Overzicht 2DL05 ( vervolg van 2DL04)

Transcriptie

1 Overzicht 2DL05 ( vervolg van 2DL04) R.R. van Hassel March 10,

2 Contents 1 Course description/material 4 2 Subjects and estimate of time 5 3 Paragraphs out of Adams 6 4 WIM and WID 6 5 2DL04, Calculus A Week WIM WID Week WIM WID Week WIM WID Week WIM WID Week WIM WID DL05, Calculus B Introduction Week WIM (290108) WIM Week WIM WID Week WIM WID Week WIM WID Week WIM WID Week

3 6.7.1 WIM Week WIM Week WIM Week WIM Week WIM Possible Parts Part Part Part Part Part Part Part Part Part Part Part Part Part Part Part Part Part Part Part Part Part Part Part Part

4 1 Course description/material Information is given in the English language. The lectures will be given in the Dutch language. The following book will be used: Calculus, A Complete Course, R.A. Adams, sixth edition, Pearson, ISBN I (René van Hassel) will start on the point, where Frans Martens ended with the course 2DL03. Every week two lectures, Wednesday 3 hours and Friday 2 hours and also two instructions, so every Week 10 hours mathematics. Before every lecture, I will try to give information about, what I have in mind (WIM) and about what the lecture has been done (WID). It are very busy weeks for me, so please forgive me, if all these promises are not always fulfilled. About the instruction groups I don t know every detail. But I hope, you can find out yourselves, when and where you can follow the instructions. It is a video-lecture, so after the lecture is given, you can repeat it on internet. May be you can also use this video-lectures, when you are busy with solving the exercises. The exam will go about the stuff, which is done during the lectures and nothing more than that. In 2DL05, Calculus B, we need the material of this course! 2DL05 will be given in Blok D and E. Nothing to do in Blok C, take Adams and keep busy with the material of this course! To bring everything to a good end, we will jump through Adams. If needed, I will put additional material on the internet. This material is most of the time not written in the English language. But it is mathematics, so most of the time it is clear, what is meant to do. If not, ask your colleagues, working together is very important. And the instructions are of importance, ask your instructor what to do, if you don t know what to do. But first, try yourself and let the instructor see, what you have done ( have tried to do) till that moment. May be just one little instruction and you can go on in solving the problem. The information given in this paper will change every week. Please, print only the essential parts of it! Last Thing: I keep the right to change from idea, about how, what and when! 4

5 2 Subjects and estimate of time There is asked to give lectures about complex numbers, differential equations (linear and separable ones): linear separable functions of more variables: graphs differentiable gradiënt extreme values integration over a two dimensional domain: also in polar coordinates To bring these questions to a good end, we will jump through Adams. Evenso I will put additional material on the internet, if needed. This material is not always written in the English language. But it is mathematics, so most of the time it is clear, what to do in certain situations. Most of the time you have to make calculations. Estimate for the time needed to do this material: complex numbers: 2 lectures differential equations: 3 lectures functions of more variables: 3 lectures integration over a two dimensional domain: 2 lectures It will go very fast. 5

6 3 Paragraphs out of Adams The sections out of Adams, that will be studied: 2DL04, Calculus A complex numbers: Appendix I and also the complex exponential function, not really mentioned in Adams. differential equations: Section: 17.1, 17.5, 17.6, ( may be: 7.9, 17.2) functions of more variables: Section: 12.1, 12.2, 12.3, 12.4, 12.5, 12.6, 12.7, ( may be 12.8) and Section: 13.1, 13.2 ( Extreme values). integration over a two dimensional domain: Section: 14.1, 14.2, 14.3, DL05, Calculus B 3-dim. integration: Section: 14.5, integration: spherical and cylindrical coordinates: Section 14.6, vectorfields, potential: Section 15.1, 15.2, line-integrals of vectorfields, simply connected domains: Section 15.4, oriented surfaces, flux across a surface: Section So you see, how we will jump. A detailed treatment of all these subjects can not be done in the time, which is given to us. But I hope to make clear, what to do in several situations, how to calculate several things and where and why, you have to be careful with your calculations. Reading yourselves in Adams is also of great importance. Understanding everything is difficult, try to understand the calculations you are doing. Try to understand the examples given in Adams. Take the time for the finer points, but on this moment they are not really of importance. Most of you know, how to work with a laptop, but have no idea, what happens inside that black box. But you know how to handle and to be careful in certain situations. Do the same with Calculus A. But I hope, for bove of us (students, myselve and instructors!) that some light is coming inside that black box, by lifting up the top of Adams. 4 WIM and WID My colleague, Bram van Asch, has made a studeerwijzer for the chemical engineering. I think, that the choices of the paragraphs out of Adams and 6

