# Calculus II MM502

## Project - Reexam

The problem sheet is now published.

Do not mix up with the mandatory exercises problem sheet.

## Mandatory Exercises - Reexam

The problem sheet is now published.

Please note: The blackboard submission frame for the mandatory exercises starts later than announced right after the deadline for the project reexam, that is, Friday 15.6.2012 13.00, and ends Monday 18.6.2012 12.00. The reason is purely technical (assignments may not overlap). If you can submit only within the earlier part of the previously announced time frame, that is, from Friday 8.6.2012 12.00 to Friday 15.6.2012 13.00, you have to hand in by sending a .pdf (following the name conventions) by email to me.

## Course Log / Time Table

Here is a protocol of the course's foreseen and actual progression, including exercise assignments. All numerical citations refer to the 7th edition of Adams' and Essex' book mentioned below. For the places and dates of the exercise classes please consult the official course description or your personal schedule.

Week Events Content
5 (30.1.)* Tue 10-12 & Thu 12-14 U55

Chapter 12.

We have discussed Sections 12.1 and 12.2, introduced (high order) partial derivatives (Section 12.3. and 12.4.), and treated a number of properties of these including Schwarz' Theorem (Theorem 12.1). Please be aware of the different notations of the partial derivative of a real function f in two variables x and y (in this order) with respect to the first variable, x. These are, for example, ∂/∂x f, fx, f1 and ∂z/∂x. We will take the second alternative whenever possible. There is also a discussion of these conventions on page 683 in the book.

Exercises There are no exercise classes this week.
6 (6.2.) Tue 10-12 & Thu 12-14 U55

Chapter 12.

We have presented several versions of the chain rule (Section 12.5, you are welcome to design more), and introduced directional derivatives (Section 12.7 as an application). The Thursday's lecture has been given by Kristian Debrabant. He has worked towards more general chain rule (cf. page 707-709) and has treated Jacobi matrices plus a little matrix algebra.

Exercises Solve the following exercises, and feel free to volunteer for a presentation in the exercise classes. Section 12.1: 5,14,18,25. Section 12.2: 4,12,16. Section 12.3: 3,16,23. Section 12.4: 7,17. Of course you are welcome to consider further exercises.
7 (13.2.) Tue 10-12 & Thu 12-14 U55

Chapter 12/13.

We have presented the general chain rule from Section 12.6. We have introduced the concepts of local and global maxima and minima of real valued functions in more than one variable and have derived a 2-dimensional analogue to the 1-dimensional criterion to the second derivative certifying a local minimum (Section 13.1). We have generalized this in terms of Hesse matrices and, moreover, looked at constrained extremal value problems (restricted domains/boundary curves/Lagrange multipliers, see Sections 13.2 and 13.3).

Exercises

Solve the following exercises, and feel free to volunteer for a presentation in the exercise classes. Section 12.4: 18,26. Section 12.5: 1,4,13,16,17,18,22. Section 12.6: 17,18,20.

The following five mandatory exercises have to be handed in week 7. Section 12.3: 17,29. Section 12.5: 3,15,23. Please solve them and hand in your written solutions at your respective exercise group - your instructors will tell you when. They will be graded pass/fail. To get a pass you need 60 percent correctness, which can be realized by 100 percent correct solutions of three out of these five, or in any other sufficiently convincing way. Best is to go for 100 percent, as we intend not to accept a second try. There will be two further such sets to be handed in week 8 and week 10. In order to get an entire pass for the exercises (worth 1 ECTS), you have to get a pass in all three.

8 (20.2.) Tue 10-12 & Thu 12-14 U55

Chapter 13/14.

We have finished the discussion of Chapter 13, having covered content of Sections 13.1, 13.2, and 13.3. We have discussed Chapter 14, with Sections 14.1, 14.2, and parts of 14.3.

Exercises

Solve the following exercises, and feel free to volunteer for a presentation in the exercise classes. Section 12.7: 17,24. Section 13.1: 1,7,20,25. Section 13.2: 3,5,20. Section 13.3: 2,5,18.

The following five mandatory exercises have to be handed in week 8. Section 12.6: 19. Section 12.7: 21. Section 13.1: 3,19. Section 13.2: 4. (For procedural details see week 7.)

9 (27.2.) Wed 12-14 & Thu 12-14 U55

Chapter 14.

