Course Learning Outcomes:

1. Understand and distinguish various topological notions for subsets of the real line and of the n-dimensional Euclidean space, such as interior and boundary points, open/closed sets, cluster points, isolated points, nowhere dense sets, compact sets, connected sets, G-delta and F-sigma sets.
2. Understand and apply the concept of limit, continuity for functions of several real variables, and their ramifications, such as the intermediate value theorem and the extreme value theorem. Understand and apply the concept of uniform continuity and its ramification, such as the Heine-Cantor theorem. Understand and distinguish sets of continuity and their properties.
3. Understand and apply the concept of differentiability for (possibly vector-valued) functions of several real variables and its ramifications. For instance, apply the chain rule, the inverse function theorem, and the implicit function theorem in concrete examples. Understand differentiability and gradients in terms of partial derivatives.
4. Understand and apply the concept of relative extrema for functions of several variables and their relation to the Jacobian and the Hessian. Understand and apply the concept of constrained extrema and the Lagrange Multiplier Theorem in concrete examples.
5. Understand and apply the Picard-Lindelöf theorem to study the existence and uniqueness of solutions of systems of ordinary differential equations.