├── basic.pdf
├── basic.png
├── manual.pdf
├── manual_examples.pdf
├── differential_calculus.pdf
├── manual_template.typ
├── LICENSE
├── basic.typ
├── README.md
├── manual_examples.typ
├── theorems.typ
├── manual.typ
└── differential_calculus.typ


/basic.pdf:
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https://raw.githubusercontent.com/sahasatvik/typst-theorems/HEAD/basic.pdf


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/basic.png:
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https://raw.githubusercontent.com/sahasatvik/typst-theorems/HEAD/basic.png


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/manual.pdf:
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https://raw.githubusercontent.com/sahasatvik/typst-theorems/HEAD/manual.pdf


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/manual_examples.pdf:
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https://raw.githubusercontent.com/sahasatvik/typst-theorems/HEAD/manual_examples.pdf


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/differential_calculus.pdf:
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https://raw.githubusercontent.com/sahasatvik/typst-theorems/HEAD/differential_calculus.pdf


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/manual_template.typ:
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 1 | #let project(title: "", authors: (), url: "", body) = {
 2 |   set page(paper: "a4", numbering: "1", number-align: center)
 3 |   set document(author: authors, title: title)
 4 |   set text(font: "Libertinus Serif", lang: "en")
 5 |   set heading(numbering: "1.1.")
 6 |   set par(justify: true)
 7 |   set list(marker: ([•], [--]))
 8 |   show heading: it => pad(bottom: 0.5em, it)
 9 |   show raw.where(block: true): it => pad(left: 4em, it)
10 |   show link: it => underline(text(fill: blue, it))
11 | 
12 | 
13 | 
14 |   align(center)[
15 |     #block(text(weight: 700, 1.75em, title))
16 |   ]
17 | 
18 |   pad(
19 |     top: 0.5em,
20 |     bottom: 2em,
21 |     x: 2em,
22 |     grid(
23 |       columns: (1fr,) * calc.min(3, authors.len()),
24 |       gutter: 1em,
25 |       ..authors.map(author => align(center)[
26 |         #author \
27 |         #link(url)
28 |       ]),
29 |     ),
30 |   )
31 | 
32 |   outline(indent: true)
33 | 
34 |   v(2em)
35 | 
36 |   body
37 | }
38 | 


--------------------------------------------------------------------------------
/LICENSE:
--------------------------------------------------------------------------------
 1 | MIT License
 2 | 
 3 | Copyright (c) 2023 Satvik Saha
 4 | 
 5 | Permission is hereby granted, free of charge, to any person obtaining a copy
 6 | of this software and associated documentation files (the "Software"), to deal
 7 | in the Software without restriction, including without limitation the rights
 8 | to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
 9 | copies of the Software, and to permit persons to whom the Software is
10 | furnished to do so, subject to the following conditions:
11 | 
12 | The above copyright notice and this permission notice shall be included in all
13 | copies or substantial portions of the Software.
14 | 
15 | THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
16 | IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
17 | FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
18 | AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
19 | LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
20 | OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
21 | SOFTWARE.
22 | 


--------------------------------------------------------------------------------
/basic.typ:
--------------------------------------------------------------------------------
 1 | #import "theorems.typ": *
 2 | #show: thmrules.with(qed-symbol: $square$)
 3 | 
 4 | #set page(width: 16cm, height: auto, margin: 1.5cm)
 5 | #set text(font: "Libertinus Serif", lang: "en")
 6 | #set heading(numbering: "1.1.")
 7 | 
 8 | #let theorem = thmbox("theorem", "Theorem", fill: rgb("#eeffee"))
 9 | #let corollary = thmplain(
10 |   "corollary",
11 |   "Corollary",
12 |   base: "theorem",
13 |   titlefmt: strong
14 | )
15 | #let definition = thmbox("definition", "Definition", inset: (x: 1.2em, top: 1em))
16 | 
17 | #let example = thmplain("example", "Example").with(numbering: none)
18 | #let proof = thmproof("proof", "Proof")
19 | 
20 | 
21 | = Prime numbers
22 | 
23 | #definition[
24 |   A natural number is called a #highlight[_prime number_] if it is greater
25 |   than 1 and cannot be written as the product of two smaller natural numbers.
26 | ]
27 | #example[
28 |   The numbers $2$, $3$, and $17$ are prime.
29 |   @cor_largest_prime shows that this list is not exhaustive!
30 | ]
31 | 
32 | #theorem("Euclid")[
33 |   There are infinitely many primes.
34 | ]
35 | #proof[
36 |   Suppose to the contrary that $p_1, p_2, dots, p_n$ is a finite enumeration
37 |   of all primes. Set $P = p_1 p_2 dots p_n$. Since $P + 1$ is not in our list,
38 |   it cannot be prime. Thus, some prime factor $p_j$ divides $P + 1$.  Since
39 |   $p_j$ also divides $P$, it must divide the difference $(P + 1) - P = 1$, a
40 |   contradiction.
41 | ]
42 | 
43 | #corollary[
44 |   There is no largest prime number.
45 | ] <cor_largest_prime>
46 | #corollary[
47 |   There are infinitely many composite numbers.
48 | ]
49 | 
50 | #theorem[
51 |   There are arbitrarily long stretches of composite numbers.
52 | ]
53 | #proof[
54 |   For any $n > 2$, consider $
55 |     n! + 2, quad n! + 3, quad ..., quad n! + n #qedhere
56 |   $
57 | ]
58 | 


--------------------------------------------------------------------------------
/README.md:
--------------------------------------------------------------------------------
  1 | # typst-theorems
  2 | 
  3 | An implementation of numbered theorem environments in
  4 | [typst](https://github.com/typst/typst).
  5 | Available as [ctheorems](https://typst.app/universe/package/ctheorems) in the
  6 | official Typst Universe.
  7 | Import with
  8 | 
  9 | ```typst
 10 | #import "@preview/ctheorems:1.1.3": *
 11 | #show: thmrules
 12 | ```
 13 | 
 14 | Alternatively, copy and import the [theorems.typ](theorems.typ) file to use in
 15 | your own projects.
 16 | 
 17 | ### Features
 18 | - Numbered theorem environments can be created and customized.
 19 | - Environments can share the same counter, via same `identifier`s.
 20 | - Environment counters can be _attached_ (just as subheadings are attached to headings) to other environments, headings, or keep a global count via `base`.
 21 | - The depth of a counter can be manually set, via `base_level`.
 22 | - Environments can be `<label>`'d and `@reference`'d.
 23 | 
 24 | ## Manual and Examples
 25 | Get acquainted with `typst-theorems` by checking out the minimal example below!
 26 | 
 27 | You can read the [manual](manual.pdf) for a full walkthrough of functionality offered by this module; flick through [manual_examples](manual_examples.pdf) and its [typ file](manual_examples.typ) to just see the examples.
 28 | 
 29 | The [differential_calculus.typ](differential_calculus.typ) ([render](differential_calculus.pdf)) project provides a practical use case. _(Hastily translated from my notes written in LaTeX)_
 30 | 
 31 | ![basic example](basic.png)
 32 | 
 33 | ### Preamble
 34 | ```typst
 35 | #import "theorems.typ": *
 36 | #show: thmrules.with(qed-symbol: $square$)
 37 | 
 38 | #set page(width: 16cm, height: auto, margin: 1.5cm)
 39 | #set text(font: "Libertinus Serif", lang: "en")
 40 | #set heading(numbering: "1.1.")
 41 | 
 42 | #let theorem = thmbox("theorem", "Theorem", fill: rgb("#eeffee"))
 43 | #let corollary = thmplain(
 44 |   "corollary",
 45 |   "Corollary",
 46 |   base: "theorem",
 47 |   titlefmt: strong
 48 | )
 49 | #let definition = thmbox("definition", "Definition", inset: (x: 1.2em, top: 1em))
 50 | 
 51 | #let example = thmplain("example", "Example").with(numbering: none)
 52 | #let proof = thmproof("proof", "Proof")
 53 | ```
 54 | 
 55 | ### Document
 56 | ```typst
 57 | = Prime numbers
 58 | 
 59 | #definition[
 60 |   A natural number is called a #highlight[_prime number_] if it is greater
 61 |   than 1 and cannot be written as the product of two smaller natural numbers.
 62 | ]
 63 | #example[
 64 |   The numbers $2$, $3$, and $17$ are prime.
 65 |   @cor_largest_prime shows that this list is not exhaustive!
 66 | ]
 67 | 
 68 | #theorem("Euclid")[
 69 |   There are infinitely many primes.
 70 | ]
 71 | #proof[
 72 |   Suppose to the contrary that $p_1, p_2, dots, p_n$ is a finite enumeration
 73 |   of all primes. Set $P = p_1 p_2 dots p_n$. Since $P + 1$ is not in our list,
 74 |   it cannot be prime. Thus, some prime factor $p_j$ divides $P + 1$.  Since
 75 |   $p_j$ also divides $P$, it must divide the difference $(P + 1) - P = 1$, a
 76 |   contradiction.
 77 | ]
 78 | 
 79 | #corollary[
 80 |   There is no largest prime number.
 81 | ] <cor_largest_prime>
 82 | #corollary[
 83 |   There are infinitely many composite numbers.
 84 | ]
 85 | 
 86 | #theorem[
 87 |   There are arbitrarily long stretches of composite numbers.
 88 | ]
 89 | #proof[
 90 |   For any $n > 2$, consider $
 91 |     n! + 2, quad n! + 3, quad ..., quad n! + n #qedhere
 92 |   $
 93 | ]
 94 | ```
 95 | 
 96 | 
 97 | ## Acknowledgements
 98 | 
 99 | Thanks to
100 | 
101 | - [MJHutchinson](https://github.com/MJHutchinson) for suggesting and
102 |   implementing the `base_level` and `base: none` features,
103 | - [rmolinari](https://github.com/rmolinari) for suggesting and
104 |   implementing the `separator: ...` feature,
105 | - [DVDTSB](https://github.com/DVDTSB) for contributing
106 |   - the idea of passing named arguments from the theorem directly to the `fmt`
107 |     function.
108 |   - the `number: ...` override feature.
109 |   - the `title: ...` override feature in `thmbox`.
110 | - [PgBiel](https://github.com/PgBiel) for fixing breaking changes in version
111 |   updates.
112 | - The awesome devs of [typst.app](https://typst.app/) for their
113 |   support.
114 | 


--------------------------------------------------------------------------------
/manual_examples.typ:
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  1 | #import "manual_template.typ": *
  2 | #import "theorems.typ": *
  3 | #show: thmrules
  4 | 
  5 | #show: project.with(
  6 |   title: "typst-theorems",
  7 |   authors: (
  8 |     "sahasatvik",
  9 |   ),
 10 |   url: "https://github.com/sahasatvik/typst-theorems"
 11 | )
 12 | 
 13 | 
 14 | 
 15 | = Introduction
 16 | 
 17 | This document only includes the examples given in the manual; each one of
 18 | these has been explained in full detail there.
