Add taylorterm() for Taylor series' n-th term; add 'big' to J/H matrices

Also, correct innerproduct to the more usual form <u,v>.
Leedehai · Dec 11, 2023 · 4434543 · 4434543
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diff --git a/README.md b/README.md
@@ -1,6 +1,6 @@
 :green_book: The [manual](https://github.com/Leedehai/typst-physics/blob/master/physica-manual.pdf).
 <p align="center">
-<img width="545" alt="logo" src="https://github.com/Leedehai/typst-physics/assets/18319900/781d5747-4842-49f6-bb41-a1fd4d17a1ff">
+<img width="545" alt="logo" src="https://github.com/Leedehai/typst-physics/assets/18319900/ed86198a-8ddb-4473-aed3-8111d5ecde60">
 </p>
 
 # The physica package for Typst
@@ -17,27 +17,21 @@ This [Typst](https://typst.app) package provides handy typesetting utilities for
 natural sciences, including:
 * Braces,
 * Vectors and vector fields,
-* Matrices,
+* Matrices, including Jacobian and Hessian,
 * Dirac braket notations,
 * Common math functions,
 * Differentials and derivatives, including partial derivatives of mixed orders with automatic order summation,
-* Familiar "h-bar", tensor abstract index notations, isotopes,
+* Familiar "h-bar", tensor abstract index notations, isotopes, Taylor series term,
 * Signal sequences i.e. digital timing diagrams.
 
-:warning: [Typst](https://typst.app) is in beta and evolving, and this package
-evolves with it. As a result, no backward compatibility is guaranteed yet.
-
-:information_source: In response to the official Typst package [guideline](https://github.com/typst/packages/tree/main#submission-guidelines)
-"Package names should not be merely descriptive to create level grounds for
-everybody", this package was renamed from `physics` to `physica`.
-
 ## A quick look
 
 See the [manual](https://github.com/Leedehai/typst-physics/blob/master/physica-manual.pdf) for more details.
 
 ![demo-quick](https://github.com/Leedehai/typst-physics/assets/18319900/4a9f40df-f753-4324-8114-c682d270e9c7)
 
-![demo-larger](https://github.com/Leedehai/typst-physics/assets/18319900/3a5cc613-bf36-4b13-ac28-790150c5fb94)
+[demo.typ](https://github.com/Leedehai/typst-physics/blob/master/demo.typ)
+![demo-larger](https://github.com/Leedehai/typst-physics/assets/18319900/ed2d95af-4bb9-4131-957f-ffba4f932a03)
 
 ## Using physica in your Typst document
 

diff --git a/demo.typ b/demo.typ
@@ -9,11 +9,15 @@ curl vb(E) = - pdv(vb(B), t),
 quad
 tensor(Lambda,+mu,-nu) = dmat(1,RR),
 quad
+dd(x,y),
+quad
+dd(vb(x),y,[3]),
+quad
 dd(x,y,2,d:Delta,p:and),
 quad
 dv(phi,t,d:upright(D)) = pdv(phi,t) + vb(u) grad phi \
 
-H(f) = hmat(f;x,y),
+H(f) = hmat(f;x,y;delim:"[",big:#true),
 quad
 vb(v^a) = sum_(i=1)^n alpha_i vu(u^i),
 quad

diff --git a/physica-manual.pdf b/physica-manual.pdf
diff --git a/physica-manual.typ b/physica-manual.typ
@@ -27,8 +27,8 @@
 
 #align(center)[
   Leedehai \
-  #linkurl("GitHub repo", "https://github.com/leedehai/typst-physics") |
-  #linkurl("Typst index", "https://typst.app/docs/packages/")
+  #linkurl("GitHub", "https://github.com/leedehai/typst-physics") |
+  #linkurl("Typst", "https://typst.app/docs/packages/")
 ]
 
 #set par(justify: true)
@@ -211,6 +211,14 @@ All symbols need to be used in *math mode* `$...$`.
   [`cprod`],
   [`a cprod b` #sym.arrow $a cprod b$],
   [cross product],
+
+  [`innerproduct`],
+  [`iprod`],
+  [
+    `iprod(u, v)` #sym.arrow $iprod(u, v)$ \
+    `iprod(sum_i a_i, b)` \ #sym.arrow $iprod(sum_i a_i, b)$
+  ],
+  [inner product],
 )
 
 == Matrix notations
@@ -291,29 +299,46 @@ All symbols need to be used in *math mode* `$...$`.
 Jacobian matrix: `jacobianmatrix(`...`)`, i.e. `jmat(`...`)`.
 
