version 3.93
//
//       Provides access to Russian and Greek guillemotleft and guillemotright
//     using the character variant tables cv3 and cv4 respectively.
//
//       The Math fonts provide the character \varnothing, an alternative to \emptyset,
//     through Character Variant cv01. The fontsetup package provides the option
//     'varnothing' to easily switch to the alternative character.

// http://mirrors.ctan.org/fonts/newcomputermodern/doc/newcm-doc.pdf
//
// Version 5.1
//
// The NewComputerModern FontFamily §14.5
// The Math fonts provide the character \varnothing (⌀, U+2300), as an alternative to \emptyset (a slashed zero), through Character Variant cv01.
// The fontsetup package provides the option ‘varnothing’ to easily switch to the alternative character.

/ Remark: Using `diameter` for `\varnothing` may cause some confusion.
  However, in #LaTeX, the `\varnothing` provided through Character Variant `cv01` is also `diameter`
  (see #link("http://mirrors.ctan.org/fonts/newcomputermodern/doc/newcm-doc.pdf")[newcm $section$14.5]).
  So a simple solution with the default math font _New Computer Modern Math_ is to define a new symbol `varnothing` with `#let varnothing = math.diameter`.
  Other solutions can be found in #link("https://sitandr.github.io/typst-examples-book/book/basics/math/symbols.html#empty-set")[Typst Examples Book].

= Decorations
#align(center, table(
  columns: 6,
  align: (right, left, right, left, right, left),
  column-gutter: (1em, 1.5em, 1em, 1.5em, 1em),
  [$f'$], [`f'`, `f prime`], [$dot(a)$], [`dot(a)`], [$tilde(a)$], [`tilde(a)`],
  [$f prime.double$], [`f prime.double`], [$diaer(a)$], [`diaer(a)`], [$macron(a)$], [`macron(a)`],
  [$Sigma^*$], [`Sigma^*`], [$hat(a)$], [`hat(a)`], [$arrow(a)$], [`arrow(a)`],
))

If the decorated letter is $i$ or $j$ then some decorations need `dotless.i` and `dotless.j`, as in $arrow(dotless.i)$ with `arrow(dotless.i)`.
Some authors use boldface for vectors: `bold(x)`.

Entering `overline(x + y)` produces $overline(x + y)$, and `hat(x + y)` gives $hat(x + y)$.
Comment on an expression as here (there is also `overbrace(..)`).

#align(center, table(
  columns: 2,
  column-gutter: 1em,
  [$ underbrace(x + y, |A|) $],
  [
    ```
    underbrace(x + y, |A|)
    ```
  ],
))

= Dots
Use low dots in a list ${0, 1, 2, ...}$, entered as `{0, 1, 2, ...}`.
Use centered dots in a sum or product $1 + dots.h.c + 100$, entered as  `1 + dots.h.c + 100`.
You can also get vertical dots `dots.v`, diagonal dots `dots.down` and anti-diagonal dots `dots.up`.

= Roman names
Just type them!

#align(center, table(
  columns: 6,
  align: (right, left, right, left, right, left),
  column-gutter: (1em, 1.5em, 1em, 1.5em, 1em),
  [$sin$], [`sin`], [$sinh$], [`sinh`], [$arcsin$], [`arcsin`],
  [$cos$], [`cos`], [$cosh$], [`cosh`], [$arccos$], [`arccos`],
  [$tan$], [`tan`], [$tanh$], [`tanh`], [$arctan$], [`arctan`],
  [$sec$], [`sec`], [$coth$], [`coth`], [$min$], [`min`],
  [$csc$], [`csc`], [$det$], [`det`], [$max$], [`max`],
  [$cot$], [`cot`], [$dim$], [`dim`], [$inf$], [`inf`],
  [$exp$], [`exp`], [$ker$], [`ker`], [$sup$], [`sup`],
  [$log$], [`log`], [$deg$], [`deg`], [$liminf$], [`liminf`],
  [$ln$], [`ln`], [$arg$], [`arg`], [$limsup$], [`limsup`],
  [$lg$], [`lg`], [$gcd$], [`gcd`], [$lim$], [`lim`],
))

