universal-formula-sheet-generator

name	universal-formula-sheet-generator
description	Generates compact, exam-ready formula sheets for ANY university worldwide. Covers engineering, mathematics, physics, chemistry, finance, statistics, and data science. Organized by unit with variable definitions, SI units, and quick-reference annotations. Optimized for closed-book recall and open-book quick lookup.

Universal Formula Sheet Generator

Overview

Creates dense, well-organized formula sheets that serve as both study aids and exam-day quick references. Every formula is presented with its variables clearly defined, applicable conditions stated, and cross-referenced to related formulas.

How This Skill Works

1. User provides: Subject, course code, university, specific units/topics, exam type (open/closed book)
2. System collects: All relevant formulas organized by unit and topic
3. System formats: In compact, scannable layout with consistent notation
4. System annotates: Variable definitions, units, applicability conditions, and common pitfalls

1. Formula Entry Format

Each formula is presented as:

Formula Name
  f(x) = expression
  WHERE: x = description [unit], f(x) = description [unit]
  WHEN: Applicable conditions or assumptions
  WATCH: Common mistakes or special cases
  SEE ALSO: Related formula #12

2. Organization Structure

Subject Name
  Unit 1 — Topic Name
    F1: Formula name
    F2: Formula name
    ...
  Unit 2 — Topic Name
    ...
  Appendix
    Constants
    Conversion Factors
    Greek Letters Used
    Index (formula name -> F-number)

3. Formula Categories

Category	Subjects	Presentation Style
Algebraic	Mathematics, Physics	Standard equation, variable definitions
Differential/Integral	Calculus, Physics	Standard notation with limits
Statistical	Statistics, ML, Data Science	Summation/probability notation
Chemical	Chemistry, Biology	Molecular structure, reaction equations
Electrical	E&TC, Electrical, Electronics	Circuit parameters, phasors
Mechanical	Mechanical, Civil, Aerospace	Force/momentum/energy equations
Financial	Finance, Economics, Business	Time value of money, ratio formulas
Algorithmic	Computer Science	Pseudocode, complexity expressions
Logical	Mathematics, CS	Set notation, logical operators

4. Sample Formula Sheet

Subject: Engineering Mathematics — Calculus (VTU, 1st Semester)

UNIT 1: Differential Calculus

F1: Derivative definition
  f'(x) = lim(h->0) [f(x+h) - f(x)] / h
  WHERE: f = function, x = variable, h = infinitesimal increment
  WATCH: Limit must exist for derivative to exist

F2: Product rule
  d/dx [u(x) * v(x)] = u'v + uv'
  WHERE: u, v = differentiable functions of x
  NOTE: "First derivative of second + second derivative of first"

F3: Quotient rule
  d/dx [u(x) / v(x)] = (u'v - uv') / v^2
  WHERE: v(x) != 0
  WATCH: Order matters — numerator is (u'v - uv') not (uv' - u'v)

F4: Chain rule
  dy/dx = dy/du * du/dx
  WHERE: y = f(u), u = g(x)
  NOTE: "Derivative of outer times derivative of inner"

UNIT 2: Integral Calculus

F5: Power rule (integration)
  int x^n dx = x^(n+1)/(n+1) + C, n != -1
  WHERE: n = real number
  WATCH: n = -1 gives int x^(-1) dx = ln|x| + C (special case)

F6: Integration by parts
  int u dv = uv - int v du
  WHERE: u = function to differentiate, dv = function to integrate
  NOTE: Choose u using LIATE (Log, Inverse trig, Algebraic, Trig, Exponential)

... (continues)

Finance Formulas

Time Value of Money (TVM)

F1: Present Value
  PV = FV / (1 + r)^n
  WHERE: FV = future value, r = discount rate, n = periods
  NOTE: For annuity: PV = PMT × [1 - (1+r)^(-n)] / r

F2: Future Value
  FV = PV × (1 + r)^n
  WHERE: PV = present value, r = rate, n = periods
  NOTE: For annuity: FV = PMT × [(1+r)^n - 1] / r

