Updated Tue Jan 21 15:20:19 PST 2014

This is a tentative calendar. I probably will have to move some things as we go along but I'll try to avoid moving the test days so you can plan ahead.

I will post homework assignments here so please check back often, at least once a week.

Chapter and section numbers refer to our textbooks

- [Lebl]:
*Basic Analysis: Introduction to Real Analysis*, by Jiří Lebl. - [Trench]:
*Introduction to Real Analysis*by William Trench.

- 1st week, Jan. 20, 22
- Mon. Jan. 22 Rev. Martin Luther King Holiday
- Outline of the course: Integration, the Existence and Uniqueness Theorem of Ordinary Differential Equations (Picard's theorem), multivariable calculus and its connection with linear algebra, the Implicit Function Theorem and the Inverse Function theorem.

- 2nd week, Jan. 27, 29
- [Lebl] 5.1 The Riemann Integral

- 3rd week, Feb. 3, 5
- [Lebl] 3.4 Uniform continuity
- Homework Sec. 3.4 exercise 3.4.8, Sec. 5.1 exercise 5.1.11. Due Monday Feb 10.
- [Lebl] 5.3 Fundamental theorem of calculus.
- [Trench] Lemma 3.2.4 p. 131: Riemann's and Darboux' integrals are the same.
- [Lebl] 5.2 Properties of the Riemann Integral (lightly)
- [Lebl] 6.1 Sequences of functions: Pointwise and Uniform convergence. Example where continuous functions converge pointwise to discontinuous function: Fourier series for a step function.
- Homework: 6.1.2, 6.1.5, 6.1.6 (I found this one really surprising), 6.1.10, 6.2.1. Due Wednesday Feb. 12

- 4th week, Feb. 10, 12
- [Lebl] 6.2 Interchange of limits.
- [Lebl] 6.3 Picard's Theorem: existence and uniqueness of solutions of ordinary differential equations
- Slope field for van der Pol equation. Also see the Wikipedia article "van der Pol oscillator".

- 5th week, Feb. 17, 19
- Mon. Feb 17 Presidents' Day Holiday

- 6th week, Feb. 24, 26
- [Lebl] 6.3 Picard's Theorem: existence and uniqueness of solutions to ordinary differential equations
- Homework: #5.1.3, 5.1.4, 5.2.4, 5.2.5, 5.2.6. Due next Wednesday.

- 7th week, Mar. 3, 5
**Cancelled: Midterm Exam Monday March 3**- Homework 6.3.4 (easy!)
- Start chapter 7 (in the same book we have been using) Metric Spaces
- More homework (due in about a week). Exercises 7.1.3, 7.1.4, 7.1.6, 7.1.7

- 8th week, Mar. 10, 12
- [Trench] 5.1 Structure of \(\mathbb{R}^n\). Vectors, open sets, Heine-Borel theorem.
- [Trench] 5.2 Continuous Real-valued functions of n variables.
- Homework: 5.1 exercises 7a, 9a, 12, 13, 19d, 24, 26, 29, 30, 5.2 exercise 3

- 9th week, Mar. 17, 19
- [Trench] 5.2 Continuous Real-valued functions of n variables.
- [Trench] 5.3 Partial derivatives and the differential. (Real-valued functions).

- 10th week, Mar. 24, 26
- [Trench] 5.3 Partial derivatives and the differential. (Real-valued functions).
- [Trench] 5.4 Chain rule, Taylor's theorem, Max-min. (Real-valued functions.)

- Mar. 31. Apr. 4 Spring Recess
- 11th week, Apr. 7, 9
- [Trench] 5.4 Chain rule, Taylor's theorem, Max-min. (Real-valued functions.)
- [Trench] 6.1 Linear Transformations and Matrices. (Linear algebra.)

- 12th week, Apr. 14, 16
- [Trench] 6.1 Linear Transformations and Matrices. (Linear algebra.)
- [Trench] 6.2 Differentiability, Chain Rule. (Vector-valued functions)
- Homework 6.2 exercises 5, 12ab, 20a

- 13th week, Apr. 21, 23
- [Trench] 6.2 Differentiability, Chain Rule. (Vector-valued functions)
- [Trench] 6.3 Inverse Function Theorem

- 14th week, Apr. 28, 30
- [Trench] 6.3 Inverse Function Theorem
- Homework Exercise 6.3 exercise 16. Hint: if you use complex variables then function is \(F(z)=z^2\) because if \(z=(x+iy)\) then \(z^2=(x^2-y^2)+i(2xy)=u(x,y)+iv(x,y)\) where \(u\) and \(v\) are the functions in this problem.

- 15th week, May 5, 7
- [Trench] 6.3 Implicit Function Theorem
- Homework Exercise 6.4 #8
- Review

- Final exams week
**Final Exam Monday May 12, 4-6pm**