#let cosec = math.op("cosec")

If an operator you wish to use does not exist, you can create one using `math.op`. For example, to create the $cosec$ operator:

```
#let cosec = math.op("cosec")
$ cosec x = 1/(sin x) $
```

$ cosec x = 1/(sin x) $

= Other symbols
#align(center, table(
  columns: 6,
  align: (right, left, right, left, right, left),
  column-gutter: (0.5em, 1em, 0.5em, 1em, 0.5em),
  [$<$], [`<`, `lt`], [$angle$], [`angle`], [$dot$], [`dot`],
  [$<=$], [`<=`, `lt.eq`], [$angle.arc$], [`angle.arc`], [$plus.minus$], [`plus.minus`],
  [$>$], [`>`, `gt`], [$ell$], [`ell`], [$minus.plus$], [`minus.plus`],
  [$>=$], [`>=`, `gt.eq`], [$parallel$], [`parallel`], [$times$], [`times`],
  [$!=$], [`!=`, `eq.not`], [$45 degree$], [`45 degree`], [$div$], [`div`],
  [$<<$], [`<<`, `lt.double`], [$tilde.equiv$], [`tilde.equiv`], [$*$], [`*`, `ast`],
  [$>>$], [`>>`, `gt.double`], [$tilde.nequiv$], [`tilde.nequiv`], [$divides$], [`divides`],
  [$approx$], [`approx`], [$tilde$], [`tilde`], [$divides.not$], [`divides.not`],
  [$\u{224D}$], [`\u{224D}` @tricky], [$tilde.eq$], [`tilde.eq`], [$n!$], [`n!`],
  [$equiv$], [`equiv`], [$tilde.not$], [`tilde.not`], [$diff$], [`diff`],
  [$prec$], [`prec`], [$plus.circle$], [`plus.circle`], [$nabla$], [`nabla`],
  [$prec.eq$], [`prec.eq`], [$minus.circle$], [`minus.cirle`], [$planck.reduce$], [`planck.reduce`],
  [$succ$], [`succ`], [$dot.circle$], [`dot.circle`], [$compose$], [`compose`],
  [$succ.eq$], [`succ.eq`], [$times.circle$], [`times.circle`], [$star$], [`star`],
  [$prop$], [`prop`], [$\u{2298}$], [`\u{2298}` @tricky], [$sqrt("")$], [`sqrt("")`],
  [$\u{2250}$], [`\u{2250}` @tricky], [$harpoon.tr$], [`harpoon.tr`], [$checkmark$], [`checkmark`],
))

Use `a divides b` for the divides relation, $a divides b$, and `a divides.not b` for the negation, $a divides.not b$.
Use `|` to get set builder notation ${a in S | a "is odd"}$ with `{a in S | a "is odd"}`.

= Arrows
#align(center, table(
  columns: 4,
  align: (right, left, right, left),
  column-gutter: (1em, 1.5em, 1em),
  [$->$], [`->`, `arrow.r`], [$|->$], [`|->`, `arrow.r.bar`],
  [$arrow.r.not$], [`arrow.r.not`], [$arrow.r.long.bar$], [`arrow.r.long.bar`],
  [$-->$], [`-->`, `arrow.r.long`], [$<-$], [`<-`, `arrow.l`],
  [$=>$], [`=>`, `arrow.r.double`], [$<->$], [`<->`, `arrow.l.r`],
  [$arrow.r.double.not$], [`arrow.r.double.not`], [$arrow.b$], [`arrow.b`],
  [$==>$], [`==>`, `arrow.r.double.long`], [$arrow.t$], [`arrow.t`],
  [$arrow.squiggly$], [`arrow.squiggly`], [$arrow.t.b$], [`arrow.t.b`],
))

The right arrows in the first column have matching left arrows, such as `arrow.l.not`, and there are some other matches for down arrows, etc.