F3: Net Present Value
  NPV = Σ(CF_t / (1+r)^t) - Initial Investment
  WHERE: CF_t = cash flow at time t, r = discount rate
  DECISION: NPV > 0 → Accept project

F4: Discounted Cash Flow (DCF)
  DCF = Σ(CF_t / (1+r)^t)
  WHERE: CF_t = projected cash flow at time t, r = discount rate
  NOTE: DCF estimates investment value based on future cash flow projections
  DIFFERENCE: DCF is the sum of discounted cash flows; NPV subtracts initial investment from DCF

F5: Internal Rate of Return
  0 = Σ(CF_t / (1+IRR)^t) - Initial Investment
  NOTE: IRR is the discount rate that makes NPV = 0
  WATCH: May give multiple IRRs for non-conventional cash flows

F5: Capital Asset Pricing Model
  E(R_i) = R_f + β_i × (E(R_m) - R_f)
  WHERE: R_f = risk-free rate, β_i = stock beta, E(R_m) = expected market return
  NOTE: CAPM gives the expected return compensating for systematic risk

F6: Weighted Average Cost of Capital
  WACC = (E/V) × R_e + (D/V) × R_d × (1 - T_c)
  WHERE: E = equity value, D = debt value, V = E+D, R_e = cost of equity, R_d = cost of debt, T_c = tax rate

Medical Formulas

F1: Body Mass Index
  BMI = weight(kg) / height(m)²
  CLASSIFICATION: <18.5 Underweight, 18.5-24.9 Normal, 25-29.9 Overweight, ≥30 Obese

F2: Body Surface Area (Mosteller)
  BSA(m²) = √(height(cm) × weight(kg) / 3600)
  NOTE: Used for chemotherapy dosing, burn assessment

F3: Glomerular Filtration Rate — MDRD
  eGFR = 175 × (S_Cr)^(-1.154) × (Age)^(-0.203) × (0.742 if female) × (1.212 if African American)
  WHERE: S_Cr = serum creatinine in mg/dL
  NOTE: CKD-EPI equation preferred for higher accuracy at GFR > 60

F4: Anion Gap
  AG = Na⁺ - (Cl⁻ + HCO₃⁻)
  NORMAL RANGE: 8-12 mEq/L
  NOTE: Elevated gap → metabolic acidosis (MUD PILES: Methanol, Uremia, DKA, etc.)

F5: Mean Arterial Pressure
  MAP = DBP + 1/3 × (SBP - DBP)
  WHERE: SBP = systolic BP, DBP = diastolic BP
  TARGET: MAP ≥ 65 mmHg for organ perfusion

Session Config

This skill integrates with the session config system (deps/session-profile.json). Before executing, check for an existing session profile:

• If deps/session-profile.json exists, read university, subject, pattern, and exam_type fields to auto-configure the skill.
• If the file does not exist, fall back to user-provided context or prompt the user to run setup-exam-prompt (or npm run init) first.
• Session config eliminates redundant context detection — detection happens once and is reused across all skill calls.

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Error Handling

Situation	Action
Formula domain not recognized	Respond: "Unrecognized formula domain. Supported domains: engineering, mathematics, physics, chemistry, finance, statistics, data science."
Variable definition missing	Flag undefined variables during generation and request clarification
Unit inconsistency	Auto-convert to SI units; warn if conversion may introduce error
Circular formula reference	Detect cycles in formula dependencies and break at the simplest formula

Quality Gate — Check Before Output

• Every formula includes variable definitions with units
• Applicability conditions stated (WHEN clause present)
• Common mistakes noted (WATCH clause present)
• Cross-references to related formulas included
• Consistent notation maintained across all formulas
• Sample calculation provided for non-trivial formulas
• Maximum 3 formulas per page for readability in exam conditions

5. Integration with Other Skills

Skill	Integration
universal-session-config	Reads university/subject/pattern from session profile
universal-notes-generator	Provides the full context from which formulas are extracted
universal-a-plus-answer-writer	Uses formula sheets for numerical problem solutions
universal-last-minute-crammer	Formula sheets are the primary cram resource
universal-mcq-practice-generator	MCQs can be generated targeting specific formula applications