From d5a7f8a1efb3d974f669afdb40e5cc26a952a7ca Mon Sep 17 00:00:00 2001 From: Leedehai Date: Sat, 9 Dec 2023 17:23:37 -0500 Subject: [PATCH] Add taylorterm(), the n-term in Taylor series; add 'tall' to J/H matrices Also, correct innerproduct to the more usual form . --- README.md | 6 +-- physica-manual.pdf | Bin 368667 -> 317140 bytes physica-manual.typ | 113 ++++++++++++++++++++++++++++++++++++--------- physica.typ | 55 +++++++++++++++++----- 4 files changed, 137 insertions(+), 37 deletions(-) diff --git a/README.md b/README.md index aba0f29..f8b17dc 100644 --- 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).

-logo +logo

# The physica package for Typst @@ -17,11 +17,11 @@ 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 diff --git a/physica-manual.pdf b/physica-manual.pdf index 90ae372d29589db6aef9677f83d4c4a854c46c95..ea390743ff36ed8b0b7ae5024a0c53820316840a 100644 GIT binary patch delta 236955 zcmV(=K-s^Wz!uc46Oc&;PDMgTR#=fwp$1MxLPu6uvGzCtvpoWM1O`q;LPu6ule-4z 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0be2ee7..4812bb1 100644 --- a/physica-manual.typ +++ b/physica-manual.typ @@ -27,8 +27,8 @@ #align(center)[ Leedehai \ - #linkurl("GitHub repo", "https://github.com/leedehai/typst-physics") | - #linkurl("Typst index", "https://typst.app/docs/packages/") + #linkurl("GitHub", "https://github.com/leedehai/typst-physics") | + #linkurl("Typst", "https://typst.app/docs/packages/") ] #set par(justify: true) @@ -211,6 +211,14 @@ All symbols need to be used in *math mode* `$...$`. [`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 +299,49 @@ All symbols need to be used in *math mode* `$...$`. Jacobian matrix: `jacobianmatrix(`...`)`, i.e. `jmat(`...`)`. #table( - columns: (1fr, 1fr), + columns: (auto, auto, auto), align: center, stroke: none, column-gutter: 1em, + [ + #hl(`inline(pdv(f,x))`) + $ inline(pdv(f,x)) $ + ], [ #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:"[") $ + ], + [ + #hl(`pdv(f,x)`) + $ pdv(f,x) $ + ], + [ + #hl(`jmat(f_1,f_2;x,y;tall:#true)`) + $ jmat(f_1,f_2;x,y;tall:#true) $ + ], + [ + #hl(`jmat(f,g;x,y,z;delim:"[",tall:#true)`) + $ jmat(f,g;x,y,z;delim:"[",tall:#true) $ ], ) Hessian matrix: `hessianmatrix(`...`)`, i.e. `hmat(`...`)`. #table( - columns: (1fr, 1fr), + columns: (auto, auto, auto), align: center, stroke: none, column-gutter: 1em, + [ + #hl(`inline(pdv(f,x,2))`) + $ inline(pdv(f,x,2)) $ + ], [ #hl(`hmat(f; x,y)`) $ hmat(f; x,y) $ @@ -322,6 +350,18 @@ Hessian matrix: `hessianmatrix(`...`)`, i.e. `hmat(`...`)`. #hl(`hmat(; x,y,z; delim:"[")`) $ hmat(; x,y,z; delim:"[") $ ], + [ + #hl(`pdv(f,x,2)`) + $ pdv(f,x,2) $ + ], + [ + #hl(`hmat(f;x,y;tall:#true)`) + $ hmat(f;x,y;tall:#true) $ + ], + [ + #hl(`hmat(;x,y,z;delim:"[",tall:#true)`) + $ hmat(; x,y,z;delim:"[",tall:#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 +439,6 @@ 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 +876,51 @@ $ 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 index b4a8271..675f49d 100644 --- 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, tall) = { + if tall { + 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:"(", tall: 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($diff#f$, $diff#x$), tall))) } 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:"(", tall: 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 { $diff #xr diff #xc$ } - )) + ), tall)) } 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) = { $attach(upright(element), tl: #a, bl: #z)$ }

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