= Variable-sized operators
The summation $sum_(j = 0)^3 j^2$ `sum_(j = 0)^3 j^2` and the integral $integral_(x = 0)^3 x^2 dif x$ `integral_(x = 0)^3 x^2 dif x` expand when displayed.

$ sum_(j = 0)^3 j^2 wide integral_(x = 0)^3 x^2 dif x $

These do the same.

#align(center, table(
  columns: 4,
  align: (right, left, right, left),
  column-gutter: (1em, 1.5em, 1em),
  row-gutter: 0.5em,
  [$integral$], [`integral`], [$integral.double$], [`integral.double`],
  [$integral.triple$], [`integral.triple`], [$integral.cont$], [`integral.cont`],
  [$union.big$], [`union.big`], [$sect.big$], [`sect.big`],
))

= Fences
#align(center, table(
  columns: 6,
  align: (right, left, right, left, right, left),
  column-gutter: (1em, 1.5em, 1em, 1.5em, 1em),
  row-gutter: 0.5em,
  [$()$], [`()`], [$angle.l angle.r$], [`angle.l angle.r`], [$abs("")$], [`abs("")`],
  [$[]$], [`[]`], [$floor("")$], [`floor("")`], [$norm("")$], [`norm("")`],
  [${}$], [`{}`], [$ceil("")$], [`ceil("")`],
))

Fix the size with the `lr` function.

#align(center, table(
  columns: 2,
  column-gutter: 1em,
  [$ lr([sum_(k = 0)^n e^(k^2)], size: #50%) $],
  [
    ```
    lr([sum_(k = 0)^n e^(k^2)], size: #50%)
    ```
  ],
))

To have them grow with the enclosed formula, also use the `lr` function.

#align(center, table(
  columns: 2,
  column-gutter: 1em,
  [$ lr(angle.l i, 2^(2^i) angle.r) $],
  [
    ```
    lr(angle.l i, 2^(2^i) angle.r)
    ```
  ],
))

Fences scale by default if entered directly as codepoints, and don't scale automatically if entered as symbol notation.

#align(center, table(
  columns: 2,
  align: (right + horizon, left + horizon),
  column-gutter: 1em,
  [$ (1 / n^(alpha)) $],
  [
    ```
    (1 / n^(alpha))
    ```
  ],
  [$ paren.l 1 / n^(alpha) paren.r $],
  [
    ```
    paren.l 1 / n^(alpha) paren.r
    ```
  ],
))

The `lr` function also allows to scale unmatched delimiters and one-side fences.

#align(center, table(
  columns: 2,
  column-gutter: 1em,
  [$ lr(frac(dif f, dif x) |)_(x_0) $],
  [
    ```
    lr(frac(dif f, dif x) |)_(x_0)
    ```
  ],
))

= Arrays, Matrices
Get a matrix with the `mat` function. You can pass an array to it.

#align(center, table(
  columns: 2,
  column-gutter: 1em,
  [$ mat(a, b; c, d) $],
  [
    ```
    $ mat(a, b; c, d) $
    ```
  ],
))

In Typst, #link("https://typst.app/docs/reference/types/array")[array] is a sequence of values,
while in #LaTeX, array is a matrix without fences, which is `$mat(delim: #none, ..)$` in Typst.

For the determinant use `|A|`, text operator $det$ `det` or `mat(delim: "|", ..)`.

Definition by cases can be easily obtained with the `cases` function.