 #table(
-  columns: (1fr, 1fr),
+  columns: (25%, auto, auto),
   align: center,
   stroke: none,
   column-gutter: 1em,
 
+  [
+    \ Typst (like LaTeX) \ cramps fractions in a matrix...
+  ],
   [
     #hl(`jmat(f_1,f_2; x,y)`)
     $ jmat(f_1,f_2;x,y) $
   ],
   [
-    #hl(`jmat(f_1,f_2; x,y,z; delim:"[")`)
-    $ jmat(f_1,f_2;x,y,z;delim:"[") $
+    #hl(`jmat(f,g; x,y,z; delim:"[")`)
+    $ jmat(f,g;x,y,z;delim:"[") $
+  ],
+  [
+    \ ...but you can uncramp them using argument #hl(`big:#true`) here
+  ],
+  [
+    #hl(`jmat(f_1,f_2;x,y;big:#true)`)
+    $ jmat(f_1,f_2;x,y;big:#true) $
+  ],
+  [
+    #hl(`jmat(f,g;x,y,z;delim:"|",big:#true)`)
+    $ jmat(f,g;x,y,z;delim:"|",big:#true) $
   ],
 )
 
 Hessian matrix: `hessianmatrix(`...`)`, i.e. `hmat(`...`)`.
 
 #table(
-  columns: (1fr, 1fr),
+  columns: (25%, auto, auto),
   align: center,
   stroke: none,
   column-gutter: 1em,
 
+  [
+    \ Typst (like LaTeX) \ cramps fractions in a matrix...
+  ],
   [
     #hl(`hmat(f; x,y)`)
     $ hmat(f; x,y) $
@@ -322,6 +347,17 @@ Hessian matrix: `hessianmatrix(`...`)`, i.e. `hmat(`...`)`.
     #hl(`hmat(; x,y,z; delim:"[")`)
     $ hmat(; x,y,z; delim:"[") $
   ],
+  [
+    \ ...but you can uncramp them using argument #hl(`big:#true`) here
+  ],
+  [
+    #hl(`hmat(f;x,y;big:#true)`)
+    $ hmat(f;x,y;big:#true) $
+  ],
+  [
+    #hl(`hmat(;x,y,z;delim:"|",big:#true)`)
+    $ hmat(; x,y,z;delim:"|",big:#true) $
+  ],
 )
 
 Matrix built with an element building function: `xmatrix(`_m, n, func_`)`, i.e. `xmat(`...`)`. The element building function _func_ takes two integers which are the row and column numbers starting from 1.
@@ -399,22 +435,6 @@ xmat(2, 2, #g)`)
   ],
   [ketbra],
 
-  [`innerproduct(`_a_, _b_`)`],
-  [`iprod`],
-  [
-    `iprod(a), iprod(u, v)` \ #sym.arrow $iprod(a), iprod(u, v)$ \
-    `iprod(a, vec(1,2))` #sym.arrow $iprod(a, vec(1,2))$
-  ],
-  [innerproduct\ = braket],
-
-  [`outerproduct(`_a_, _b_`)`],
-  [`oprod`],
-  [
-    `oprod(a), oprod(u, v)` \ #sym.arrow $oprod(a), oprod(u, v)$ \
-    `oprod(a, vec(1,2))` #sym.arrow $oprod(a, vec(1,2))$
-  ],
-  [outerproduct\ = ketbra],
-
   [`matrixelement(`_n_, _M_, _m_`)`],
   [`mel`],
   [
@@ -852,6 +872,51 @@ $ isotope("Bi",a:211,z:83) --> isotope("Tl",a:207,z:81) + isotope("He",a:4,z:2)
 *(4)* #hl(`isotope("Tl",a:207,z:81) --> isotope("Pb",a:207,z:82) + isotope(e,a:0,z:-1)`)
 $ isotope("Tl",a:207,z:81) --> isotope("Pb",a:207,z:82) + isotope(e,a:0,z:-1) $
 
+=== The n-th term in Taylor series
+
+#v(1em)
+
+Function: `taylorterm(`_func_, _x_, _x0_, _idx_`)`.
+- _func_: the function e.g. `f`, `(f+g)`,
+- _x_: the variable name e.g. `x`,
+- _x0_: the variable value at the expansion point e.g. `x_0`, `(1+a)`,
+- _idx_: the index of the term, e.g. `0`, `1`, `2`, `n`, `(n+1)`.
+If _x0_ or _idx_ is in a parenthesis e.g. `(1+k)`, then the function
+automatically removes the outer parenthesis where appropriate.
+
+#align(center, [*Examples*])
+
+#grid(
+  columns: (50%, 50%),
+  row-gutter: 1em,
+  column-gutter: 2em,
+  [
+    *(1)* #hl(`taylorterm(f,x,x_0,0)`) \
+    $ taylorterm(f,x,x_0,0) $
+  ],
+  [
+    *(2)* #hl(`taylorterm(f,x,x_0,1)`) \
+    $ taylorterm(f,x,x_0,1) $
+  ],
+  [
+    *(3)* #hl(`taylorterm(f,x,(1+a),2)`) \
+    $ taylorterm(f,x,(1+a),2) $
+  ],
+  [
+    *(4)* #hl(`taylorterm(f,x,x_0,n)`) \
+    $ taylorterm(f,x,x_0,n) $
+  ],
+  [
+    *(5)* #hl(`taylorterm(F,x^nu,x^nu_0,n)`) \
+    $ taylorterm(F,x^nu,x^nu_0,n) $
+  ],
+  [
+    *(6)* #hl(`taylorterm(f_p,x,x_0,(n+1))`) \
+    $ taylorterm(f_p,x,x_0,(n+1)) $
+  ],
+)
+
+
 === Signal sequences (digital timing diagrams)
 
