Lecturer

Prof. Boris Tsirelson (School of Mathematical Sciences).

Instructors

Adi Glücksam

Yonatan Shelah

Prerequisites

Analysis 2; Linear algebra 2a

Grading policy

First semester exam (01.02; 03.03)

Final exam (01.07; 08.09)

LECTURE NOTES

Preliminaries

0. Conventions, notation, terminology etc.
1. Euclidean space Rⁿ.
• Appendix A: Structures and isomorphisms in general (if you are interested).
2. Basics of differentiation.

Differentiation

3. Open mappings and constrained optimization.
4. Inverse function theorem.
5. Implicit function theorem.
• What is the Implicit Function Theorem good for? (A discussion on Mathoverflow).

Integration

6. Basics of integration.
7. Iterated integral.
8. A glimpse into Lebesgue's theory.
9. Change of variables.
10. Improper integral.

Manifolds and differential forms

11. From path functions to differential forms.
12. Manifolds in Rⁿ.
13. Integration, from local to global.
14. Divergence, flux, Laplacian.
15. From boundary to exterior derivative.
16. Stokes' theorem.

Summary

• Sections 1-8
• Sections 9-15

TEXTBOOKS

• Jerry SHURMAN, "Multivariable calculus". (Online only, not printed.)
• Reyer SJAMAAR, "Manifolds and differential forms lecture notes". (Online only, not printed.)

ADDITIONAL LITERATURE

• Theodore SHIFRIN, "Multivariable mathematics" (2005 Wiley).
• John H. HUBBARD, Barbara Burke HUBBARD, "Vector calculus, linear algebra, and differential forms" (2002 Prentice).
• Vladimir A. ZORICH, "Mathematical analysis II" (2004 Springer). (Our library has 8 copies.)
• Wendell FLEMING, "Functions of several variables" (1977 Springer). (Our library has 3 copies.)
• (History, if you like) Hans Samelson, "Differential Forms, the Early Days; or the Stories of Deahna's Theorem and of Volterra's Theorem", The American Mathematical Monthly, Vol. 108, No. 6 (Jun. - Jul., 2001), pp. 522-530.

Solutions to selected exercises

• 1b14.

EXAMS (in Hebrew)

• 01.02.2015; see also typical errors, comments etc.
• 03.03.2015; see also typical errors, comments etc.

• 2015, sample
• 01.07.2015; see also typical errors, comments etc.
• 08.09.2015; see also typical errors, comments etc.

A quote:

The world is not one-dimensional, and calculus doesn't stop with a single independent variable.

James Nearing, Chapter 8 "Multivariable Calculus" of the course "Mathematical Tools for Physics".