This is a beta course, so its structure, chapters, and examples may continue to change.
Multivariable Differential Calculus
Multivariable differential calculus carries the core ideas of single-variable calculus—limits, continuity, derivatives, and extrema—into settings with two or more independent variables. Almost every physical quantity depends on several factors, so these tools are indispensable in physics, engineering, economics, and machine learning.
- Basic Concepts — what a multivariable function is, its domain and geometric meaning, limits along different paths, and continuity.
- Partial Derivatives and Total Differentials — partial derivatives, higher-order partials, the total differential, and how they relate to differentiability.
- Directional Derivatives and Gradients — the rate of change in arbitrary directions, the gradient vector, tangent planes, and normal lines.
- Extrema and Constrained Extrema — critical points, the second-derivative test, Lagrange multipliers, and practical optimization.
| Order | Page | Covers |
|---|---|---|
| 1 | Basic Concepts | definition, domain, limits, continuity |
| 2 | Partial Derivatives and Total Differentials | partials, higher-order derivatives, total differential |
| 3 | Directional Derivatives and Gradients | directional derivatives, gradient, tangent plane |
| 4 | Extrema and Constrained Extrema | critical points, second test, Lagrange multipliers |
Prerequisites: single-variable differential calculus (especially limits, continuity, and derivatives) and basic linear algebra / vector concepts. Start with Basic Concepts to build intuition, then proceed in order.