Tutorial for Mathematical Analysis II (summer term 2017/2018)

Lecture: Lecturer Mgr. Tereza Klimošová, Ph.D. Webpage of the lecture is here.

Office hours: After a mutual agreement; preferably via email. My office is on the fifth floor at Malá Strana.

Conditions to get a course credit:

• In order to get a credit you need to obtain at least 60 points in total (max is 100).
• Homeworks: There will be 10 series, each worth 3 points. In total you can get at most 30 points.
• Deadline: The beginning of the next tutorial. Solutions delivered after the deadline get 0 points automatically.
• How to hand in your solution: Personally at the beginning of the next tutorial or via email in pdf (but please do not send me a photo of your handwritten solutions).
• If you intend to write your solutions on a paper, please, use the A4 size and do not bend it (I have to make a copy of all your solutions).
• Test: There will be two tests (one around the midterm, second at the end), worth 35 points each.
• Attendance: Compulsory, you can be absent at most 3 times.
• Activity during the tutorials: You can get 1 point for being active during the session (max 1 point per person and session). These points add to your total count, but are not included in the maximal count. They are a bonus.

• If you do not manage to write a test to your satisfaction, you will have (at most) two additional attempts.
• Who will not have at least 60 points at the end of semester, but will have at least 50 points, will get a chance to obtain the remaining points for additional homeworks (but expect them to be hard and laborious).

1. session (22.2.2018) - Organization. Primitive function (indefinite integral) and its basic properties. Integration by parts.

Exercise sheet (in pdf): here
Homework (in pdf): here

2. session (1.3.2018) - Integration by substitution (change of variables), finding primitive function on a maximal domain (gluing). Integration of rational function.

Exercise sheet (in pdf): here
Homework (in pdf): here

3. session (8.3.2018) - Tips and tricks to calculate partial fraction decomposition. Standard substitutions (rational functions in sine and cosine, trigonometric substitutions)

Exercise sheet (in pdf): here
Homework (in pdf): here

4. session (15.3.2018) - Standard substitutions continued (rational functions in roots of x and roots of (ax+b)/(cx+d), Euler's substitutions). Riemann integral. Bonus: a definition of Henstock-Kurzweil integral.

Exercise sheet (in pdf): here
Homework (in pdf): here Note the corrected version of the assignment of the Task 1. Deadline for Task 1 shifted to 29.3.2018

5. session (22.3.2018) - Newton's and Riemann's integrals, the first and the second Fundamental theorem of Calculus, integration by parts and by substitution for Riemann's integral as well as for Newton's integral. We will write a test in 2 weeks (5.4.2018).

Exercise sheet (in pdf): here
Homework (in pdf): here

6. session (29.3.2018) - Applications of definite integral: area determined by curve(s), length of a curve in R^n, volume of a solid of revolution, surface area of revolution. Integral criterion for convergence of series. Approximation of definitie integral via Taylor's expansion. We will write a test the next week (5.4.2018).

Exercise sheet (in pdf): here
No homework! (Prepare for the test.)

7. session (5.4.2018) - Test (everything we have done so far except for approximation of integral using Taylor's expansion and Henstock-Kurzweil integral)
No homework!

8. session (13.4.2018) - Solutions of the test. Multiple Riemann integral (only very briefly, to be continued).

Exercise sheet (in pdf): here
No homework! The second attempt to write the first test will be on Friday April 20 at 17:00 in S6.

9. session (19.4.2018) - Multiple Riemann integral, Fubini's theorem. A generalized n-dimensional volume of a set in R^n.

Exercise sheet is the same as in the previous session.
Homework (in pdf): here

10. session (26.4.2018) - An introduction to calculus of vector-valued mappings of several variables. All norms on R^n are equivalent. A notion of a limit, continuity and boundedness for mappings. Open, closed and bounded set.

Exercise sheet (in pdf): here
Homework (in pdf): here

11. session (3.5.2018) - Partial and directional derivatives, (total) derivative and the relations between them, necessary and sufficient conditions for existence. A matrix of derivative, algebra of derivatives. Tangent hyperplane. Mixed partial derivatives od second order.

Exercise sheet (in pdf): here
Homework (in pdf): here

12. session (10.5.2018) - Derivative of a composition of two mappings, chain rule. Linear approximation of given function using derivative, second order approximation using Taylor's series. Extrema of functions of several variables on an open set. Stationary points, non-zero directional derivative implies no extreme. Sufficient conditions using the Hessian matrix.
For now, ignore Problem 4 from the exercise sheet.

Exercise sheet (in pdf): here
Homework (in pdf): here

13. session (17.5.2018) - Conditional extrema of functions of several variables, Lagrange multipliers, parameterization. Implicit function theorem.
We will write a test the next week (24.5.2018) - topics: everything starting with integral of functions of several variables.

Exercise sheet (in pdf): here
Homework (in pdf): here

14. session (24.5.2018) - We wrote Test 2.