 19 | 
 20 | 
 21 | = Feature demonstration
 22 | 
 23 | #let theorem = thmbox(
 24 |   "theorem",
 25 |   "Theorem",
 26 |   fill: rgb("#e8e8f8")
 27 | )
 28 | 
 29 | #theorem("Euclid")[
 30 |   There are infinitely many primes.
 31 | ] <euclid>
 32 | 
 33 | 
 34 | #let lemma = thmbox(
 35 |   "theorem",
 36 |   "Lemma",
 37 |   fill: rgb("#efe6ff")
 38 | )
 39 | 
 40 | #lemma[
 41 |   If $n$ divides both $x$ and $y$, it also divides $x - y$.
 42 | ]
 43 | 
 44 | #let corollary = thmbox(
 45 |   "corollary",
 46 |   "Corollary",
 47 |   base: "theorem",
 48 |   fill: rgb("#f8e8e8")
 49 | )
 50 | 
 51 | #corollary(numbering: "1.1")[
 52 |   If $n$ divides two consecutive natural numbers, then $n = 1$.
 53 | ]
 54 | 
 55 | 
 56 | == Proofs
 57 | 
 58 | #let proof = thmproof("proof", "Proof")
 59 | 
 60 | #proof([of @euclid])[
 61 |   Suppose to the contrary that $p_1, p_2, dots, p_n$ is a finite enumeration
 62 |   of all primes. Set $P = p_1 p_2 dots p_n$. Since $P + 1$ is not in our list,
 63 |   it cannot be prime. Thus, some prime factor $p_j$ divides $P + 1$.  Since
 64 |   $p_j$ also divides $P$, it must divide the difference $(P + 1) - P = 1$, a
 65 |   contradiction.
 66 | ]
 67 | 
 68 | #theorem[
 69 |   There are arbitrarily long stretches of composite numbers.
 70 | ]
 71 | #proof[
 72 |   For any $n > 2$, consider $
 73 |     n! + 2, quad n! + 3, quad ..., quad n! + n #qedhere
 74 |   $
 75 | ]
 76 | 
 77 | 
 78 | == Suppressing numbering
 79 | 
 80 | #let example = thmplain(
 81 |   "example",
 82 |   "Example"
 83 | ).with(numbering: none)
 84 | 
 85 | #example[
 86 |   The numbers $2$, $3$, and $17$ are prime.
 87 | ]
 88 | 
 89 | #lemma(numbering: none)[
 90 |   The square of any even number is divisible by $4$.
 91 | ]
 92 | #lemma[
 93 |   The square of any odd number is one more than a multiple of $4$.
 94 | ]
 95 | 
 96 | #lemma(number: "42")[
 97 |   The square of any natural number cannot be two more than a multiple of 4.
 98 | ]
 99 | 
100 | 
101 | == Limiting depth
102 | 
103 | #let definition = thmbox(
104 |   "definition",
105 |   "Definition",
106 |   base_level: 1,
107 |   stroke: rgb("#68ff68") + 1pt
108 | )
109 | #definition("Prime numbers")[
110 |   A natural number is called a _prime number_ if it is greater than $1$ and
111 |   cannot be written as the product of two smaller natural numbers. <prime>
112 | ]
113 | 
114 | #definition("Composite numbers")[
115 |   A natural number is called a _composite number_ if it is greater than $1$
116 |   and not prime.
117 | ]
118 | 
119 | #example(base_level: 4, numbering: "1.1")[
120 |   The numbers $4$, $6$, and $42$ are composite.
121 | ]
122 | 
123 | 
124 | == Custom formatting
125 | 
126 | #let proof-custom = thmplain(
127 |   "proof",
128 |   "Proof",
129 |   base: "theorem",
130 |   titlefmt: smallcaps,
131 |   bodyfmt: body => [
132 |     #body #h(1fr) $square$
133 |   ]
134 | ).with(numbering: none)
135 | 
136 | #lemma[
137 |   All even natural numbers greater than 2 are composite.
138 | ]
139 | #proof-custom[
140 |   Every even natural number $n$ can be written as the product of the natural
141 |   numbers $2$ and $n\/2$. When $n > 2$, both of these are smaller than $2$
142 |   itself.
143 | ]
144 | 
145 | #let notation = thmenv(
146 |   "notation",                 // identifier
147 |   none,                       // base - do not attach, count globally
148 |   none,                       // base_level - use the base as-is
149 |   (name, number, body, color: black) => [
150 |                               // fmt - format content using the environment name, number, body, and an optional color
151 |     #text(color)[#h(1.2em) *Notation (#number) #name*]:
152 |     #h(0.2em)
153 |     #body
154 |     #v(0.5em)
155 |   ]
156 | ).with(numbering: "I")        // use Roman numerals
157 | 
158 | #notation[
159 |   The variable $p$ is reserved for prime numbers.
160 | ]
161 | #notation("for Reals", color: green)[
162 |   The variable $x$ is reserved for real numbers.
163 | ]
164 | 
165 | #lemma(title: "Lem.", stroke: 1pt)[
166 |   All multiples of 3 greater than 3 are composite.
167 | ]
168 | 
169 | 
170 | == Labels and references <references>
171 | 
172 | #pad(
173 |   left: 1.2em,
174 |   [
175 |     Recall that there are infinitely many prime numbers via @euclid.
176 |   ]
177 | )
178 | 
179 | 
180 | #pad(
181 |   left: 1.2em,
182 |   [
183 |     You can reference future environments too, like @oddprime[Cor.].
184 |   ]
185 | )
186 | 
187 | #lemma(supplement: "Lem.", refnumbering: "(1.1)")[
188 |   All primes apart from $2$ and $3$ are of the form $6k plus.minus 1$.
189 | ] <primeform>
190 | 
191 | You can modify the supplement and numbering to be used in references, like @primeform.
192 | 
193 | 
194 | == Overriding `base`
195 | 
196 | 
197 | #let remark = thmplain("remark", "Remark", base: "heading")
198 | #remark[
199 |   There are infinitely many composite numbers.
200 | ]
201 | #corollary[
202 |   All primes greater than $2$ are odd.
203 | ] <oddprime>
204 | #remark(base: "corollary")[
205 |   Two is a _lone prime_.
206 | ]
207 | 


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/theorems.typ:
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  1 | // Store theorem environment numbering
  2 | 
  3 | #let thmcounters = state("thm",
  4 |   (
  5 |     "counters": ("heading": ()),
  6 |     "latest": ()
  7 |   )
  8 | )
  9 | 
 10 | 
 11 | #let thmenv(identifier, base, base_level, fmt) = {
 12 | 
 13 |   let global_numbering = numbering
 14 | 
 15 |   return (
 16 |     ..args,
 17 |     body,
 18 |     number: auto,
 19 |     numbering: "1.1",
 20 |     refnumbering: auto,
 21 |     supplement: identifier,
 22 |     base: base,
 23 |     base_level: base_level
 24 |   ) => {
 25 |     let name = none
 26 |     if args != none and args.pos().len() > 0 {
 27 |       name = args.pos().first()
 28 |     }
 29 |     if refnumbering == auto {
 30 |       refnumbering = numbering
 31 |     }
 32 |     let result = none
 33 |     if number == auto and numbering == none {
 34 |       number = none
 35 |     }
 36 |     if number == auto and numbering != none {
 37 |       result = context {
 38 |         let heading-counter = counter(heading).get()
 39 |         return thmcounters.update(thmpair => {
 40 |           let counters = thmpair.at("counters")
 41 |           // Manually update heading counter
 42 |           counters.at("heading") = heading-counter
 43 |           if not identifier in counters.keys() {
 44 |             counters.insert(identifier, (0, ))
 45 |           }
 46 | 
 47 |           let tc = counters.at(identifier)
 48 |           if base != none {
 49 |             let bc = counters.at(base)
 50 | 
 51 |             // Pad or chop the base count
 52 |             if base_level != none {
 53 |               if bc.len() < base_level {
 54 |                 bc = bc + (0,) * (base_level - bc.len())
 55 |               } else if bc.len() > base_level{
 56 |                 bc = bc.slice(0, base_level)
 57 |               }
 58 |             }
 59 | 
 60 |             // Reset counter if the base counter has updated
 61 |             if tc.slice(0, -1) == bc {
 62 |               counters.at(identifier) = (..bc, tc.last() + 1)
 63 |             } else {
 64 |               counters.at(identifier) = (..bc, 1)
 65 |             }
 66 |           } else {
 67 |             // If we have no base counter, just count one level
 68 |             counters.at(identifier) = (tc.last() + 1,)
 69 |             let latest = counters.at(identifier)
 70 |           }
 71 | 
 72 |           let latest = counters.at(identifier)
 73 |           return (
 74 |             "counters": counters,
 75 |             "latest": latest
 76 |           )
 77 |         })
 78 |       }
 79 | 
 80 |       number = context {
 81 |         global_numbering(numbering, ..thmcounters.get().at("latest"))
 82 |       }
 83 |     }
 84 | 
 85 |     return figure(
 86 |       result +  // hacky!
