# Calendar: MAT 403 Advanced Analysis II. Spring 2013. Jennings.

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

Chapter and section numbers refer to our textbooks

### Lectures and assignments

• 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