Resources

Here are some resources that you might use to understand things better. If you run into other resources that you find helpful, please share them with me so that I can share them with the class! I’d especially be interested in hearing about reliable freely available resources, and non-English language resources.

You might consider using the following to supplement your understanding of single variable and multivariable derivatives.

• Protter and Morrey. A First Course in Real Analysis. 2nd Edition.
  • You probably used this book for real analysis 1. Chapter 4 is single variable derivatives. Chapter 7 is multivariable derivatives (except that the inverse function theorem is in Chapter 14).
• Callahan. Advanced Calculus: A Geometric View.
  • This is a nice book with lots of pictures. I highly recommend checking it out. Chapters 1 through 5 cover single variable and multivariable derivatives. If I had to pick one more topic from this book to include in our course, it would be Chapter 7, “Critical Points.”
• Munkres. Analysis on Manifolds.
  • Chapter 2 covers multivariable differentiation.
• Rudin. Principles of Mathematical Analysis. 3rd Edition.
  • This is the classical real analysis textbook. It’s very challenging to read, but well worth the effort. Chapter 5 is single variable derivatives, and chapter 9 is multivariable derivatives.
• Spivak. Calculus on Manifolds.
  • Chapter 2 covers multivariable differentiation.
• Trench. Introduction to Real Analysis.
  • Freely available online. Chapter 2 talks about single variable derivatives, and chapters 5-6 talk about multivariable derivatives.

And you might consider using the following to supplement your readings on manifolds and tangent spaces.

• Gadea, Muñoz Masqué, and Mykytyuk. Analysis and Algebra on Differentiable Manifolds. 2nd Edition.
  • This is a nice book with lots of problems. Chapter 1 is the one that’s most relevant to our course.
• Munkres. Analysis on Manifolds.
  • This was also listed above. Chapters 4 and 5 cover manifolds, though only manifolds that live inside \(\mathbb{R}^n\). Chapter 6 talks about tangent spaces, but its interspersed with things about multilinear algebra that we won’t be discussing.
• Spivak. A Comprehensive Introduction to Differential Geometry. Volume 1, 3rd Edition.
  • Chapters 1, 3, and 5 overlap significantly with the latter part of our class. If I had to pick one more topic out of this book to include in our course, it would be Chapter 5, “Vector fields and differential equations.”
• Tu. An Introduction to Manifolds. 2nd Edition.
  • Sections 1–2, 5–9, and 11–13 are the parts that overlap with our course, but this book contains much, much more information than we’ll be discussing. Section 14 overlaps with chapter 5 of Spivak’s book cited above.

You may also find Serre’s talk “How to Write Mathematics Badly” useful for improving your proof writing.