 87 |       fmt(name, number, body, ..args.named()) +
 88 |       [#metadata(identifier) <meta:thmenvcounter>],
 89 |       kind: "thmenv",
 90 |       outlined: false,
 91 |       caption: name,
 92 |       supplement: supplement,
 93 |       numbering: refnumbering,
 94 |     )
 95 |   }
 96 | }
 97 | 
 98 | 
 99 | #let thmbox(
100 |   identifier,
101 |   head,
102 |   ..blockargs,
103 |   supplement: auto,
104 |   padding: (top: 0.5em, bottom: 0.5em),
105 |   namefmt: x => [(#x)],
106 |   titlefmt: strong,
107 |   bodyfmt: x => x,
108 |   separator: [#h(0.1em):#h(0.2em)],
109 |   base: "heading",
110 |   base_level: none,
111 | ) = {
112 |   if supplement == auto {
113 |     supplement = head
114 |   }
115 |   let boxfmt(name, number, body, title: auto, ..blockargs_individual) = {
116 |     if not name == none {
117 |       name = [ #namefmt(name)]
118 |     } else {
119 |       name = []
120 |     }
121 |     if title == auto {
122 |       title = head
123 |     }
124 |     if not number == none {
125 |       title += " " + number
126 |     }
127 |     title = titlefmt(title)
128 |     body = bodyfmt(body)
129 |     pad(
130 |       ..padding,
131 |       block(
132 |         width: 100%,
133 |         inset: 1.2em,
134 |         radius: 0.3em,
135 |         breakable: false,
136 |         ..blockargs.named(),
137 |         ..blockargs_individual.named(),
138 |         [#title#name#separator#body]
139 |       )
140 |     )
141 |   }
142 |   return thmenv(
143 |     identifier,
144 |     base,
145 |     base_level,
146 |     boxfmt
147 |   ).with(
148 |     supplement: supplement,
149 |   )
150 | }
151 | 
152 | 
153 | #let thmplain = thmbox.with(
154 |   padding: (top: 0em, bottom: 0em),
155 |   breakable: true,
156 |   inset: (top: 0em, left: 1.2em, right: 1.2em),
157 |   namefmt: name => emph([(#name)]),
158 |   titlefmt: emph,
159 | )
160 | 
161 | 
162 | // Track whether the qed symbol has already been placed in a proof
163 | #let thm-qed-done = state("thm-qed-done", false)
164 | 
165 | // The configured QED symbol
166 | #let thm-qed-symbol = state("thm-qed-symbol", $qed$)
167 | 
168 | // Show the qed symbol, update state
169 | #let thm-qed-show = {
170 |   thm-qed-done.update(true)
171 |   context [#thm-qed-symbol.get()]
172 | }
173 | 
174 | // If placed in a block equation/enum/list, place a qed symbol to its right
175 | #let qedhere = metadata("thm-qedhere")
176 | 
177 | // Checks if content x contains the qedhere tag
178 | #let thm-has-qedhere(x) = {
179 |   if x == "thm-qedhere" {
180 |     return true
181 |   }
182 | 
183 |   if type(x) == content {
184 |     for (f, c) in x.fields() {
185 |       if thm-has-qedhere(c) {
186 |         return true
187 |       }
188 |     }
189 |   }
190 | 
191 |   if type(x) == array {
192 |     for c in x {
193 |       if thm-has-qedhere(c) {
194 |         return true
195 |       }
196 |     }
197 |   }
198 | 
199 |   return false
200 | }
201 | 
202 | 
203 | // bodyfmt for proofs
204 | #let proof-bodyfmt(body) = {
205 |   thm-qed-done.update(false)
206 |   body
207 |   context {
208 |     if thm-qed-done.at(here()) == false {
209 |       h(1fr)
210 |       thm-qed-show
211 |     }
212 |   }
213 | }
214 | 
215 | #let thmproof(..args) = thmplain(
216 |     ..args,
217 |     namefmt: emph,
218 |     bodyfmt: proof-bodyfmt,
219 |     ..args.named()
220 | ).with(numbering: none)
221 | 
222 | 
223 | #let thmrules(qed-symbol: $qed$, doc) = {
224 | 
225 |   show figure.where(kind: "thmenv"): set block(breakable: true)
226 |   show figure.where(kind: "thmenv"): set align(start)
227 |   show figure.where(kind: "thmenv"): it => it.body
228 | 
229 |   show ref: it => {
230 |     if it.element == none {
231 |       return it
232 |     }
233 |     if it.element.func() != figure {
234 |       return it
235 |     }
236 |     if it.element.kind != "thmenv" {
237 |       return it
238 |     }
239 | 
240 |     let supplement = it.element.supplement
241 |     if it.citation.supplement != none {
242 |       supplement = it.citation.supplement
243 |     }
244 | 
245 |     let loc = it.element.location()
246 |     let thms = query(selector(<meta:thmenvcounter>).after(loc))
247 |     let number = thmcounters.at(thms.first().location()).at("latest")
248 |     return link(
249 |       it.target,
250 |       [#supplement~#numbering(it.element.numbering, ..number)]
251 |     )
252 |   }
253 | 
254 |   show math.equation.where(block: true): eq => {
255 |     if thm-has-qedhere(eq) and thm-qed-done.at(eq.location()) == false {
256 |       grid(
257 |         columns: (1fr, auto, 1fr),
258 |         [], eq, align(right + horizon)[#thm-qed-show]
259 |       )
260 |     } else {
261 |       eq
262 |     }
263 |   }
264 | 
265 |   show enum.item: it => {
266 |     show metadata.where(value: "thm-qedhere"): {
267 |       h(1fr)
268 |       thm-qed-show
269 |     }
270 |     it
271 |   }
272 | 
273 |   show list.item: it => {
274 |     show metadata.where(value: "thm-qedhere"): {
275 |       h(1fr)
276 |       thm-qed-show
277 |     }
278 |     it
279 |   }
280 | 
281 |   thm-qed-symbol.update(qed-symbol)
282 | 
283 |   doc
284 | }
285 | 


--------------------------------------------------------------------------------
/manual.typ:
--------------------------------------------------------------------------------
  1 | #import "manual_template.typ": *
  2 | #import "theorems.typ": *
  3 | #show: thmrules
  4 | 
  5 | #show: project.with(
  6 |   title: "typst-theorems",
  7 |   authors: (
  8 |     "sahasatvik",
  9 |   ),
 10 |   url: "https://github.com/sahasatvik/typst-theorems"
 11 | )
 12 | 
 13 | 
 14 | 
 15 | = Introduction
 16 | 
 17 | The `typst-theorems` package provides `Typst` functions that help create
 18 | numbered `theorem` environments. This is heavily inspired by the `\newtheorem`
 19 | functionality of `LaTeX`.
 20 | 
 21 | A _theorem environment_ lets you wrap content together with automatically
 22 | updating _numbering_ information. Such environments use internal `state`
 23 | counters for this purpose. Environments can
 24 | 
 25 | - share the same counter (_Theorems_ and _Lemmas_ often do so)
 26 | - keep a global count, or be attached to
 27 |   - other environments (_Corollaries_ are often numbered based upon the parent _Theorem_)
 28 |   - headings
 29 | - have a numbering level depth fixed (for instance, use only top level heading
 30 |   numbers)
 31 | - be referenced elsewhere in the document, via `label`s
 32 | 
 33 | 
 34 | = Using `typst-theorems`
 35 | 
 36 | Import all functions provided by `typst-theorems` using
 37 | ```typst
 38 | #import "theorems.typ": *
 39 | #show: thmrules
 40 | ```
 41 | The second line is crucial for displaying `thmenv`s and references correctly!
 42 | 
 43 | The core of this module consists of `thmenv`.
 44 | The functions `thmbox`, `thmplain`, and `thmproof` provide some simple
 45 | defaults for the appearance of `thmenv`s.
 46 | 
 47 | 
 48 | = Feature demonstration <feat>
 49 | 
 50 | Create box-like _theorem environments_ using `thmbox`, a wrapper around
 51 | `thmenv` which provides some simple defaults.
 52 | 
 53 | ```typst
 54 | #let theorem = thmbox(
 55 |   "theorem",              // identifier
 56 |   "Theorem",              // head
 57 |   fill: rgb("#e8e8f8")
 58 | )
 59 | ```
 60 | #let theorem = thmbox(
 61 |   "theorem",
 62 |   "Theorem",
 63 |   fill: rgb("#e8e8f8")
 64 | )
 65 | 
 66 | Such definitions are convenient to place in the preamble or a template; use
 67 | the environment in your document via
 68 | ```typst
 69 | #theorem("Euclid")[
 70 |   There are infinitely many primes.
 71 | ] <euclid>
 72 | ```
 73 | 
 74 | This produces the following.
 75 | #theorem("Euclid")[
 76 |   There are infinitely many primes.
 77 | ] <euclid>
 78 | 
 79 | Note that the `name` is optional. This `theorem` environment will be numbered
 80 | based on its parent `heading` counter, with successive `theorem`s
 81 | automatically updating the final index.
 82 | 
 83 | The `<euclid>` label can be used to refer to this Theorem via the reference
 84 | `@euclid`. Go to @references to read more.
 85 | 
 86 | You can create another environment which uses the same counter, say for
 87 | _Lemmas_, as follows.
 88 | 
 89 | ```typst
 90 | #let lemma = thmbox(
 91 |   "theorem",              // identifier - same as that of theorem
 92 |   "Lemma",                // head
 93 |   fill: rgb("#efe6ff")
 94 | )
 95 | 
 96 | #lemma[
 97 |   If $n$ divides both $x$ and $y$, it also divides $x - y$.
 98 | ]
 99 | ```
100 | #let lemma = thmbox(
101 |   "theorem",
102 |   "Lemma",
103 |   fill: rgb("#efe6ff")
104 | )
105 | #lemma[
106 |   If $n$ divides both $x$ and $y$, it also divides $x - y$.
107 | ]
108 | 
109 | You can _attach_ other environments to ones defined earlier. For instance,
110 | _Corollaries_ can be created as follows.
111 | 
112 | ```typst
113 | #let corollary = thmbox(
114 |   "corollary",            // identifier
115 |   "Corollary",            // head
116 |   base: "theorem",        // base - use the theorem counter
117 |   fill: rgb("#f8e8e8")
118 | )
119 | 
120 | #corollary(numbering: "1.1")[
121 |   If $n$ divides two consecutive natural numbers, then $n = 1$.
122 | ]
123 | ```
124 | #let corollary = thmbox(
125 |   "corollary",
126 |   "Corollary",
127 |   base: "theorem",
128 |   fill: rgb("#f8e8e8")
129 | )
130 | 
131 | #corollary(numbering: "1.1")[
132 |   If $n$ divides two consecutive natural numbers, then $n = 1$.
133 | ]
134 | 
135 | Note that we have provided a `numbering` string; this can be any valid
136 | numbering pattern as described in the
137 | #link("https://typst.app/docs/reference/meta/numbering/")[numbering]
138 | documentation.
139 | 
140 | 
141 | == Proofs
142 | 
143 | The `thmproof` function gives nicer defaults for formatting proofs.
144 | ```typst
145 | #let proof = thmproof("proof", "Proof")
146 | 
147 | #proof([of @euclid])[
148 |   Suppose to the contrary that $p_1, p_2, dots, p_n$ is a finite enumeration
149 |   of all primes. Set $P = p_1 p_2 dots p_n$. Since $P + 1$ is not in our list,
150 |   it cannot be prime. Thus, some prime factor $p_j$ divides $P + 1$.  Since
151 |   $p_j$ also divides $P$, it must divide the difference $(P + 1) - P = 1$, a
152 |   contradiction.
153 | ]
154 | ```
155 | #let proof = thmproof("proof", "Proof")
156 | 
157 | #proof([of @euclid])[
158 |   Suppose to the contrary that $p_1, p_2, dots, p_n$ is a finite enumeration
159 |   of all primes. Set $P = p_1 p_2 dots p_n$. Since $P + 1$ is not in our list,
160 |   it cannot be prime. Thus, some prime factor $p_j$ divides $P + 1$.  Since
161 |   $p_j$ also divides $P$, it must divide the difference $(P + 1) - P = 1$, a
162 |   contradiction.
163 | ]
164 | 
165 | If your proof ends in a block equation, or a list/enum, you can place
166 | `qedhere` to correctly position the qed symbol.
167 | ```typst
168 | #theorem[
169 |   There are arbitrarily long stretches of composite numbers.
