Lecturer

Prof. Boris Tsirelson (School of Mathematical Sciences).

Instructors

Michael Bromberg and Dmitry Faifman

Daniel Rosen

Prerequisites

Analysis 2; Linear algebra 2.

Grading policy

First semester exam (26.01; 12.09)

Final exam (11.07; 19.09)

LECTURE NOTES

Preliminaries

0. Conventions, notation, terminology etc.
1. Euclidean space Rⁿ.
0. Appendix: If spaces are not a joy to you.

Differentiation

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

Integration

6. Riemann integral.
7. Iterated integral.
8. Change of variables.
9. Convergence of volumes and integrals, and a correction to it.

Differential forms

10. From path functions to differential forms.
11. From boundary to exterior derivative; Stokes' theorem.
12. Low dimensions, vector fields.
13. Exact forms, closed forms, loops and electromagnetism.
14. Higher order forms; divergence theorem.

Manifolds

15. Chart, orientation, volume form.
16. Integration: from single-chart to many-chart.

Summary

• Sections 2-8.
• Sections 9-16.

Solutions to selected exercises

• 2f2, 3b10, 3b12.
• 3e3, 3i1, 4c9.
• Homework: 6#1(b,d), 6#3, 8#2, 8#4.

TEXTBOOKS

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

ADDITIONAL LITERATURE

• Vladimir A. Zorich, "Mathematical analysis II". (Our library has 16 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.

EXAMS (in Hebrew)

• 2014, sample
• 26.01.2014; see also typical errors, comments etc.
• 12.09.2014; see also typical errors, comments etc.

• 2014, sample
• 11.07.2014; see also typical errors, comments etc and some solutions.
• 19.09.2014; 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".