7 also exercises are good for us to take as a guide. But the order of doing some things will not always be the same. This because of the fact, that me is asked, to do some specific subjects in this blok. The word: week is changed in Part and in WIM and WID will be refered to these Part(s), see Possible Parts. ( click on the reference) Everything we skip, is already done, or will be done in the next course Calculus B, in blok D and E. 7

8 5 2DL04, Calculus A 5.1 Week WIM Click on the references and you come to the place, where you can read the information: Part 1 Part 2 See for Aanvulling Aanvulling. A short introduction to the complex numbers: ComplexNumbers. In blok A there were 4 groups, named Group 1,2,3 and 4. See on the internet side of Frans Martens (2DL03). Group 1, the instructor was Peter van Leeuwen, Peter wants to keep this group. We have, besides Peter van Leeuwen, also the instructors Peter van Liesdonk, Wim van Beers and Hennie Wilbrink. Proposal: Group 2 goes to Peter van Liesdonk, Group 3 to Wim van Beers and Group 4 is going to Hennie Wilbrink. We have no idea, how great these groups will be. Tomorrow ( ) we will see, some students are not coming anymore. Wednesday, there are not enough rooms for four groups! There are some popular times to give lectures and instructions, wednesday is one of them. Proposal: the two smallest groups in Aud. 2. The idea of Frans Martens is that Group 1 and Group 4 will be small. Group 1, because 24 students stop and Group 4 was already a small group. Friday: we have also Aud.1 and Aud. 2 ( just arranged), besides Aud. 10 and Aud. 15. Tomorrow , we start and we will see how everything is going WID Very much: calculations with complex numbers, polar coördinates of complex numbers, complex exponential function exp(iφ) with φ R (formula of Euler), and examples of how to calculate zeros of second order polynomials. Going further with these complexe numbers next time. If you need some information, written in the English language, look Complex Numbers. Much of the material, handled in this course, can be found on the website of Wikipedia. 8

9 See for instance for the linear differential equations, next week on the program, linear differential equations (Wikipedia). Not everything will be done, but you can find a lot of information on Wiki, which will also be given on the lectures. 5.2 Week WIM Chapter 17 is also of importance, see Sections: 17.1, 17.5, 17.6, ( may be: 7.9, 17.2). The following is meant with Aanvulling. Other material to read and may be also nice to make a short summary of the linear differential equations, that is the first part of Old Lecture Notes (postscript file!). Part 3 Part WID Friday, the last part of the differential equations will be done. ( Second order linear diff.eq. with constant coëffcients.) The formulas as staying in Adams (sixth ed.) on page 426 I have not done. I don t like to learn formulas. The methods of variation of constants is a nice way to solve these kind of diff.eq.. Adams have also found these formulas with the method of variation of constants. Look for instance or search with Google: variation of constants (Searching: first order differential equations variation of constants?) But look also to the video made of the lecture! Try to imitate me, step after step. 5.3 Week WIM Diff. equations, one lecture (May be, we start directly with Chapter 12 of Adams?) Start with Part WID Differential equations nothing to do, last week everything done. Started with Part 5. Also done: Part 7 9

10 5.4 Week WIM Not complete, but parts of it. Part 6, Part 8, May be, chapter 12 of Adams completely done, as far as needed WID? 5.5 Week WIM A start with integration, and using other coördinates (polar coördinates, see complex numbers!). Part 9 Part WID? 10