170 | ]
171 | #proof[
172 |   For any $n > 2$, consider $
173 |     n! + 2, quad n! + 3, quad ..., quad n! + n #qedhere
174 |   $
175 | ]
176 | ```
177 | #theorem[
178 |   There are arbitrarily long stretches of composite numbers.
179 | ]
180 | #proof[
181 |   For any $n > 2$, consider $
182 |     n! + 2, quad n! + 3, quad ..., quad n! + n #qedhere
183 |   $
184 | ]
185 | 
186 | *Caution*: The `qedhere` symbol does not play well with numbered/multiline
187 | equations!
188 | 
189 | You can set a custom qed symbol (say $square$) by setting the appropriate
190 | option in `thmrules` as follows.
191 | ```typst
192 | #show: thmrules.with(qed-symbol: $square$)
193 | ```
194 | 
195 | == Suppressing numbering
196 | Supplying `numbering: none` to an environment suppresses numbering for that
197 | block, and prevents it from updating its counter.
198 | 
199 | ```typst
200 | #let example = thmplain(
201 |   "example",
202 |   "Example"
203 | ).with(numbering: none)
204 | 
205 | #example[
206 |   The numbers $2$, $3$, and $17$ are prime.
207 | ]
208 | ```
209 | #let example = thmplain(
210 |   "example",
211 |   "Example"
212 | ).with(numbering: none)
213 | 
214 | #example[
215 |   The numbers $2$, $3$, and $17$ are prime.
216 | ]
217 | 
218 | Here, we have used the `thmplain` function, which is identical to `thmbox` but
219 | sets some plainer defaults. You can also write
220 | ```typst
221 | #lemma(numbering: none)[
222 |   The square of any even number is divisible by $4$.
223 | ]
224 | #lemma[
225 |   The square of any odd number is one more than a multiple of $4$.
226 | ]
227 | ```
228 | #lemma(numbering: none)[
229 |   The square of any even number is divisible by $4$.
230 | ]
231 | #lemma[
232 |   The square of any odd number is one more than a multiple of $4$.
233 | ]
234 | 
235 | Note that the last _Lemma_ is _not_ numbered 3.1.2!
236 | 
237 | You can also override the automatic numbering as follows.
238 | ```typst
239 | #lemma(number: "42")[
240 |   The square of any natural number cannot be two more than a multiple of 4.
241 | ]
242 | ```
243 | #lemma(number: "42")[
244 |   The square of any natural number cannot be two more than a multiple of 4.
245 | ]
246 | 
247 | Note that this does _not_ affect the counters either!
248 | 
249 | 
250 | == Limiting depth
251 | 
252 | You can limit the number of levels of the `base` numbering used as follows.
253 | ```typst
254 | #let definition = thmbox(
255 |   "definition",
256 |   "Definition",
257 |   base_level: 1,          // take only the first level from the base
258 |   stroke: rgb("#68ff68") + 1pt
259 | )
260 | 
261 | #definition("Prime numbers")[
262 |   A natural number is called a _prime number_ if it is greater than $1$ and
263 |   cannot be written as the product of two smaller natural numbers. <prime>
264 | ]
265 | ```
266 | #let definition = thmbox(
267 |   "definition",
268 |   "Definition",
269 |   base_level: 1,
270 |   stroke: rgb("#68ff68") + 1pt
271 | )
272 | #definition("Prime numbers")[
273 |   A natural number is called a _prime number_ if it is greater than $1$ and
274 |   cannot be written as the product of two smaller natural numbers. <prime>
275 | ]
276 | 
277 | Note that this environment is _not_ numbered 3.2.1!
278 | 
279 | ```typst
280 | #definition("Composite numbers")[
281 |   A natural number is called a _composite number_ if it is greater than $1$
282 |   and not prime.
283 | ]
284 | ```
285 | #definition("Composite numbers")[
286 |   A natural number is called a _composite number_ if it is greater than $1$
287 |   and not prime.
288 | ]
289 | 
290 | Setting a `base_level` higher than what `base` provides will introduce padded
291 | zeroes.
292 | 
293 | ```typst
294 | #example(base_level: 4, numbering: "1.1")[
295 |   The numbers $4$, $6$, and $42$ are composite.
296 | ]
297 | ```
298 | #example(base_level: 4, numbering: "1.1")[
299 |   The numbers $4$, $6$, and $42$ are composite.
300 | ]
301 | 
302 | 
303 | == Custom formatting
304 | The `thmbox` function lets you specify rules for formatting the `title`, the
305 | `name`, and the `body` individually. Here, the `title` refers to the `head`
306 | and `number` together.
307 | 
308 | ```typst
309 | #let proof-custom = thmplain(
310 |   "proof",
311 |   "Proof",
312 |   base: "theorem",
313 |   titlefmt: smallcaps,
314 |   bodyfmt: body => [
315 |     #body #h(1fr) $square$    // float a QED symbol to the right
316 |   ]
317 | ).with(numbering: none)
318 | 
319 | #lemma[
320 |   All even natural numbers greater than 2 are composite.
321 | ]
322 | #proof-custom[
323 |   Every even natural number $n$ can be written as the product of the natural
324 |   numbers $2$ and $n\/2$. When $n > 2$, both of these are smaller than $2$
325 |   itself.
326 | ]
327 | ```
328 | #let proof-custom = thmplain(
329 |   "proof",
330 |   "Proof",
331 |   base: "theorem",
332 |   titlefmt: smallcaps,
333 |   bodyfmt: body => [
334 |     #body #h(1fr) $square$
335 |   ]
336 | ).with(numbering: none)
337 | 
338 | #lemma[
339 |   All even natural numbers greater than 2 are composite.
340 | ]
341 | #proof-custom[
342 |   Every even natural number $n$ can be written as the product of the natural
343 |   numbers $2$ and $n\/2$. When $n > 2$, both of these are smaller than $2$
344 |   itself.
345 | ]
346 | 
347 | You can go even further and use the `thmenv` function directly. It accepts an
348 | `identifier`, a `base`, a `base_level`, and a `fmt` function.
349 | ```typst
350 | #let notation = thmenv(
351 |   "notation",                 // identifier
352 |   none,                       // base - do not attach, count globally
353 |   none,                       // base_level - use the base as-is
354 |   (name, number, body, color: black) => [
355 |                               // fmt - format content using the environment
356 |                               // name, number, body, and an optional color
357 |     #text(color)[#h(1.2em) *Notation (#number) #name*]:
358 |     #h(0.2em)
359 |     #body
360 |     #v(0.5em)
361 |   ]
362 | ).with(numbering: "I")        // use Roman numerals
363 | 
364 | #notation[
365 |   The variable $p$ is reserved for prime numbers.
366 | ]
367 | #notation("for Reals", color: green)[
368 |   The variable $x$ is reserved for real numbers.
369 | ]
370 | ```
371 | #let notation = thmenv(
372 |   "notation",                 // identifier
373 |   none,                       // base - do not attach, count globally
374 |   none,                       // base_level - use the base as-is
375 |   (name, number, body, color: black) => [
376 |                               // fmt - format content using the environment name, number, body, and an optional color
377 |     #text(color)[#h(1.2em) *Notation (#number) #name*]:
378 |     #h(0.2em)
379 |     #body
380 |     #v(0.5em)
381 |   ]
382 | ).with(numbering: "I")        // use Roman numerals
383 | 
384 | #notation[
385 |   The variable $p$ is reserved for prime numbers.
386 | ]
387 | #notation("for Reals", color: green)[
388 |   The variable $x$ is reserved for real numbers.
389 | ]
390 | 
391 | Note that the `color: green` named argument supplied to the theorem
392 | environment gets passed to the `fmt` function. In general, all extra named
393 | arguments supplied to the theorem will be passed to `fmt`.
394 | On the other hand, the positional argument `"for Reals"` will always be
395 | interpreted as the `name` argument in `fmt`.
396 | 
397 | ```typst
398 | #lemma(title: "Lem.", stroke: 1pt)[
399 |   All multiples of 3 greater than 3 are composite.
400 | ]
401 | ```
402 | #lemma(title: "Lem.", stroke: 1pt)[
403 |   All multiples of 3 greater than 3 are composite.
404 | ]
405 | 
406 | Here, we override the `title` (which defaults to the `head`) as well as the
407 | `stroke` in the `fmt` produced by `thmbox`. All `block` arguments can be
408 | overridden in `thmbox` environments in this way.
409 | 
410 | 
411 | == Labels and references <references>
412 | 
413 | You can place a `<label>` outside a theorem environment, and reference it
414 | later via `@` references! For example, go back to @euclid.
415 | 
416 | ```typst
417 | Recall that there are infinitely many prime numbers via @euclid.
418 | ```
419 | #pad(
420 |   left: 1.2em,
421 |   [
422 |     Recall that there are infinitely many prime numbers via @euclid.
423 |   ]
424 | )
425 | 
426 | 
427 | ```typst
428 | You can reference future environments too, like @oddprime[Cor.].
429 | ```
430 | #pad(
431 |   left: 1.2em,
432 |   [
433 |     You can reference future environments too, like @oddprime[Cor.].
434 |   ]
435 | )
436 | 
437 | ```typst
438 | #lemma(supplement: "Lem.", refnumbering: "(1.1)")[
439 |   All primes apart from $2$ and $3$ are of the form $6k plus.minus 1$.
440 | ] <primeform>
441 | 
442 | You can modify the supplement and numbering to be used in references, like @primeform.
443 | ```
444 | #lemma(supplement: "Lem.", refnumbering: "(1.1)")[
445 |   All primes apart from $2$ and $3$ are of the form $6k plus.minus 1$.
446 | ] <primeform>
447 | 
448 | You can modify the supplement and numbering to be used in references, like @primeform.
449 | 
450 | *Caution*: Links created by references to `thmenv`s will be styled according
451 | to `#show link:` rules.
452 | 
453 | 
454 | == Overriding `base`
455 | 
456 | ```typst
457 | #let remark = thmplain("remark", "Remark", base: "heading")
458 | #remark[
459 |   There are infinitely many composite numbers.
460 | ]
461 | 
462 | #corollary[
463 |   All primes greater than $2$ are odd.
464 | ] <oddprime>
465 | #remark(base: "corollary")[
466 |   Two is a _lone prime_.
467 | ]
468 | ```
469 | 
470 | #let remark = thmplain("remark", "Remark", base: "heading")
471 | #remark[
472 |   There are infinitely many composite numbers.
473 | ]
474 | #corollary[
475 |   All primes greater than $2$ are odd.
476 | ] <oddprime>
477 | #remark(base: "corollary")[
478 |   Two is a _lone prime_.
479 | ]
480 | 
481 | This `remark` environment, which would normally be attached to the current
482 | _heading_, now uses the `corollary` as a base.
483 | 
484 | 
485 | #v(4em)
486 | 
487 | = Function reference
488 | == `thmenv`
489 | 
490 | The `thmenv` function produces a _theorem environment_.
