# Math 425 – Honors Advanced Analysis – Spring 2016

Differential an integral calculus form the foundations of modern day geometry and analysis. This course offers a theoretical treatment of differential and integral calculus in higher dimensions. Topics to be discussed include the inverse and implicit function theorems, submanifolds, the theorems of Green, Gauss and Stokes, differential forms, and applications. As part of the honors sequence, this course will be rigorous and abstract.

Lecturer:Rui Loja Fernandes
Email: ruiloja (at) illinois.edu
Office: 346 Illini Hall
Office Hours: TTh 11:00AM-11.50AM (or by appointment);
Class meets: MWF 12:00PM-12:50PM, 241 Altgeld Hall;
Prerequisites: Prerequisite: MATH 424 and either MATH 415 or MATH 416, and consent of the department.

## Syllabus:

• Differential Calculus in Several Variables. Differentiable maps, chain rule, partial derivatives, extrema. Inverse Function Theorem and Implicit Function Theorem.
• Integral Calculus in Several Variables. Integral of a real valued function, measure zero and content zero, integrable functions. Fubini’s Theorem. Partition of unity. Change of variable formula. Sard’s Theorem.
• Differential Forms. Differential forms and vector fields in R^n. Differential of forms and PoincarÃ© Lemma. Integration over chains. The Fundamental Theorem of Calculus.
• Integration on Manifolds. Manifolds. Fields and forms on manifolds. Integration on manifolds and Stokes Formula. The classical theorems of calculus: Green’s Theorem, Stokes Theorem and the Divergence Theorem.

## Textbooks:

In the course we will follow the little book below by Spivak. A good supplement, which contains many more details and examples is the book by Munkrees.

• James R. Munkres, Analysis on Manifolds, Addison-Wesley (1991), Westview Press (1997).
• Michael Spivak, Calculus on Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus, Westview Press (1971).

There will be weekly homework, 2 midterms and a final exam. All exams/midterms will be closed book.

• Homework and in class participation (20% of the grade): Homework problems are to be assigned every week. They are due on the Friday class, at the beginning of the class. No late homework will be accepted. The two worst homework grades will be dropped. On the Friday classes, some homework problems may be discussed in class, with the participation of the students and this will be taken into account for the grade.
• Midterms (40% of the grade): The midterms will take place on February 26 and April 11, in the regular classroom (the dates are subject to change).
• Final Exam (40% of the grade): According to the non-combined final examination schedule it will take place Thursday, May 12, 1:30-4:30 PM, in the regular classroom.

## Homework Assignments and Sections covered so far:

• Homework #1: Read Section “Norms and Inner Product” (Chapter 1) from Spivak’s book and solve exercises 1.1, 1.2, 1.3, 1.4, 1.6, 1.7, 1.8, 1.10.
• Homework #2: Read Section “Subsets of Euclidean space” (Chapter 1) from Spivak’s book and solve exercises 1.14, 1.16, 1.17, 1.19, 1.20, 1.21, 1.22.
• Homework #3: Read Section “Functions and Continuity” (Chapter 1) from Spivak’s book and solve exercises 1.23, 1.24, 1.25, 1.26, 1.28, 1.29, 1.30.
• Homework #4: Read Sections “Basic Definitions” and “Basic Theorems” (Chapter 2) from Spivak’s book and solve exercises 2.1, 2.4, 2.7, 2.10, 2.11 (b), 2.12, 2.13.
• Homework #5: Read Sections “Partial Derivatives” and “Derivatives” (Chapter 2) from Spivak’s book and solve exercises 2.17 (g), (h), (i), 2.18 (d), 2.19, 2.21, 2.22, 2.23, 2.24, 2.25, 2.26.
• Homework #6: Read Sections “Inverse Functions” and “Implicit Functions” (Chapter 2) from Spivak’s book and solve exercises 2.31, 2.32, 2.34, 2.35, 2.36, 2.37, 2.39, 2.41.
• Homework #7: Read Sections “Basic Definitions” and “Measure Zero and Content Zero” (Chapter 3) from Spivak’s book and solve exercises 3.2, 3.3, 3.6, 3.7, 3.9, 3.10, 3.11, 3.12.
• Homework #8: Read Sections “Integrable Functions” and “Fubini’s Theorem” (Chapter 3) from Spivak’s book and solve exercises 3.14, 3.16, 3.18, 3.21, 3.22, 3.26, 3.27, 3.28, 3.30, 3.32, 3.34, 3.35.
• Homework #9: Read Sections “Partitions of Unity” and “Change of Variable” (Chapter 3) from Spivak’s book and solve exercises 3.37, 3.38, 3.39, 3.40, 3.41.
• Homework #10: Read Section “Algebraic Preliminarities” (Chapter 4) from Spivak’s book and solve exercises 4.1, 4.2, 4.3, 4.4, 4.5, 4.8, 4.10, 4.11.
• Homework #11: Read Section “Fields and Forms” (Chapter 4) from Spivak’s book and solve exercises 4.13, 4.14, 4.16, 4.17, 4.18, 4.20, 4.21 and the following two extra problems.
• Homework #12: Read Sections “Geometric Preliminarities” and “Fundamental Theorem of Calculus” (Chapter 4) from Spivak’s book and solve exercises 4.22, 4.23, 4.24, 4.27, 4.29, 4.30, 4.33, 4.34.
• Homework #13: Read Sections “Manifolds” and “Fields and Form on Manifolds” (Chapter 5) from Spivak’s book and solve exercises 5.1, 5.2, 5.3, 5.7, 5.9, 5.10, 5.11, 5.12, 5.14.
• Homework #14: Read Sections “Stokes’ Theorem on Manifolds” and “The Volume Element” (Chapter 5) from Spivak’s book and solve exercises 5.18, 5.19, 5.22, 5.21, 5.26, 5.27, 5.28, 5.31.

Sections of the book covered:

• Chapter 1: Functions on Euclidean Space
• Chapter 2: Differentiation
• Chapter 3: Integration
• Chapter 4: Integration on Chains
• Chapter 5: Integration on Manifolds:
• Manifolds
• Fields and Forms on Manifolds
• Stokes’ Theorem on Manifolds
• The Volume Element
• The Classical Theorems