Chapter 1 of the course notes.
Chapter 2 of the course notes.
Chapter 3 of the course notes.
Chapter 4 of the course notes.
Homework 1, due in class Wed., January 31: Exercises 1,2,6,7,8 from the notes (Chapter 1)
Homework 2, due in class Mon., February 19: Exercises 9,10,11,12,14 from the notes (Chapters 1&2)
Homework 3, due in class, Mon., March 26: Exercises 15,17,19,22,23 from Chapter 2 of the notes.
Homework 4, due in class, Mon., April 9: Exercises 26,27,28,30,31 from Chapter 3 of the notes.
Homework 5, due 12 noon, Mon., May 7: Exercises 38,46,49,56,59,61,62 from Chapter 4 of the notes.
A note on the connectedness of intervals.
An old version of the course notes: Course notes (New version to come...)
The real definition of a smooth manifold
| Instructor | Prof. John Loftin, 323 Smith, Phone: 5156 ext. 23. |
| loftin AT andromeda DOT rutgers DOT edu | |
| Website | www.andromeda.rutgers.edu/~loftin (this syllabus is attached to the website) |
| Department Website | http://math.newark.rutgers.edu/ |
| Text | Notes and references will be provided. |
| Supplemental reading | TBA |
| Prerequisites | Real Analysis I (metric spaces, Lebesgue theory, basic Fourier analysis) |
| Overview | The main focus will be on differential equations. We will develop tools from functional analysis along the way as needed. |
| Topics | Multivariable calculus, ordinary differential equations, the calculus of variations, weak solutions and distributions |
| Homework | Homework will be collected about every 2 weeks. |
| Tests | There will be an in-class midterm exam, to be held on Wednesday, March 21. There may also be an in-class or take-home final exam. |
Tentative outline of topics:
1. Multivariable Calculus
a. Differentiable functions
b. Spaces of functions
c. Contraction mappings: Inverse Function
Theorem
2. Ordinary Differential Equations
a. Existence and uniqueness
b. Some techniques for explicitly solving ODEs
c. Vector fields and flows
3. The Calculus of Variations
a. Example: Geodesics on Riemannian
manifolds
b. Distributions and weak derivatives
c. Fourier series and Hilbert space methods
d. The direct method of the calculus of
variations: Finding a
length-minimizing curve in a free homotopy class on a compact manifold