491 | ```typst
492 | #let thmenv(
493 |   identifier,             // environment counter name
494 |   base,                   // base counter name, can be "heading" or none
495 |   base_level,             // number of base number levels to use
496 |   fmt                     // formatting function of the form
497 |                           // (name, number, body, ..args) -> content
498 | ) = { ... }
499 | ```
500 | 
501 | The `fmt` function must accept a theorem `name`, `number`, `body`, and produce
502 | formatted content. It may also accept additional positional arguments, via
503 | `args`.
504 | 
505 | A _theorem environment_ is itself a map of the following form.
506 | ```typst
507 | (
508 |   ..args,
509 |   body,                   // body content
510 |   number: auto,           // number, overrides numbering if present
511 |   numbering: "1.1",       // numbering style, can be a function
512 |   refnumbering: auto,     // numbering style used in references,
513 |                           // defaults to "numbering"
514 |   supplement: identifier, // supplement used in references
515 |   base: base,             // base counter name override
516 |   base_level: base_level  // base_level override
517 | ) -> content
518 | ```
519 | 
520 | Positional arguments in `args` are as follows
521 | - `name`: The name of the theorem, typically displayed after the title.
522 | 
523 | All additional named arguments in `args` will be passed on to the associated
524 | `fmt` function supplied in `thmenv`.
525 | 
526 | 
527 | == `thmbox` and `thmplain`
528 | 
529 | The `thmbox` wraps `thmenv`, supplying a box-like `fmt` function.
530 | ```typst
531 | #let thmbox(
532 |   identifier,             // identifier
533 |   head,                   // head - common name, used in the title
534 |   ..blockargs,            // named arguments, passed to #block
535 |   supplement: auto,       // supplement for references, defaults to "head"
536 |   padding: (top: 0.5em, bottom: 0.5em),
537 |                           // box padding, passed to #pad
538 |   namefmt: x => [(#x)],   // formatting for name
539 |   titlefmt: strong,       // formatting for title (head + number)
540 |   bodyfmt: x => x,        // formatting for body
541 |   separator: [#h(0.1em):#h(0.2em)],
542 |                           // separator inserted between name and body
543 |   base: "heading",        // base - defaults to using headings
544 |   base_level: none,       // base_level - defaults to using base as-is
545 | ) = { ... }
546 | ```
547 | 
548 | The `thmbox` function sets the following defaults for the `block`.
549 | ```typst
550 | (
551 |   width: 100%,
552 |   inset: 1.2em,
553 |   radius: 0.3em,
554 |   breakable: false,
555 | )
556 | ```
557 | 
558 | The `thmplain` function is identical to `thmbox`, except with plainer
559 | defaults.
560 | 
561 | ```typst
562 | #let thmplain = thmbox.with(
563 |   padding: (top: 0em, bottom: 0em),
564 |   breakable: true,
565 |   inset: (top: 0em, left: 1.2em, right: 1.2em),
566 |   namefmt: name => emph([(#name)]),
567 |   titlefmt: emph,
568 | )
569 | ```
570 | 
571 | 
572 | == `thmproof`, `proof-bodyfmt` and `qedhere`
573 | 
574 | The `thmproof` function is identical to `thmplain`, except with defaults
575 | appropriate for proofs.
576 | 
577 | ```typst
578 | #let thmproof(..args) = thmplain(
579 |     ..args,
580 |     namefmt: emph,
581 |     bodyfmt: proof-bodyfmt,
582 |     ..args.named()
583 | ).with(numbering: none)
584 | ```
585 | 
586 | The `proof-bodyfmt` function is a `bodyfmt` function that automatically places
587 | a qed symbol at the end of the body.
588 | 
589 | You can place `#qedhere` inside a block equation, or at the end of a list/enum
590 | item to place the qed symbol on the same line.
591 | 
592 | 
593 | == `thmrules`
594 | 
595 | The `thmrules` show rule sets important styling rules for theorem
596 | environments, references, and equations in proofs.
597 | 
598 | ```typst
599 | #let thmrules(
600 |   qed-symbol: $qed$,      // QED symbol used in proofs
601 |   doc
602 | ) = { ... }
603 | ```
604 | 
605 | 
606 | = Acknowledgements
607 | 
608 | Thanks to
609 | - #link("https://github.com/MJHutchinson")[MJHutchinson] for suggesting and
610 |   implementing the `base_level` and `base: none` features,
611 | - #link("https://github.com/rmolinari")[rmolinari] for suggesting and
612 |   implementing the `separator: ...` feature,
613 | - #link("https://github.com/DVDTSB")[DVDTSB] for contributing
614 |   - the idea of passing named arguments from the theorem directly to the `fmt`
615 |     function.
616 |   - the `number: ...` override feature.
617 |   - the `title: ...` override feature in `thmbox`.
618 | - The awesome devs of #link("https://typst.app/")[typst.app] for their
619 |   support.
620 | 


--------------------------------------------------------------------------------
/differential_calculus.typ:
--------------------------------------------------------------------------------
  1 | #import "theorems.typ": *
  2 | #show: thmrules
  3 | 
  4 | // Define theorem environments
  5 | 
  6 | #let theorem = thmbox(
  7 |   "theorem",
  8 |   "Theorem",
  9 |   fill: rgb("#e8e8f8")
 10 | )
 11 | #let lemma = thmbox(
 12 |   "theorem",            // Lemmas use the same counter as Theorems
 13 |   "Lemma",
 14 |   fill: rgb("#efe6ff")
 15 | )
 16 | #let corollary = thmbox(
 17 |   "corollary",
 18 |   "Corollary",
 19 |   base: "theorem",      // Corollaries are 'attached' to Theorems
 20 |   fill: rgb("#f8e8e8")
 21 | )
 22 | 
 23 | #let definition = thmbox(
 24 |   "definition",         // Definitions use their own counter
 25 |   "Definition",
 26 |   fill: rgb("#e8f8e8")
 27 | )
 28 | 
 29 | #let exercise = thmbox(
 30 |   "exercise",
 31 |   "Exercise",
 32 |   stroke: rgb("#ffaaaa") + 1pt,
 33 |   base: none,           // Unattached: count globally
 34 | ).with(numbering: "I")  // Use Roman numerals
 35 | 
 36 | // Examples and remarks are not numbered
 37 | #let example = thmplain("example", "Example").with(numbering: none)
 38 | #let remark = thmplain(
 39 |   "remark",
 40 |   "Remark",
 41 |   inset: 0em
 42 | ).with(numbering: none)
 43 | 
 44 | // Proofs are attached to theorems, although they are not numbered
 45 | #let proof = thmproof(
 46 |   "proof",
 47 |   "Proof",
 48 |   base: "theorem",
 49 | )
 50 | 
 51 | #let solution = thmplain(
 52 |   "solution",
 53 |   "Solution",
 54 |   base: "exercise",
 55 |   inset: 0em,
 56 | ).with(numbering: none)
 57 | 
 58 | 
 59 | // Template
 60 | 
 61 | #let project(title: "", authors: (), body) = {
 62 |   set document(author: authors, title: title)
 63 |   set text(font: "Libertinus Serif", lang: "en")
 64 |   set heading(numbering: "1.1.", )
 65 |   set par(justify: true)
 66 | 
 67 |   show heading: it => [
 68 |     #v(1em)
 69 |     #it
 70 |   ]
 71 | 
 72 |   align(center)[
 73 |     #block(text(weight: 700, 1.75em, title))
 74 |     #v(2em)
 75 |   ]
 76 | 
 77 |   outline(fill: none, indent: true)
 78 | 
 79 |   body
 80 | }
 81 | 
 82 | // Shorthand for vectors
 83 | #let va = $bold(a)$
 84 | #let vb = $bold(b)$
 85 | #let vx = $bold(x)$
 86 | #let vy = $bold(y)$
 87 | #let vz = $bold(z)$
 88 | #let vv = $bold(v)$
 89 | #let vw = $bold(w)$
 90 | #let ve = $bold(e)$
 91 | 
 92 | // Mapping arrow
 93 | #let mapsto = $arrow.r.bar$
 94 | 
 95 | // Operators
 96 | #let grad = (x) => $nabla #x$
 97 | #let ip = (x, y) => $angle.l #x, #y angle.r$
 98 | #let pp = (x, y) => $(diff #x) / (diff #y)$
 99 | #let dd = (x, y) => $(d #x) / (d #y)$
100 | #let Df = $D f$
101 | #let Dg = $D g$
102 | #let DT = $D T$
103 | 
104 | 
105 | // Document starts here
106 | 
107 | #show: project.with(
108 |   title: "Notes on Differential Calculus"
109 | )
110 | 
111 | = Differentiability
112 | 
113 | #definition[
114 |     Let $f : (a, b) -> RR^n$, and let $f_i = pi_i compose  f$ be its
115 |     components. Then, $f$ is differentiable at $t_0  in  (a, b)$ if the
116 |     following limit exists. $ 
117 |         f'(t_0) =  lim _(h -> 0)  frac(f(t_0 + h) - f(t_0), h)  .
118 |     $
119 |     #remark[
120 |         The vector $f'(t_0)$ represents the tangent to the curve $f$ at the
121 |         point $f(t_0)$. The full tangent line is the parametric curve $f(t) +
122 |         f'(t_0)(t - t_0)$.
123 |     ]
124 | ]
125 | 
126 | #definition[
127 |     Let $U  subset.eq  RR^n$ be open, and let $f: U -> RR^m$. Then, $f$ is
128 |     differentiable at $x  in  U$ if there exists a linear transformation $
129 |     lambda : RR^n -> RR^m$ such that $ 
130 |          lim _(h -> 0)   frac(f(x + h) - f(x) -  lambda  h,  norm(h))  = 0.
131 |     $ The derivative of $f$ at $x$ is denoted by $lambda  = Df(x)$.
132 |     #remark[
133 |         In a neighbourhood of $x$, we may approximate $ 
134 |             f(x + h)  approx  f(x) + Df(x)(h).
135 |         $
136 |     ]
137 |     #remark[
138 |         The statement that this quantity goes to zero means that each of the
139 |         $m$ components must also go to zero. For each of these limits, there
140 |         are $n$ axes along which we can let $h -> 0$. As a result, we obtain
141 |         $m times n$ limits, which allow us to identify the $m times n$
142 |         components of the matrix representing the linear transformation
143 |         $lambda$ (in the standard basis). These are the partial derivatives of
144 |         $f$, and the matrix of $lambda$ is the Jacobian matrix of $f$
145 |         evaluated at $x$.
146 |     ]
147 | ]
148 | 
149 | #example[
150 |     Let $T: RR^n -> RR^m$ be a linear map. By choosing $ lambda = T$, we see
151 |     that $T$ is differentiable everywhere, with $DT(x) = T$ for every choice
152 |     of $x  in  RR^n$. This is made obvious by the fact that the best linear
153 |     approximation of a linear map at some point is the map itself; indeed, the
154 |     'approximation' is exact.
