Multivariate Real Analysis

A multivariable course taught at UCR as Cálculo en varias variables, extending the rigorous real-analysis treatment of single-variable calculus from \(\mathbb{R}\) to \(\mathbb{R}^n\). The notes follow Dr. Juan Gabriel Calvo Alpízar’s lectures and cover the topology of \(\mathbb{R}^n\) (norms, open and closed sets, compactness, connectedness), continuity and limits of functions of several variables, differentiation (partial derivatives, the total derivative, the chain rule, the inverse and implicit function theorems, Taylor’s theorem in several variables), multiple integration (Jordan measurability, Fubini’s theorem, change of variables) and an introduction to vector calculus.

These are personal notes I took as a student. They were written in Spanish and translated into English; both versions are available and you can switch between them at any point. They are shared as-is, may contain errors, and are not affiliated with the instructor.