--- azemark: 2 title: Mathematics notation examples author: - AzeForge examples --- # Mathematics notation examples Equations and derivations share one readable mathematics grammar, so the same spellings appear in either directive. This document starts from the smallest relation, adds bound variables, and closes with derivations whose annotations stay prose rather than mathematics. ## Equation The `equation` directive typesets one readable expression, and numbers or aligns it without any markup of its own. The simplest form is a single relation with no numbering. :::: equation id: ohm-relation ---- V = I * R :::: Greek names, subscripts, and superscripts pass through unchanged in a counted expression. :::: equation id: sample-variance number: true syntax: readable ---- sigma^2 = frac(1, n) sum i=1..n of (x_i - mu)^2 :::: Multi-index subscripts carry tensor indices in one subscript group, and the physics symbols read as registered names. :::: equation id: field-equation-form ---- G_(mu, nu) = R_(mu, nu) - 1 / 2 R g_(mu, nu) :::: A bracketed operator is one group under the same grammar, so Hamiltonians and commutators keep their square delimiters in the rendered artifact, and a comma inside a group lists arguments in evaluation order. :::: equation id: bracketed-hamiltonian ---- i hbar frac(partial, partial t) Psi(r, t) = [-frac(hbar^2, 2 m) nabla^2 + V(r, t)] Psi(r, t) :::: A dense scalar form binds continuous and discrete variables in one relation and mixes a root, a fraction and a piecewise branch. :::: equation id: hybrid-mode-weight number: true align: left ---- integral x=0..L of (sqrt(x) + frac(1, 1 + x)) dx = sum k=1..n of cases(w_k x^k when k < m; 0 otherwise) :::: ### Native notation grammar Readable mathematics is a closed expression grammar, not LaTeX with the backslashes removed. Operators have fixed precedence. Parentheses and square brackets group expressions. A relation may form a left-to-right chain. Registered symbols include lower and upper Greek names and physics names such as `hbar`, `partial`, `nabla`, `infinity` and `emptyset`. One base may carry a subscript, a power, or up to two primes. Put the base in parentheses before combining those forms, as in `(x')^2`. This Schrödinger relation combines registered symbols, grouped function arguments, a square-bracket group, a fraction, derivatives and a relation chain. :::: equation id: schrodinger-relation-chain number: true ---- i hbar frac(partial, partial t) Psi(x, t) = [-frac(hbar^2, 2 m) nabla^2 + V(x)] Psi(x, t) = -frac(hbar^2, 2 m) frac(partial^2 Psi, partial x^2) + V(x) Psi(x, t) :::: Function-like constructs use fixed names and argument shapes. The grammar registers `frac`, `sqrt`, `root`, `abs`, `vector`, `matrix`, `pmatrix`, `vmatrix` and `cases`. Matrix rows are bracket groups. A one-item `vector` is an arrow vector, while a multi-item `vector` is a bold tuple. :::: equation id: rotation-matrix-vector number: true ---- y_i = R_(i, j) x_j = pmatrix [[cos theta, -sin theta], [sin theta, cos theta]] vector [x, y] = vector [x cos theta - y sin theta, x sin theta + y cos theta] :::: The binder forms are `sum`, `product`, `integral`, `limit`, `forall` and `exists`. A bounded sum, product or integral writes `name=from..to`, then `of`, then its body. An integral closes with a differential whose one-letter name matches the bound name. A quantifier writes `name in set of body`. :::: equation id: binder-catalog align: left ---- S_n = sum i=1..n of i^2 + product k=1..m of k + integral x=0..infinity of exp(-x^2) dx :::: Quantifiers may nest. Set relations and operations include `in`, `notin`, `subset`, `supset`, `subseteq`, `union`, `intersect` and `equiv`. :::: equation id: nested-quantifiers ---- forall e in R of exists M in R of (abs(x) <= M) :::: Registered aliases are reserved and cannot become identifiers. Unknown two-letter words are read as juxtaposed one-letter names, while unknown words of three or more letters are diagnostics. This keeps a typo from silently turning into a new function name. ## Derivation A `derivation` holds ordered `- expression:` steps, each optionally followed by an indented `annotation:` line of plain prose, and the renderer aligns the expressions as one chain. Two steps are enough to show how a chain reads. :::: derivation id: compound-interest-chain ---- - expression: A_1 = P_0 (1 + r) - expression: A_n = P_0 (1 + r)^n :::: Every step may carry an annotation, and `align` moves the column the relations sit in. :::: derivation id: enzyme-rate-linearized number: true align: center ---- - expression: v = k_2 E_0 S / (K_m + S) annotation: steady-state rate before any rearrangement - expression: v (K_m + S) = k_2 E_0 S annotation: clear the denominator - expression: frac(K_m + S, S) = frac(k_2 E_0, v) annotation: collect the rate on one side - expression: 1 / v = frac(K_m, k_2 E_0) (1 / S) + frac(1, k_2 E_0) annotation: invert both sides, and the reciprocal rate is affine in the reciprocal substrate :::: A long chain finishes by taking a limit, which is where the prose of an annotation stays visibly outside the mathematics. :::: derivation id: annuity-present-value number: true ---- - expression: V_0 = sum i=1..n of C / (1 + r)^i annotation: present value as a bounded sum of discounted payments - expression: V_0 = C / (1 + r) sum i=0..n-1 of (1 + r)^(-i) annotation: shift the index and factor out the first payment - expression: S = sum i=0..n-1 of q^i annotation: name the geometric factor q = 1 / (1 + r) - expression: S - q S = 1 - q^n annotation: subtract the shifted copy of the same sum - expression: S = (1 - q^n) / (1 - q) annotation: divide by 1 - q, which is nonzero for every positive rate - expression: limit n->infinity of V_0 = C / r annotation: the tail vanishes only while abs(q) < 1, stated here in prose ::::