Mathematical Tools

Y. Eddie Lu, Summer 2026

These pages rebuild the course as a sequence of arguments and calculations. They are not a substitute for doing problems from memory. Reading records exposure.

With the page closed, mastery means that you can:

1. state definitions;
2. audit a theorem’s assumptions;
3. use the result on a new problem.

What this course is building

Economics repeatedly asks whether an object exists, how it changes after a shock, and whether a candidate is optimal. Linear algebra describes directions and maps. Topology supplies the conditions for limits and existence. Differentiation gives local approximations. Optimization and convexity convert those approximations into economic conclusions.

Dependency map

FoundationsLogic, sets, functions, and notation used everywhere.

Linear algebra: objectsSpaces, linear maps, kernels, images, and dimension.

Linear algebra: matricesCoordinates, rank, inverses, eigenvalues, and quadratic forms.

TopologyOpen sets, compactness, continuity, and correspondences.

DifferentiationLocal linear approximation, gradients, Hessians, and local inversion.

OptimizationExistence, first- and second-order conditions, equality constraints, and KKT conditions.

Maximum and envelope theoremsStability of optima and sensitivity of optimized values.

Convexity and global optimizationConvex geometry, global optimality, uniqueness, and KKT sufficiency.

The dependency is not merely chronological:

1. Foundations supports linear algebra and topology.
2. Linear-algebra objects support matrices.
3. Topology supports differentiation, optimization, and the maximum theorem.
4. Differentiation supports optimization and the envelope theorem.
5. Convexity strengthens optimization: it can turn local conditions into global ones.

Reading routes

First pass

Read the motivation, definitions, examples, and theorem statements. On a first pass, do not confuse recognizing a statement with being able to use it.

Second pass

Open a proof only after you can state the theorem and explain why its conclusion is plausible. Keep the proof closed on the first pass. At each theorem, say why each assumption appears before reading the argument.

Retrieval pass

Answer the collapsed checks before opening them. Then work the exit tickets on paper. Revisit any item that requires a prompt from the page.

Pages

1. Foundations: logic, sets, functions, domains, codomains, and coordinatewise order.
2. Linear algebra: objects: vector spaces, subspaces, bases, linear maps, kernels, images, and rank-nullity.
3. Linear algebra: matrices: coordinates, rank, inversion, eigenvalues, and definiteness.
4. Topology: compactness, continuity, and correspondences.
5. Differentiation: derivatives, Jacobians, gradients, and Hessians.
6. Optimization: unconstrained, equality-constrained, and inequality-constrained conditions, including KKT conditions.
7. Maximum and envelope theorems: Berge’s theorem, argmax stability, and unconstrained and constrained envelope formulas.
8. Convexity and global optimization: convex sets and functions, separating and supporting hyperplanes, global optimality, and KKT sufficiency. The page then extends the chapter to quasiconcavity and quasiconvexity.

How to use a page

1. Read a retrieval question and write an answer before opening it.
2. For every purple theorem, identify the domain, assumptions, and conclusion.
3. For every red counterexample, name the precise inference that fails.
4. Complete the exit tickets without notes on a later day.

Exam preparation

The August 14 in-class exam rewards exact distinctions: a vector versus a scalar, a domain versus an image, local versus global, and necessary versus sufficient. When you study a theorem, write its implication direction explicitly.

Scope

The eight linked pages cover the course sequence through Chapter 4, Sections 4.1–4.7, and Chapter 5, Sections 5.1–5.3. The Chapter 5 page also includes a clearly marked extension on quasiconcavity and quasiconvexity. The pages use original explanations and do not reproduce homework solutions.