155 | ]
156 | 
157 | #lemma[
158 |     If $f: RR^n -> RR^m$ is differentiable at $x  in  RR^n$, with derivative
159 |     $Df(x)$, then
160 |         + $f$ is continuous at $x$.
161 |         + The linear transformation $Df(x)$ is unique.
162 | ]
163 | #proof[
164 |     We prove the second part. Suppose that $lambda$, $mu$ satisfy the
165 |     requirements for $Df(x)$; it can be shown that $ lim _(h -> 0) ( lambda  -
166 |     mu )h \/  norm(h)  = 0$. Now, if $ lambda  v  !=   mu  v$ for some
167 |     non-zero vector $v  in  RR^n$, then $ 
168 |          lambda  v -  mu  v =   frac(lambda (t v) - mu(t v), norm(t v))
169 |          dot.c norm(v) -> 0,
170 |     $ a contradiction.
171 | ]
172 | 
173 | = Chain rule
174 | 
175 | #exercise[
176 |     Let $T: RR^n -> RR^m$ be a linear transformation. Then, there exists $M
177 |     > 0$ such that for all $ vx   in  RR^n$, we have $ 
178 |          norm(T vx )   <=  M norm( vx ) .
179 |     $
180 |     #solution[
181 |         Set $ vv _i = T( ve _i)$ where $ ve _i$ are the standard unit basis
182 |         vectors of $RR^n$. Then, $ 
183 |              norm(T vx )
184 |              =   norm( sum _(i) x_i vv _i)
185 |              <=  sum _(i) norm(x_i vv _i) 
186 |              <=  max  norm( vv _i)   sum _i |x_i|.
187 |         $  Since each $|x_i|  <=   norm( vx ) $, set $M = n max  norm( vv _i)
188 |         $ and write $ 
189 |              norm(T vx )
190 |              <=  max  norm( vv _i)  sum _i |x_i|
191 |              <=  max  norm( vv _i)  dot.c  n norm( vx )
192 |              = M norm( vx ) .
193 |         $
194 |     ]
195 | ]
196 | #theorem("Chain Rule")[
197 |     Let $f: RR^n -> RR^m$, $g: RR^m -> RR^k$ where $f$ is differentiable at $a
198 |     in  RR^n$ and $g$ is differentiable at $f(a)  in  RR^m$.  Then, $g compose
199 |     f$ is differentiable, with $D(g compose  f)(a) = Dg(f(a))  compose Df(a)$.
200 |     Note that this means that the Jacobian matrices simply multiply.
201 | ]
202 | #proof[
203 |     Set $b = f(a)  in  RR^m$, $ lambda  = Df(a)$, $ mu  = Dg(f(a))$. Define 
204 |     $
205 |          phi : RR^n -> RR^m, quad   phi (x) &= f(x) - f(a) -  lambda (x
206 |         - a), \
207 |          psi : RR^m -> RR^k, quad   psi (y) &= g(y) - g(b) -  mu (y - b).
208 |     $
209 |     We claim that $ 
210 |          lim _(x -> a) frac(g compose  f(x) - g compose  f(a) -  mu  compose   lambda (x
211 |         - a),  norm(x - a) ) = 0.
212 |     $  Write the numerator as $ 
213 |         g compose  f(x) - g compose  f(a) -  mu  compose   lambda (x - a) =  psi (f(x)) +
214 |          mu ( phi (x)).
215 |     $  Note that $ 
216 |          lim _(x -> a)   frac( phi (x),  norm(x - a)  )  = 0, quad
217 |          lim _(y -> b)   frac( psi (y),  norm(y - b)  )  = 0.
218 |     $  Thus, find $M > 0$ such that $ norm( mu ( phi (x)))   <= 
219 |      norm( phi (x)) $ for all $x  in  RR^n$, hence $ 
220 |          lim _(x -> a)   frac( norm( mu ( phi (x))) ,  norm(x - a)  )   <=
221 |          lim _{x -> a}   frac(M norm( phi (x)) ,  norm(x - a)  )  = 0.
222 |     $  Now write $ 
223 |          lim _(f(x) -> b)   frac( psi (f(x)),  norm(f(x)  - b) )  = 0, 
224 |     $  hence for any $ epsilon  > 0$, there is a neighbourhood of $b$ on which $ 
225 |          norm( psi (f(x)))   <=   epsilon   norm(f(x) - b)  =  epsilon 
226 |          norm( phi (x) +  lambda (x - a)) .
227 |     $  Apply the triangle inequality and find $M' > 0$ such that $ 
228 |          norm( psi (f(x)))   <= 
229 |          epsilon  norm( phi (x))  +  epsilon  M' norm(x - a) .
230 |     $  Thus, $ 
231 |          lim _(x -> a)   frac( norm( psi (f(x))) ,  norm(x - a)  )   <= 
232 |          lim _(x -> a)   frac( epsilon  norm( phi (x)) ,  norm(x - a)  )  +  epsilon 
233 |         M' =  epsilon  M'.
234 |     $  Since $ epsilon  > 0$ was arbitrary, this limit is zero, completing the proof.
235 | ] 
236 | 
237 | 
238 | = Partial derivatives
239 | 
240 | #definition[
241 |     Let $U  subset.eq  RR^n$ be open, and let $f: U -> RR$. The partial
242 |     derivative of $f$ with respect to the coordinate $x_j$ at some $a  in  U$
243 |     is defined by the following limit, if it exists. $ 
244 |          pp(f, x_j) (a) =  lim _(h -> 0)   frac(f(a + h ve_j) - f(a), h)  .
245 |     $  
246 | ]
247 | #lemma[
248 |     If $f: U -> RR$ is differentiable at a point $a  in  RR^n$, then $ 
249 |         Df(a)(x_1,  dots , x_n) = x_1 \, pp(f, x_1) (a) +  dots  + x_n
250 |         \, pp(f, x_n) (a).
251 |     $  
252 | ]
253 | #example[
254 |     Consider $ 
255 |         f: RR^2 -> RR,  quad  (x, y)  mapsto  cases(
256 |              (x y) \/ (x^2 + y^2)\, & " if "  (x, y)  !=  (0, 0),
257 |             0\, & " if "  (x, y) = (0, 0).
258 |         )
259 |     $  Note that $f$ is not differentiable at $(0, 0)$; it is not even continuous
260 |     there. However, both partial derivatives of $f$ exist at $(0, 0)$.
261 | ]
262 | 
263 | #lemma[
264 |     If $f: RR^n -> RR^m$ is differentiable at $a  in  RR^n$, then the matrix
265 |     representation of $Df(a)$ in the standard basis is given by $ 
266 |         [Df(a)] = [ pp(f_i, x_j) (a)]_(i j).
267 |     $  
268 | ]
269 | 
270 | #lemma[
271 |     Let $f: RR^n -> RR^m$ be differentiable at $a  in  RR^n$, and let $g: RR^m
272 |     -> RR^k$ be differentiable at $f(a)  in  RR^m$. Then, the matrix
273 |     representation of $D(g compose  f)(a)$ in the standard basis is the
274 |     product $ 
275 |         [D(g compose  f)(a)] = [Dg(f(a))][Df(a)] = [ sum _( ell  = 1)^m
276 |          pp(g_i, y_ ell )  pp(f_ ell , x_j) ]_(i j).
277 |     $  In other words, $ 
278 |          pp(, x_j) (g compose  f)_i(a) =  sum _( ell  = 1)^m  pp(g_i, y_ ell ) (f(a))
279 |          pp(f_ ell , x_j) (a).
280 |     $  
281 | ]
282 | 
283 | #example[
284 |     Let $f: RR^2-> RR$ be differentiable, and let $ Gamma (f) = {(x, y, f(x,
285 |     y)): x, y  in  RR}$ be the graph of $f$. Now, let $gamma : [-1, 1] ->
286 |     Gamma (f)$ be a differentiable curve, represented by $ 
287 |          gamma (t) = (g(t), h(t), f(g(t), h(t))).
288 |     $  Then, we can compute the derivative $ 
289 |          gamma '(a) = (g'(a), h'(a), lr(g'(a) pp(f, x)  + h'(a) pp(f, y) 
290 |             \|)_((g(a), h(a))))
291 |     $  
292 | ]
293 | 
294 | #exercise[
295 |     Consider the inner product map, $ ip( dot.c ,  dot.c ) : RR^n  times RR^n
296 |     -> RR$. What is its derivative?
297 |     #solution[
298 |         We treat the inner product as a map $g: RR^(2n) -> RR$, which acts
299 |         as $ 
300 |              ip( vx ,  vy )  colon.eq g(x_1,  dots , x_n, y_1,  dots , y_n) = x_1y_1 +
301 |              dots  + x_n y_n.
302 |         $  Now, note that $ 
303 |              pp(g, x_i)  = y_i,  quad   pp(g, y_i)  = x_i.
304 |         $  Thus, $
305 |             Dg( va ,  vb )( vx ,  vy ) &=  sum _(i = 1)^n x_i  pp(g, x_i) ( va ,  vb ) +
306 |              sum _(i = 1)^n y_i  pp(g, y_i) ( va ,  vb ) \
307 |             &=  sum _(i = 1)^n x_i b_i +  sum _(i = 1)^n y_i a_i \
308 |             &=  ip( vx ,  vb )  +  ip( vy ,  va ) .
309 |         $
310 |         In other words, the matrix representation of the derivative of the inner
311 |         product map at the point $( va ,  vb )$ is given by $[ vb^top
312 |          va^top]$.
313 |     ]
314 | ]
315 | 
316 | #exercise[
317 |     Let $gamma : RR -> RR^n$ be a differentiable curve. What is the
318 |     derivative of the real map $t  mapsto   norm( gamma (t)) ^2$?
319 |     #solution[
320 |         We write this map as $t  mapsto   ip( gamma (t),  gamma (t)) $. Consider the
321 |         scheme $
322 |             RR -> RR^(2n) -> RR, quad
323 |             t  mapsto  mat(
324 |                  gamma (t) ;  gamma (t)
325 |             )  mapsto   ip( gamma (t),  gamma (t)) .
326 |         $  Pick a point $t in  RR$, whence the derivative of the map at $t$ is $ 
327 |             mat(
328 |                  gamma (t)^top ,  gamma (t)^top
329 |             ) mat(
330 |                  gamma '(t) ;  gamma '(t)
331 |             ) = 2 ip( gamma (t),  gamma '(t)) .
332 |         $
333 |     ]
334 |     #remark[
335 |         Consider the surface $S^(n - 1)  subset  RR^n$, and pick an arbitrary
336 |         differentiable curve $gamma : RR -> S^(n - 1)$. Now, the tangent
337 |         vector $gamma '(t)$ is tangent to the sphere $S^(n - 1)$ at any point
338 |         $gamma (t)$. We claim that this tangent drawn at $ gamma (t)$ is always
339 |         perpendicular to the position vector $ gamma (t)$. This is made trivial by
340 |         our exercise: the map $t  mapsto   norm( gamma (t)) ^2 = 1$ is a constant
341 |         map since $gamma$ is a curve on the unit sphere. This means that it has
342 |         zero derivative, forcing $ ip( gamma (t),  gamma '(t))  = 0$.
