generate-theory

name	generate-theory
description	Generate a complete, NASA-level theory section for any mechanical engineering topic. 11+ sections, all formulas in KaTeX, term tables, SVG diagrams, worked example. Shigley/Hibbeler depth.
metadata	{"priority":8,"promptSignals":{"phrases":["generate theory","write theory","theory section","theory for","explain the theory","add theory"],"minScore":5}}

Generate Theory — Complete Skill

Standards (Non-Negotiable)

Every theory section must meet the NASA/MIT standard:

• People's lives depend on correct information
• Teach age-10 curiosity → master's-level rigor in the SAME section
• Never shortcut — walk the full textbook chapter
• Sources: Shigley, Hibbeler, Popov, Beer & Johnston, Roark's

Mandatory Section Checklist

Generate ALL of these, customized for the topic:

Section 1: Introduction

• What physical phenomenon this describes
• When/why it matters (real failure examples)
• Assumptions and validity range
• Relationship to other phenomena (how it connects to adjacent theory)

Section 2: Governing Equation

<TheorySection title="Governing Equation">
  <p>Plain English: what the equation says physically.</p>
  <MathBlock>{String.raw`\sigma = \frac{Mc}{I}`}</MathBlock>
  <TermTable rows={[
    { symbol: 'σ', name: 'Normal Stress', meaning: 'Stress at distance c from NA', imperial: 'psi or ksi', si: 'Pa or MPa' },
    { symbol: 'M', name: 'Bending Moment', meaning: 'Internal moment at section', imperial: 'lbf·in', si: 'N·m' },
    ...
  ]} />
</TheorySection>

Section 3: Derivation (condensed — show key steps)

• Start from first principles or fundamental law
• Show the key simplification steps
• State every assumption made

Section 4: Cross-Section Properties

• I, S, Z formulas for rectangle, circle, hollow circle, standard sections
• How to find centroid (first moment of area)
• Parallel axis theorem: I = I_c + Ad²
• Include SVG cross-section diagram with v3 labels

Section 5: Material Properties

• Which properties matter for this analysis
• Typical values table (Imperial and SI)
• Temperature effects
• When to worry about anisotropy

Section 6: Failure Modes (comprehensive)

• List ALL failure modes that can occur
• Formula for each
• How to check — inequality to verify
• Factor of safety for each mode

Section 7: Stress Concentrations

• Where they occur (geometric discontinuities)
• Kt charts or equations for common geometries
• Fatigue reduction factor Kf
• How to apply in design

Section 8: Combined Loading (if applicable)

• How this stress combines with axial, shear, torsional
• Superposition principle (valid for linear elastic only)
• Von Mises or maximum shear stress criterion

Section 9: Design Codes

• Applicable codes: AISC, ASME, ASCE, AWS, API, ASHRAE
• Safety factors required by code
• Code check format (LRFD vs. ASD)
• Citation: code edition, section number

Section 10: Special Cases / Edge Cases

• What happens at limiting geometries
• Plastic analysis (vs. elastic assumed above)
• Residual stresses
• Thermal effects
• Dynamic vs. static loading differences

Section 11: Worked Example

<WorkedExample
  title="Example: Calculate stress in W12×26 at midspan"
  given={['P = 10 kips', 'L = 15 ft', 'W12×26 (Ix = 204 in⁴, c = 6.11 in)']}
  find={['Maximum bending stress σ_max', 'Factor of safety vs. yield (Fy = 36 ksi)']}
  solution={[
    { step: '1. Find M_max', equation: 'M = PL/4 = (10)(15×12)/4 = 450 kip·in' },
    { step: '2. Calculate σ', equation: 'σ = Mc/I = (450)(6.11)/204 = 13.5 ksi' },
    { step: '3. Factor of safety', equation: 'n = Fy/σ = 36/13.5 = 2.67 ✓' },
  ]}
/>

Section 12+: Advanced Topics (add as needed)

• Buckling / instability interaction
• Fatigue under this loading
• Fracture mechanics consideration
• FEA modeling tips for this problem type
• Historical failures and lessons learned

KaTeX Syntax Rules

• Fractions: \frac{numerator}{denominator}
• Square root: \sqrt{expression} or \sqrt[n]{expression}
• Subscripts: \sigma_{allow}, \tau_{xy}
• Superscripts: E^{1/2}, \sigma^2
• Greek: \sigma \tau \epsilon \gamma \phi \nu \rho \mu \omega \alpha \beta \delta \Delta
• Always use String.raw for complex expressions: String.raw\sigma = \frac{Mc}{I}``
• Inline: <Math>{\sigma}</Math>
• Block: <MathBlock>{String.raw\sigma = \frac{Mc}{I}}</MathBlock>

SVG Diagrams in Theory

Every theory section should have ≥ 2 diagrams showing:

1. The geometry being analyzed (cross-section, loading diagram)
2. The stress distribution (color gradient or zone diagram)

Use v3 layered pattern — SVG shapes, HTML labels.