Chapter 1 - Part 3

This is the content for Fri, Jan 16, 2026.

Covered in this lecture

• Topological concepts

  • Definition of a topology: D2L Video

  • Definition of a metric space: D2L Video

  • Definition of a Metric topology: D2L Video

  • Open and closed sets: D2L Video

• Differentiability

  • Gradient

  • Directional derivative

  • Hessian

• Approximation Theorems

  • Linear approximation theorem D2L Video

  • Quadratic approximation theorem D2L Video

Todos

Before class

• Read basic topological concepts in Chapter 1.5 of the textbook.

  • Watch the videos linked above for further information, but note that in our class we will only focus on the case of open and closed sets in \(\R^n\), and these videos above give a more general context than we need.

• Read differentiability section in Chapter 1.5.1 of the textbook.

  • We don’t have videos for this.

In class

• Lec 1-3 Slides

• Worksheet 1-3

After class

• Upload worksheet to Crowdmark by 5pm.

• No class Monday for university holiday

• Note there is a quiz next class, Weds Jan 21! Come prepared with a handwritten, 8.5”x11” cheat sheet.