343 |     ]
344 | ]
345 | 
346 | == Directional derivatives
347 | 
348 | #definition[
349 |     Let $U  subset.eq  RR^n$ be open, and let $f: U -> RR$. The directional
350 |     derivative of $f$ along a direction $ vv   in  RR^n$ at a point $a  in  U$ is
351 |     defined by the following limit, if it exists. $ 
352 |          nabla _v f(a) =  lim _(h -> 0)   frac(f(a + h vv ) - f(a), h)  .
353 |     $  
354 | ]
355 | 
356 | #example[
357 |     Consider $ 
358 |         f: RR^2 -> RR,  quad  (x, y)  mapsto  cases(
359 |             x^3\/(x^2 + y^2)\, & " if "  (x, y)  !=  (0, 0),
360 |             0\, & " if "  (x, y) = (0, 0).
361 |         )
362 |     $  Note that $f$ is not differentiable at $(0, 0)$. However, all
363 |     directional derivatives derivatives of $f$ exist at $(0, 0)$. Indeed,
364 |     consider a direction $( cos  theta , sin theta )$, and examine the limit
365 |     $ 
366 |          lim_(t -> 0)   frac(1, t)  [f(t cos  theta , t sin theta ) - f(0,
367 |         0)] =  cos ^3 theta .
368 |     $  
369 | ]
370 | 
371 | #definition[
372 |     Let $f: RR^n -> RR$ be differentiable. The gradient of $f$ is defined
373 |     as the map $ 
374 |          grad(f) : RR^n -> RR^n,  quad  x  mapsto 
375 |         [ pp(f, x_i) (x)]_i.
376 |     $  
377 |     #remark[
378 |         The gradient at a point $x  in  RR^n$ is thought of as a vector. In
379 |         contrast, the derivative is thought of as a linear transformation.
380 |         Otherwise, we see that $ grad(f) (x) = [Df(x)]$.
381 |     ]
382 | ]
383 | 
384 | #definition[
385 |     Let $C^1(RR^n)$ be the set of real-valued differentiable functions on $RR^n$.
386 |     Fix a point $a  in  RR^n$, then fix a tangent vector $v  in  RR^n$. Then, the
387 |     map $ 
388 |          nabla _v: C^1(RR^n) -> RR,  quad  f mapsto  Df(a)(v)
389 |     $  is a linear functional. The quantity $ nabla _v f$ is called the
390 |     directional derivative of $f$ in the direction $v$ at the point $a$.
391 |     #remark[
392 |         We can represent $ nabla _v$ as the operator $ 
393 |              nabla _v ( dot.c ) = D( dot.c )(a)(v) =  sum _i v_i  lr(pp(, x_i)\|)_a =
394 |             v dot.c   nabla  ( dot.c ).
395 |         $ 
396 |     ]
397 | ]
398 | 
399 | #lemma[
400 |     The directional derivatives $ nabla _v$ form a vector space called the tangent
401 |     space, attached to the point $a  in  RR^n$. This can be identified with the
402 |     vector space $RR^n$ by the natural map $ nabla _v  mapsto  v$. The standard
403 |     basis can be informally denoted by the vectors $ 
404 |          nabla_( ve _1) colon.eq  pp(, x_1) ,  quad dots quad,  nabla _( ve _n) colon.eq  pp(, x_n) .
405 |     $  
406 | ]
407 | 
408 | 
409 | == Differentiation on manifolds \*
410 | 
411 | #definition[
412 |     A homeomorphism is a continuous, bijective map whose inverse is also continuous.
413 | ]
414 | 
415 | #lemma[
416 |     Let $f: RR^n -> RR$ be continuous. Denote the graph of $f$ as $ 
417 |          Gamma (f) = \{(x, f(x)): x  in  RR^n\}.
418 |     $  Then, $ Gamma (f)$ is a smooth manifold.
419 | ]
420 | #proof[
421 |     Consider the homeomorphism $ 
422 |          phi :  Gamma (f) -> RR^n,  quad  (x, f(x))  mapsto  x.
423 |     $  This is clearly bijective, continuous (restriction of a projection map),
424 |     with a continuous inverse (from the continuity of $f$). Call this
425 |     homeomorphism $phi$ a coordinate map on $Gamma (f)$.
426 | ]
427 | 
428 | #definition[
429 |     Let $f: M -> RR$ where $M$ is a smooth manifold, with a coordinate map
430 |     $phi : M -> RR^n$. We say that $f$ is differentiable at a point $a
431 |      in  M$ if $f compose   phi ^(-1): RR^n -> RR$ is differentiable at
432 |     $ phi (a)$.
433 | ]
434 | 
435 | #definition[
436 |     Let $f: M -> RR$ where $M$ is a smooth manifold, let $ phi : M
437 |     -> RR^n$ be a coordinate map, and let $a  in  M$. Let $gamma : RR -> M$
438 |     be a curve such that $ gamma (0) = a$, and further let $gamma$ be
439 |     differentiable in the sense that $phi  compose   gamma : RR -> RR^n$ is
440 |     differentiable. The directional derivative of $f$ at $a$ along $gamma$ is
441 |     defined as $ 
442 |          dd(, t)  f( gamma (t)) \|_ (t = 0) =  lim _(h -> 0) lr(frac(f( gamma (t +
443 |         h)) - f( gamma (t)), h) \|)_ (t = 0).
444 |     $  Note that we are taking the derivative of $f compose   gamma : RR -> RR$
445 |     in the conventional sense.
446 | ]
447 | 
448 | #lemma[
449 |     Let $ gamma _1$ and $gamma _2$ be two curves in $M$ such that $gamma_1(0) =
450 |      gamma _2(0) = a$, and $ 
451 |          dd(, t) lr( phi  compose  gamma _1(t) \|)_ (t = 0) 
452 |          = dd(, t)  lr( phi  compose  gamma _2(t)  \|)_ (t = 0).
453 |     $  In other words, $gamma _1$ and $ gamma _2$ pass through the same point $a$
454 |     at $t = 0$, and have the same velocities there. Then, the directional
455 |     derivatives of $f$ at $a$ along $gamma _1$ and $ gamma _2$ are the same.
456 | ]
457 | 
458 | #definition[
459 |     Let $M$ be a smooth manifold, and let $a  in  M$. Consider the following
460 |     equivalence relation on the set of all curves $gamma$ in $M$ such that
461 |     $gamma (0) = a$. $ 
462 |          gamma_1 tilde.op gamma_2  quad arrow.r.double.long quad
463 |          dd(, t) lr( phi  compose gamma _1(t)  \|)_ (t = 0) 
464 |          = dd(, t) lr( phi  compose   gamma _2(t) \|)_ (t = 0).
465 |     $ Each resultant equivalence class of curves is called a tangent vector at
466 |     $a  in  M$. Note that all these curves in a particular equivalence class pass
467 |     through $a$ with the same velocity vector.
468 | 
469 |     The collection of all such tangent vectors, i.e. the space of all curves
470 |     through $a$ modulo the equivalence relation which identifies curves with the
471 |     same velocity vector through $a$, is called the tangent space to $M$ at $a$,
472 |     denoted $T_a M$.
473 | 
474 |     #remark[
475 |         Each tangent vector $v in  T_a M$ acts on a differentiable function
476 |         $f: M -> RR$ yielding a (well-defined) directional derivative at
477 |         $a$. $ 
478 |             v: C^1(M) -> RR,  quad  f mapsto   dd(, t) 
479 |             lr(f( gamma _v(t)) \|)_ (t = 0).
480 |         $ 
481 |         Thus, the tangent space represents all the directions in which taking a
482 |         derivative of $f$ makes sense.
483 |     ]
484 |     #remark[
485 |         The tangent space $T_a M$ is a vector space. Upon fixing $f$, the map
486 |         $Df(a): T_a M -> RR$, $v  mapsto  v f(a)$ is a linear functional on
487 |         the tangent space.
488 |     ]
489 |     #remark[
490 |         Given a tangent vector $v  in  T_a M$, it can be identified with its
491 |         corresponding velocity vector in $RR^n$. Thus, the tangent space $T_a M$
492 |         can be identified with the geometric tangent plane drawn to the manifold
493 |         $M$ at the point $a$.
494 |     ]
495 | ]
496 | 
497 | 
498 | = Mean value theorem
499 | Consider a differentiable function $f: RR^n -> RR$, and fix $a  in  RR^n$.
500 | Define the functions $ 
501 |     g_i: RR -> RR,  quad  g_i(x) = f(a_1,  dots , a_(i - 1), x, a_(i + 1),
502 |      dots , a_n).
503 | $  Then, each $g_i$ is differentiable, with $ 
504 |     g_i'(x) =  pp(f, x_i) (a_1,  dots , a_(i - 1), x, a_(i + 1),  dots , a_n).
505 | $  By applying the Mean Value Theorem on some interval $[c, d]$, we can find
506 | $ alpha  in  (c, d)$ such that $g_i(d) - g_i(c) = g_i'( alpha )(d - c)$. In other
507 | words,
508 | $ 
509 |     f( dots , d,  dots ) - f( dots , c,  dots ) =  pp(f, x_i) ( dots ,  alpha ,  dots )(d
510 |     - c).
511 | $
512 | 
513 | #theorem[
514 |     Let $f: RR^n -> RR^m$ and $a  in  RR^n$. Then, $f$ is differentiable at
515 |     $a$ if all the partial derivatives $diff  f\/diff  x_j$ exist in a
516 |     neighbourhood of $a$ and are continuous at $a$.
517 | ]
518 | #proof[
519 |     Without loss of generality, let $m = 1$. We claim that $ 
520 |          lim _(h -> 0)     frac(1,  norm(h)  )  norm(f(a + h) - f(a) -  sum _(i = 0) ^n
521 |          pp(f, x_i) (a)h_i) = 0.
522 |     $  Examine $
523 |         f(a + h) - f(a) &= f(a_1 + h_1,  dots , a_n + h_n) - f(a_1,  dots ,
524 |         a_n) \
525 |         &= f(a_1 + h_1,  dots , a_n + h_n) - f(a_1 + h_1,  dots , a_(n - 1) +
526 |         h_(n - 1), a_n) + \
527 |         &   quad  f(a_1 + h_1,  dots , a_(n - 1) + h_(n - 1), a_n) - f(a_1 + h_1,
528 |          dots , a_(n - 1), a_n) + \
529 |         &   quad   dots.h  \
530 |         &   quad  f(a_1 + h_1, a_2,  dots , a_n) - f(a_1,  dots , a_n) \
531 |         &=  pp(f, x_n) (c_n)h_n +  dots  +  pp(f, x_1) (c_1)h_1.