 In engineering, people often need to draw digital timing diagrams for signals, like $signals("1|0|1|0")$.

diff --git a/physica.typ b/physica.typ
@@ -206,8 +206,25 @@
 #let crossproduct = $times$
 #let cprod = crossproduct
 
+#let innerproduct(u, v) = {
+  $lr(angle.l #u, #v angle.r)$
+}
+#let iprod = innerproduct
+
 // == Matrices
 
+// Display matrix element in display/inline style. The latter vertically
+// compresses a tall content (e.g. a fraction) while the former doesn't.
+// In Typst and LaTeX, a matrix element is automatically cramped, even if
+// the matrix is in a standalone math block.
+#let __mate(content, big) = {
+  if big {
+    math.display(content)
+  } else {
+    math.inline(content)
+  }
+}
+
 #let matrixdet(..sink) = {
   math.mat(..sink, delim:"|")
 }
@@ -276,20 +293,20 @@
 }
 #let zmat = zeromatrix
 
-#let jacobianmatrix(fs, xs, delim:"(") = {
+#let jacobianmatrix(fs, xs, delim:"(", big: false) = {
   assert(type(fs) == array, message: "expecting an array of function names")
   assert(type(xs) == array, message: "expecting an array of variable names")
   let arrays = ()  // array of arrays
   for f in fs {
-    arrays.push(xs.map((x) => math.frac($diff#f$, $diff#x$)))
+    arrays.push(xs.map((x) => __mate(math.frac($diff#f$, $diff#x$), big)))
   }
   math.mat(delim: delim, ..arrays)
 }
 #let jmat = jacobianmatrix
 
-#let hessianmatrix(fs, xs, delim:"(") = {
-  assert(type(fs) == array, message: "expecting a one-element array")
-  assert(fs.len() == 1, message: "expecting only one function name")
+#let hessianmatrix(fs, xs, delim:"(", big: false) = {
+  assert(type(fs) == array, message: "usage: hessianmatrix(f; x, y...)")
+  assert(fs.len() == 1, message: "usage: hessianmatrix(f; x, y...)")
   let f = fs.at(0)
   assert(type(xs) == array, message: "expecting an array of variable names")
   let row_arrays = ()  // array of arrays
@@ -299,10 +316,10 @@
     let xr = xs.at(r)
     for c in range(order) {
       let xc = xs.at(c)
-      row_array.push(math.frac(
+      row_array.push(__mate(math.frac(
         $diff^2 #f$,
         if xr == xc { $diff #xr^2$ } else { $diff #xr diff #xc$ }
-      ))
+      ), big))
     }
     row_arrays.push(row_array)
   }
@@ -364,11 +381,6 @@
   $ lr(bar.v ket#h(0pt)mid(angle.r#h(0pt)angle.l)#h(0pt)bra bar.v) $
 })
 
-#let innerproduct = braket
-#let iprod = innerproduct
-#let outerproduct = ketbra
-#let oprod = outerproduct
-
 #let matrixelement(n, M, m) = style(styles => {
   $ lr(angle.l #n#h(0pt)mid(bar.v)#h(0pt)#M#h(0pt)mid(bar.v)#h(0pt)#m angle.r) $
 })
@@ -663,6 +675,25 @@
   math.attach((T,hphantom(sym.zwj)).join(), t: uppers.join(), b: lowers.join())
 }
 
+#let taylorterm(fn, xv, x0, idx) = {
+  let noparen(expr) = {
+    if type(expr) == content and expr.func() == math.lr {
+      let children = expr.at("body").at("children")
+      children.slice(1, children.len() - 1).join()
+    } else {
+      expr
+    }
+  }
+
+  if idx == [0] or idx == 0 {
+    $fn (noparen(x0))$
+  } else if idx == [1] or idx == 1 {
+    $fn^((1)) (noparen(x0))(xv - x0)$
+  } else {
+    $frac(fn^((noparen(idx))) (noparen(x0)), idx !)(xv - x0)^noparen(idx)$
+  }
+}
+
 #let isotope(element, /*atomic mass*/a: none, /*atomic number*/z: none) = {
   $attach(upright(element), tl: #a, bl: #z)$
 }