Brief Lecture Notes on Vector Analysis

These are brief lecture notes on Vector Analysis developed when I (Gautam Iyer) taught Vector Analysis at Carnegie Mellon University. These notes only list statements of results covered in lectures, and sometimes brief sketches of proofs. I do motivation, intuition, and complete proofs in class, and don’t plan to include them in these notes.

Note – I initially incorrectly assumed students who took this class would be familiar with rigorous one variable calculus, in which case it makes sense to start with metric spaces. When I realized this wasn’t the case, I backtracked which is why the section on real numbers comes after the section on metric spaces. Future iterations of this course, will likely start with an introduction to real numbers before Metric spaces.

Contents

• Metric spaces
• Limits and continuity
• The real numbers
• Sequences and series
• Compactness
• Derivatives in one dimension
• Derivatives in higher dimensions
• Higher order derivatives
• Inverse and implicit functions
• Manifolds and tangent spaces
• Constrained optimization
• Riemann integrals in one dimension
• Integrals in higher dimensions
• Line integrals
• Greens theorem, Curvature and Winding
• Surface integrals and Stokes theorem
• The divergence theorem

Source Code

The source code is available on Codeberg.

The notes are written in Markdown, and converted to HTML5 using the Python script md-to-html. It should be viewable on most modern browsers, but I have only tested it on Firefox.

Licence

Brief Lecture Notes on Vector Analysis © 2026 by Gautam Iyer is licensed under Creative Commons Attribution-NonCommercial 4.0 International .

You may adapt and share this work for non-commercial purposes, provided you cite it appropriately. The full terms are in the licence deed.