Multivariable Analysis, Spring 2014

Instructor

Grader

General Information

Course description

This is a rigorous-style graduate-level analysis course meant to introduce Master's students to differentiation and integration for vector-valued functions of one and several variables. Covered topics include curves, surfaces, manifolds, inverse and implicit function theorems, integration on manifolds, Stokes’ theorem, applications. There will also be additional material related to convex analysis and optimization.

Lectures

  1. Jan 27: Convex sets (Ch 1) and overview of Ch 1.

  2. Feb 3: Non-euclidean norms and overview of Ch 2. Minkowski functional.

  3. Feb 10: Directional derivatives, differentiation (Ch 3.1 - 3.3).

  4. Feb 17: President's Day

  5. Feb 24: Functions of class C^q, relative extrema.

  6. Mar 3: Ch 3.6 - 4.1. Convex functions and linear transformations

  7. Mar 10: Ch 4.1 - 4.5. Affine transformations. Inverse function theorem.

  8. Mar 17: Spring break

  9. Mar 24: Midterm

  10. Mar 31: Ch 4.5 - 4.8. Inverse/Implicit Function Theorem, Lagrange multipliers.

  11. Apr 7: Manifolds (skipped intersection of manifolds), Ch 6.1 - 6.4.

  12. Apr 14: Ch 7.1 - 7.2.

  13. Apr 21: Ch 7.3 -

  14. Apr 28: Selected topics from Ch 8 and 5

  15. May 5: Selected topics from Ch 8 and 5

  16. May 12: Final exam

Homeworks and exams

Note: For HW submissions after the deadline, please reach out to our grader Insuk directly. Late HWs will be assessed a late penalty of 10% per day. Solutions posted here are selected from submitted homeworks and might contain some typos/minor errors.

Suggested reading

  1. Convexity (Ch 1 and 2): Convex Optimization by Boyd and Vandenberghe

  2. Differentiation, Riemann Integration : Elementary Classical Analysis by Marsden and Hoffman

  3. Implicit/Inverse Function Theorem: Advanced Calculus by Buck.