Analysis II
Course facts
- Payment: Free, no certificate
- Credits: Non Credit
- Learning style: Self-paced, start anytime
- Resources: Lecture Notes
MIT OpenCourseWare details
- As taught in: Fall 2005
- Department: Mathematics
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A rigorous course on multivariable calculus, differential forms and analysis on manifolds, ending with degree theory and Stokes’ theorem.
Continuing from a first analysis course in the direction of manifolds and global analysis, this class first works in several variables. Differentiation, the inverse function theorem and Riemann integrals in several variables are treated rigorously. The second half develops the theory of differential forms on n-dimensional vector spaces and curved spaces, and it ends with degree theory and Stokes’ theorem.
What You Will Achieve
- Prove results on differentiability and the inverse function theorem
- Integrate over general regions using Fubini’s theorem
- Work with tensors, the wedge product and differential forms
- Apply Stokes’ theorem and integration on manifolds
Program Overview
Multivariable Calculus
The first half gives a rigorous treatment of several variables.
- Metric spaces and compactness
- Differentiation in n dimensions
- Inverse function theorem
- Fubini theorem and partitions of unity
Multilinear Algebra and Forms
The next lectures introduce the algebraic tools.
- Tensors and alternating tensors
- Wedge product
- Tangent spaces and k-forms
- Poincare theorem
Manifolds
The closing lectures carry these tools to curved spaces.
- Proper maps and degree
- Definition and examples of manifolds
- Orientations of manifolds
- Stokes’ theorem
Course Completion
Job Outlook
Rigorous analysis on manifolds supports further study in mathematics and in theoretical areas that depend on geometry.
Good for: Mathematics students who have finished a first course in analysis and linear algebra and want to reach analysis on manifolds.
Keep in mind: Lecture notes are the listed material, and most problems come from Munkres and Spivak, which are not supplied. No solutions are listed.
Bottom line: A proof-based route to differential forms and Stokes’ theorem, with a daily problem plan keyed to its textbooks.
Frequently Asked Questions
Whose book does it draw on most?
Analysis on Manifolds by J. Munkres is the main textbook. Some material and homework come from Calculus on Manifolds by M. Spivak, along with notes on multi-linear algebra.
How are the problems organized?
There are daily problems, which are not graded, and graded assignments, which usually include one problem from each day. Not all lectures have assigned daily problems.
What does the first half cover?
Metric spaces, compactness, differentiation in n dimensions, the chain rule, the inverse function theorem, integration of several variables and partitions of unity.
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