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. 21, 23
**Martin Luther King Jr. Holiday**Monday Jan. 21- 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. 28, 30
- [Lebl] 5.1 The Riemann Integral
- [Lebl] 3.4 Uniform continuity
- [Lebl] 5.3 Fundamental theorem of calculus.
- Homework Sec. 3.4 exercise 3.4.8, Sec. 5.1 exercise 5.1.11. Due Wednesday Feb 6.

- 3rd Week: Feb. 4, 6.
- [Trench] Lemma 3.2.4 p. 131: Riemann's and Darboux' integrals are the same.
- [Lebl] 5.2 Properties of the Riemann Integral
- [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
- [Lebl] 6.2 Interchange of limits.

- 4th Week: Feb. 11, 13
- [Lebl] 6.2 Interchange of limits. (cont.)
- [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. 18, 20
**Presidents' Day Holiday**Monday Feb. 18

- 6th Week: Feb. 25, 27
- [Lebl] 6.3 Picard's Theorem: existence and uniqueness of solutions to ordinary differential equations
- Review

- 7th Week: Mar. 4, 6
**Midterm Exam Monday March 4>**Solutions- [Trench] 5.1 Structure of \(\mathbb{R}^n\). Vectors, open sets, Heine-Borel theorem.

- 8th Week: Mar. 11, 13
- [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. 18, 20
- [Trench] 5.2 Continuous Real-valued functions of n variables.
- [Trench] 5.3 Partial derivatives and the differential. (Real-valued functions).

- 10th Week: Mar. 25, 27
- [Trench] 5.3 Partial derivatives and the differential. (Real-valued functions).
- [Trench] 5.4 Chain rule, Taylor's theorem, Max-min. (Real-valued functions.)

**Spring Break April 1-6**- 11th Week: Apr. 8, 10
- [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. 15, 17
- [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. 22, 24
- [Trench] 6.2 Differentiability, Chain Rule. (Vector-valued functions)
- [Trench] 6.3 Inverse Function Theorem

- 14th Week: Apr. 29, May 1
- [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 6, 8
- [Trench] 6.3 Implicit Function Theorem
- Homework Exercise 6.4 #8
- Review

**Final Exam Monday May 13 4-6pm**