We have discussed the remaining part of Section 14.3 and have introduced polar coordinates. The integration formulae obtained is a special case of the following, more general statement: If D and R are open subsets of ℝd and g: R→D is a bijection and both g and its inverse mapping are continuously differientable then the function f: D→ℝ is integrable if and only if the function (fog)⋅|det Dg|: R→ℝ is integrable, and in that case ∫...∫D f dV = ∫...∫R (fog)⋅|det Dg| dV. Here Dg is the Jacobi matrix of g (do not mix it up with the domain D of f). As before when we have discussed the chain rule, one could R consider to be a part of some abstract coordinate space, whereas D is in most cases a part of the `normal', Euclidean d-space. The application is often as follows: f and D are given but it is difficult to calculate the integral of f. One then decides for a transformation g, together with the correct R in order to simplify integration. Examples are transformations to polar coordinates, spherical coordinates, and cylinder coordinates (Sections 14.4 to 14.6). It is a good exercise to apply the transformation formula for a proof that the volume of a body D in 3-space does not change if it is moved by some constant vector c.

Exercises

Solve the following exercises, and feel free to volunteer for a presentation in the exercise classes. Section 13.2: 12. Section 13.3: 13,20. Section 14.1: 8,9,10,14,20. Section 14.2: 1,9,20,28.

10 (5.3.) Tue 10-12 & Thu 12-14 U55

Chapter 15.

We have introduced vector fields and have discussed the representation of a vector field in polar coordinates (Section 15.1). Thursday we will consider line integrals and conservative vector fields.

Exercises

The following five mandatory exercises have to be handed in not necessarily in week 10 but in the beginning of week 11. Section 14.2: 4. Section 14.3: 6. Section 14.4: 27. Section 14.5: 10. Section 14.6: 1. (For procedural details see week 7.)

Solve the following exercises, and feel free to volunteer for a presentation in the exercise classes. Section 14.3: 13,22. Section 14.4: 15,20,29,34. Section 14.5: 1,6,17. Section 14.6: 2,5,10.

11 (12.3.) Tue 10-12

Chapter 15/9.

We have closed Chapter 15 by discussing two important characterizations of conservative vector fields in terms of curve integrals. - We have introduced convergency of sequences and series. We have looked at the harmonic series and the geometric series. We have introduced power series and, as a special case, Taylor series. This way, we found representations of sin x and ex as power series.

Exercises

Solve the following exercises, and feel free to volunteer for a presentation in the exercise classes. Section 15.1: 1,5,6,10,14,18. Section 15.2: 2,3,4. Section 15.3: 2,3. Section 15.4: 3.

12 (19.3.) Second Try Exercises

If you failed in one or more of the three sets of mandatory exercises, you are prompted to hand in solutions to the corresponding alternative sets directly to your teaching instructor, or in the drop box nearby the teaching instructor's offices at IMADA ("dueslag") not later than Tuesday, 27.3.2012, 12.00. (You will know by the end of week 12 whether you failed with the third assignment.) Here are the alternative sets:

Alternative set for those who failed in week 7: Section 12.3: 18,28. Section 12.4: 10. Section 12.5: 2,19.

Alternative set for those who failed in week 8: Section 12.7: 18. Section 13.1: 6,9. Section 13.2: 6. Section 13.3: 4.

Alternative set for those who failed in week 10: Section 14.2: 2. Section 14.3: 5. Section 14.4: 22. Section 14.5: 4. Section 14.6: 3.

It may be the case that you have to hand in more than one of these: If, for example, you failed in the first and third assignment but passed the second, then you have to hand in solutions to the alternative sets of first and to the third assignment, respectively (but not to the second).

13 (26.3.)* Project

The project problems are now published, see top of this page for viewing/downloading. You can submit/upload your solutions via blackboard in the time frame from Monday 26.3.2012 12.00 to Friday 30.3.2012 12.00. You have to solve them on your own, i.e. you are not allowed to do it together in groups and you will not receive any help of the teaching staff. To get a pass (worth 4 ECTS) you need 50 percent of correctness.

*: The date (dd.mm.) indicates monday of the respective week.

## Contents

Functions of several variables, partial derivatives, chain rule, Taylor's formula for functions of several variables, extreme values of functions in two variables. Double integrals, Riemann sums, calculation by iteration, polar coordinates, triple integrals. Vector fields, line integrals of functions and vector fields, existence conditions for potential functions of a vector field, parametric surfaces, surface integrals, areas on surfaces. Sequences, convergency of sequences and infinite series, absolute convergency, quotient series, convergency tests for infinite series, power series, differentiation and integration of power series.

## Literature

The course covers material of some chapters of the book

Calculus: A complete course by R. A. Adams and C. Essex, 7th edition, Pearson Canada, Toronto (2010).

Further books are in the MM502 slot in IMADA's library.

10.6.2012