Here is a list of topics that are covered in class, in the order they were covered. All theorems on differentiation (including inverse and implicit function theorems) will be proved rigorously. In the interest of time, however, a few proofs on integration will be skipped.

- Distance in .
- - definition of limits
- Uniqueness, sums and products.
- Continuity. Continuity of composites.
- Open sets and domains.

- Sequences
- Infimum and Supremum
- Bolzano-Weierstrass theorem
- Proof of the extreme value theorem
- Continuity of inverses
- Uniform continuity

- Derivatives
- Mean value theorems

- Partial and directional derivatives
- The derivative as a linear transformation
- Continuity of partials
- The chain rule
- Mean value theorem
- Higher order partials and Clairut’s theorem.
- Taylor’s theorem
- Maxima and minima

- Proof of the inverse function theorem
- Coordinate changes
- The implicit function theorem
- Parametric curves.
- Surfaces, Manifolds, and tangent spaces.
- Lagrange multipliers

- Double and triple integrals
- Iterated integrals and Fubini’s theorem
- Change of variables

- Line integrals
- Parametrization invariance and arc length
- The fundamental theorem
- Greens theorem

- Surface integrals
- Surface integrals of vector functions
- Stokes theorem
- Conservative and potential forces
- Divergence theorem

- Lecture Schedule
- 268 website (2015)
- 269 website (2012)
- Midterm 1 (2012) (and solutions)
- Midterm 2 (2012) (and solutions)
- Your midterm 1 (and solutions).
- A few “exam like” questions
- Your midterm 2 (and solutions).
- Lagrange multipliers
- Screencasts
- Your final (and solutions).