532 |     $
533 |     The last step follows from the Mean Value Theorem. As $h -> 0$, each $c_i
534 |     -> a$. Thus, $
535 |           frac(1,  norm(h))  norm(f(a + h) - f(a) -  sum _(i = 0)^n
536 |          pp(f, x_i) (a)h_i) 
537 |         &=   frac(1,  norm(h))  norm( sum _(i = 0) ^n (pp(f, x_i)(c_i) -
538 |          pp(f, x_i) (a))h_i) \
539 |         & <=   sum _(i = 0)^n | pp(f, x_i) (c_i) -  pp(f, x_i) (a) |
540 |           frac(|h_i|,  norm(h)  )  \
541 |         & <=   sum _(i = 0)^n | pp(f, x_i) (c_i) -  pp(f, x_i) (a) |.
542 |     $
543 |     Taking the limit $h -> 0$, observe that $(diff  f\/diff  x_i) (c_i) ->
544 |     (diff  f \/ diff  x_i) (a)$ by the continuity of the partial derivatives,
545 |     completing the proof.
546 | ]
547 | 
548 | #corollary[
549 |     All polynomial functions on $RR^n$ are differentiable.
550 | ]
551 | 
552 | #theorem[
553 |     Let $f: RR^n -> RR$ be differentiable with continuous partial
554 |     derivatives, and let $a  in  RR^n$ be a point of local maximum. Then, $Df(a) =
555 |     0$.
556 | ]
557 | #proof[
558 |     We need only show that each $ 
559 |          pp(f, x_i) (a) = 0.
560 |     $  This must be true, since $a$ is also a local maximum of each of the
561 |     restrictions $g_i$ as defined earlier.
562 | ]
563 | 
564 | = Inverse and implicit function theorems
565 | #theorem([Inverse function theorem])[
566 |     Let $f: RR^n -> RR^n$ be continuously differentiable on a neighbourhood
567 |     of $a  in  RR^n$, and let $"det" (Df(a))  !=  0$. Then, there exist neighbourhoods
568 |     $U$ of $a$ and $W$ of $f(a)$ such that the restriction $f: U -> W$ is
569 |     invertible. Furthermore, $f^(-1)$ is continuous on $U$ and differentiable on
570 |     $U$.
571 | ]
572 | 
573 | #lemma[
574 |     Consider a continuously differentiable function $f: RR^n -> RR$, and
575 |     let $M$ denote the surface defined by the zero set of $f$. Then, $M$ can be
576 |     represented as the graph of a differentiable function $h: RR^(n - 1) ->
577 |     RR$ at those points where $Df  !=  0$.
578 | ]
579 | #proof[
580 |     Without loss of generality, suppose that $diff  f \/ diff  x_n  !=  0$
581 |     at some point $a  in  M$. It can be shown that the map $ 
582 |         F: RR^n -> RR^n ,  quad  x  mapsto  (x_1, x_2,  dots , x_(n - 1),
583 |         f(x))
584 |     $  is invertible in a neighbourhood $W$ of $a$, with a continuous and
585 |     differentiable inverse of the form
586 |     $ 
587 |         G: RR^n -> RR^n,  quad  u  mapsto  (u_1, u_2,  dots , u_(n - 1),
588 |         g(u)).
589 |     $
590 |     Since $F compose G$ must be the identity map on $W$, we demand $ 
591 |         (x_1, x_2,  dots , x_(n - 1), f(x_1, x_2,  dots , x_(n - 1), g(x))) = (x_1,
592 |         x_2,  dots , x_(n - 1), x_n).
593 |     $  Thus, the zero set of $f$ in this neighbourhood of $a$ satisfies $x_n =
594 |     0$, hence $ 
595 |         f(x_1, x_2,  dots , x_(n - 1), g(x_1, x_2,  dots , x_(n - 1), 0)) = 0.
596 |     $  In other words, the part of the surface $M$ in the neighbourhood of $a$ is
597 |     precisely the set of points $ 
598 |         (x_1, x_2,  dots , x_(n - 1), g(x_1, x_2,  dots , x_(n - 1), 0)).
599 |     $  Simply set $ 
600 |         h: RR^(n - 1) -> RR,  quad  x  mapsto  g(x_1, x_2,  dots , x_(n - 1),
601 |         0),
602 |     $  whence the surface $M$ is locally represented by the graph of $h$.
603 | ]
604 | 
605 | #block(inset: 1em)[
606 | #remark()[
607 |     Note that by using $ 
608 |         f(x_1,  dots , x_(n - 1), h(x_1,  dots , x_(n - 1))) = 0
609 |     $  on the surface, we can use the chain rule to conclude that for all $1  <= 
610 |     i < n$, we have $ 
611 |          pp(f, x_i) (a) +  pp(f, x_n) (a) pp(h, x_i) (a_1,  dots , a_(n - 1)) = 0.
612 |     $  
613 | ]
614 | ]
615 | 
616 | #theorem([Implicit function theorem])[
617 |     Let $f: RR^n times RR^m -> RR^m$ be continuously differentiable in an open
618 |     set containing $(a, b)$, with $f(a, b) = 0$. Let $"det" (diff  f^j \/ diff
619 |     x_(n + k) (a, b))  !=  0$. Then, there exists an open set $U  subset RR^n$
620 |     containing $a$, an open set $V  subset  RR^m$ containing $b$, and a
621 |     differentiable function $g: U -> V$ such that $f(x, g(x)) = 0$.
622 | 
623 |     #remark[
624 |         The condition on the determinant can be rephrased as
625 |         $"rank" Df(a, b) = m$.
626 |     ]
627 | ]
628 | 
629 | #theorem[
630 |     Let $f: RR^n -> RR$ be continuously differentiable, and let $M$ be the
631 |     surface defined by its zero set. Furthermore, let $ grad(f) (a)  !=  0$ for
632 |     some $a  in  M$; thus, $M$ can be locally represented by a graph on $RR^(n
633 |     - 1)$. Then, $ grad(f) (a)$ is normal to the tangent vectors drawn at $a$
634 |     to $M$; in fact, the perpendicular space of $ grad(f) (a)$ is precisely
635 |     the tangent space $T_a M$.
636 | ]
637 | #proof[
638 |     Consider a tangent vector drawn at $a$ to $M$, represented by the
639 |     differentiable curve $gamma : RR -> M$, $ gamma (0) = a$; note that
640 |     we use the identification $gamma '(0) = v  in  RR^n$. Then, calculate $ 
641 |          dd(, t) lr(f( gamma (t))  \|)_ (t = 0) = Df( gamma (0))( gamma '(0))
642 |          = Df(a)(v).
643 |     $  On the other hand, we have $f( gamma (t)) = 0$ identically. Thus, $ 
644 |         v dot.c   grad(f) (a) = Df(a)(v) = 0
645 |     $ as claimed.
646 | ]
647 | 
648 | = Taylor's theorem
649 | 
650 | #theorem("Clairaut")[
651 |     Let $f: RR^n -> RR$ have continuous second order partial derivatives.
652 |     Then, $ 
653 |           frac(diff ^2 f, diff  x_i diff  x_j)   = frac(diff ^2
654 |         f, diff  x_j diff  x_i).
655 |     $
656 | ]
657 | 
658 | #theorem("Taylor")[
659 |     Let $f: RR^2 -> RR$ have continuous second order partial derivatives,
660 |     and let $(x_0, y_0)  in  RR^2$. Then, there exists $ epsilon  > 0$ such that
661 |     for all $norm((x - x_0, y - y_0))  <  epsilon$,
662 |     $
663 |         f(x, y)   =   &f(x_0, y_0) +  pp(f, x) (x - x_0) +  pp(f, y) (y - y_0) \
664 |         & + space frac(1, 2)  frac(diff^2 f, diff x^2)(x - x_0)^2 + frac(1, 2)
665 |         frac(diff^2 f, diff y^2)(y - y_0)^2 \
666 |         & + space frac(diff ^2 f, diff  x diff  y)  (x - x_0)(y - x_0) + R(x,
667 |         y),
668 |     $
669 |     where as $(x, y) -> (x_0, y_0)$, the remainder term vanishes as $ 
670 |           frac(|R(x, y)|,  norm((x - x_0, y - y_0) ) ^2)  -> 0.
671 |     $  All partial derivatives here are evaluated at $(x_0, y_0)$.
672 | ]
673 | #proof[
674 |     This follows from applying the Taylor's Theorem in one variable to the real
675 |     function $g: RR -> RR$, $t  mapsto  f((1 - t)(x_0, y_0) + t(x, y))$.
676 | ]
677 | 
678 | 
679 | = Critical points and extrema
680 | #definition[
681 |     We say that $a  in  RR^n$ is a critical point of $f: RR^n -> RR$ if all
682 |     $diff  f \/ diff  x^j = 0$ there.
683 | ]
684 | 
685 | #lemma[
686 |     All points of extrema of a differentiable function are critical points.
687 | ]
688 | #proof[
689 |     We already know that $Df(a) = 0$ where $a$ is either a point of maximum or
690 |     minimum.
691 | ]
692 | 
693 | #example[
694 |     In order to find a point of extrema of a $C^2$-smooth function $f: RR^2
695 |     -> RR$, we first identify a critical point $(x_0, y_0)$. Next, we must find
696 |     a neighbourhood of $(x_0, y_0)$ which contains no other critical points -- to
697 |     do this, apply Taylor's Theorem. Indeed, we see that $ 
698 |         f(x, y) = f(x_0, y_0) + A(x - x_0)^2 + 2B(x - x_0)(y - y_0) + C(y -
699 |         y_0)^2 + R_2.
700 |     $  For non-degeneracy of solutions, we demand $A C - B^2  !=  0$, i.e. at
701 |     $(x_0, y_0)$, we want $ 
702 |         [  frac(diff ^2 f, diff  x diff  y)  ]^2  != 
703 |           frac(diff ^2 f, diff  x^2)  
704 |           frac(diff ^2 f, diff  y^2)  .
705 |     $  
706 | 
707 |     If $A C - B^2 > 0$ and $diff ^2f \/ diff  x^2 > 0$, then we have found a
708 |     point of minima; if $diff ^2 f \/ diff  x^2 < 0$, then we have found a
709 |     point of maximum. If $A C - B^2 < 0$, then we have found a saddle point.
710 | ]
711 | 
712 | #example[
713 |     Suppose that we wish to maximize the function $f: RR^2 -> RR$, given an
714 |     equation of constraint $g = 0$, where $g: RR^2 -> RR$. Using the method
715 |     of Lagrange multipliers, we look for solutions of the system $ 
716 |         cases(
717 |             grad(f) (x, y) +  lambda   grad(&g) (x, y) &= 0,
718 |             &g(x, y) &= 0.
719 |         )
720 |     $
721 | ]
722 | 


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