#align(center, table(
  columns: 2,
  column-gutter: 1em,
  [
    $ f_n = cases(
      a &"if" n = 0,
      r dot f_(n - 1) &"else"
    ) $
  ],
  [
    ```
    $ f_n = cases(
      a &"if" n = 0,
      r dot f_(n - 1) &"else"
    ) $
    ```
  ]
))

= Spacing in mathematics
Improve $sqrt(2) x$ to $sqrt(2) thin x$ with a thin space, as in `sqrt(2) thin x`.
Slightly wider are `med` and `thick` (the three are in ratio $3 : 4 : 5$).
Bigger space are: `quad` for $-> quad <-$ and `wide` for $-> wide <-$, which are useful between parts of a display.
Get arbitrary space with the `h` function.
For example, use `#h(-0.1667em)` for `\!` in #LaTeX.

= Displayed equations
Display equations in a block level using `$ ... $` with at least one space separating the math content and the `$`.

#align(center, table(
  columns: 2,
  column-gutter: 1em,
  [$ S = k dot lg W $],
  [
    ```
    $ S = k dot lg W $
    ```
  ],
))

You can break into multiple lines.

#align(center, table(
  columns: 2,
  column-gutter: 1em,
  [
    $ sin(x) = x - x^3 / 3! \
      + x^5 / 5! - dots.h.c $
  ],
  [
    ```
    $ sin(x) = x - x^3 / 3! \
        + x^5 / 5! - dots.h.c $
    ```
  ],
))

Align equations using `&`

#align(center, table(
  columns: 2,
  column-gutter: 1em,
  [
    $ nabla dot bold(D) &= rho \
      nabla dot bold(B) &= 0 $
  ],
  ```
  $ nabla dot bold(D) &= rho \
    nabla dot bold(B) &= 0 $
  ```,
))

(the left or right side of an alignment can be empty).
Get a numbered version by `#set math.equation(numbering: ..)`.

= Calculus examples
The last three here are display style.

#align(center, table(
  columns: 2,
  column-gutter: 1em,
  [$ f: RR -> RR $],
  [
    ```
    f: RR -> RR
    ```
  ],
  [$ 9.8 thin "m/s"^2 $],
  [`9.8 thin "m/s"^2` @tricky],
  [$ lim_(h->0) (f(x+h)-f(x))/h $],
  [
    ```
    lim_(h -> 0) (f(x + h) - f(x)) / h
    ```
  ],
  [$ integral x^2 dif x = x^3 \/ 3 + C $],
  [
    ```
    integral x^2 dif x = x^3 \/ 3 + C
    ```
  ],
  [$ nabla = bold(i) dif / (dif x) + bold(j) dif / (dif y) + bold(k) dif / (dif z) $],
  [
    ```
    nabla = bold(i) dif / (dif x) + bold(j) dif / (dif y) + bold(k) dif / (dif z)
    ```
  ],
))

= Discrete mathematics examples
For modulo, there is a symbol $equiv$ from `equiv` and a text operator $mod$ from `mod`.

For combinations the binomial symbol $binom(n, k)$ is from `binom(n, k)`.
This resizes to be bigger in a display.

For permutations use $n^(underline(r))$ from `n^(underline(r))` (some authors use $P(n, r)$, or $""_n P_r$ from `""_n P_r`).

= Statistics examples
#align(center, table(
  columns: 2,
  column-gutter: 1em,
  [$ sigma^2 = sqrt(sum(x_i - mu)^2 \/ N) $],
  [
    ```
    sigma^2 = sqrt(sum(x_i - mu)^2 \/ N)
    ```
  ],
  [$ E(X) = mu_X = sum(x_i - P(x_i)) $],
  [
    ```
    E(X) = mu_X = sum(x_i - P(x_i))
    ```
  ],
  [$ 1 / sqrt(2 sigma^2 pi) e^(- (x - mu)^2 / (2 sigma^2)) $],
  [
    ```
    1 / sqrt(2 sigma^2 pi) e^(- (x - mu)^2 / (2 sigma^2))
    ```
  ]
))

= For more
See also the Typst Documentation at #link("https://typst.app/docs").