Research

Masters Thesis

Abstract

This project extends calculus from Euclidean spaces to higher-dimensional objects known as manifolds. We develop a coordinate-free approach to calculus on \(\mathbb{R}^n\) using differential forms and derivations. After rigorously defining topological and smooth manifolds, we explore smooth maps, tangent spaces, and the differential as a generalization of the derivative. Key results include the inverse function theorem and criteria for submanifolds. We also study immersions, submersions, embeddings, and introduce differential 1-forms. Finally, we present partitions of unity, which connect local and global analysis, enabling smooth constructions across charts on manifolds.

Current Work