11 6 2DL05, Calculus B 6.1 Introduction Asked is to do the following: to use the book: Calculus, a complete approach, R.A. Adams, 9th edition, ISBN , to discuss: Three dimensional integration: spherical and cylindrical-coördinates. Vectorfields: conservative vectorfields, potential. Line- and surface-integrals. Integral theorems ( Theorems of Green, Gauss and Stokes). For certainty: the order of the subjects, which will be done, can be another one then written in the tabel above. On this moment is not known if every subject will be discussed. This information comes out of: Wiskundevakken in de doorstroomminor en in minor academische oriëntatie, Versie 4, 18 juni The course will be given in 2 blokken, of every 5 weeks and every week a lecture of 2 hours and 2 hours instruction. There is assumed that you are familar with the course 2DL04. Information will be given in the same way as done in the course 2DL04. During the last weeks there are not so much exercises. The chance exists there will be given extra exercises. 11

12 6.2 Week 1 The best to do, I think, is to repeat the last two lectures of 2DL04. Integration over a two dimensional domain in the xy-plane. During the last lecture of 2DL04, this is done very fast, and nothing is asked about it on the exam of 2DL04, I mean, the integration in the xyplane, using polar coördinates and naturally the song which belongs to it: dx dy = R dr dϕ. May be I will tell something about a general coördinate transformation, see within par of Adams. How to calculate the correction term, for some coördinate transformation you want to use on a certain moment? Sometimes the W IM changes, so read this paper regularly and don t print it out! WIM (290108) First will be spend some time to the cross product and determinants, see Sections 10.2 and 10.3 of Adams. See also the exercises : 2 [part a) till e) and extra: calculate cross product] and : see above. Keep in mind that the cross product is a vector with 3 coördinates. The cross product can not be calculated in R 2. But R 2 is as a part of R 3. And two-dimensional vectors can be made three-dimensional, by putting an extra 0 behind the two dimensional vector. This is an artifical intervention, not more than that, such that we can do, what we want to do, calculate the cross product. The change of variables, see 14.4, is also of importance. ( But I don t know? We have not so much time for the lecture.) Part WIM Everything what I had in mind, the WIM is done. I hope that the exercises are also done, not only at the instruction but also at home! Try to make them once more, without looking at the notes made during the instruction. 12

13 6.3 Week WIM We will look to triple integrals, so look to integrals about three dimensional regions. It is very important to have an idea about outline of such a region. My advice is to make a drawing of the regions, given in the exercises. Witout an idea of the shape it is difficult to determine the values of the boundaries of the areas you have to integrate. Part WID Almost everything I wanted is done, but at the beginning a little bit too long spoken about these integrals. Somewhere during this course I will may be explain how to calculate the correction term needed if you use other variables then the Euclidean variables (x,y,z). I mean the Jacobian for more then two variables. I have done it for two variables, almost the same has to be done when you have more then two variables. Now you will still get some melodies like: polar coördinates: dxdy = RdRdφ cylindrical coördinates: dxdydz = RdRdφdz spherical coördinates: dxdydz = R 2 sin(θ) drdφdθ. Be careful! I use x = R cos(φ) sin(θ), y = R sin(φ) sin(θ), z = R cos(θ), compare with the formulas in Adams!! Learn them! I have done also, but first I would not do it, because I know how they can be calculated. But on a certain moment it tooks too much time to do such a thing over and over again. 6.4 Week WIM Part WID Almost everything done what I wanted. But the calculation of the last integral is not copmpletely done, do it yourself! And I have also given on the black-board that 1 cos(x) dx = sin 2 (x) sin(x) + C. I used also that sin 3 (x) = (1 cos 2 (x)) sin(x), that function is easy to integrate, something with a cos(x) and cos 3 (x) is coming out of it, some 13

14 factors are not well, but you can calculate them. 2 π 0 5 π/6 π/6 2a a/ sin(θ) (R 2 sin 2 (θ)) (R 2 sin(θ)) dr dθ dφ Next time I will speak about vector fields and so on. 6.5 Week WIM Vector fields, differentiation and integration is an important tool to use. Train yourself in integration and differentation. I mean with differentiation, the partial derivatives! Differentiation of the function f(x, y, z) to the variables x, y or z, or other parameters. If you have some interest look at Wiki Vector Fields Looking on the internet I found also a nice alternative for Adams: Look here, examples etc.! Search at that last page Veteran s day number 11-17, that is what we are going to do. But be careful in using their theorems! Compare example 5, page 816 of Adams, with: Vector fields and then the Theorem: Testing for Conservativeness. Don t use that theorem, from the right to the left side! Part WID Everything I wanted, that is done. Searching on internet for something, I saw the following site java-applets vectorvelden. Click and wait a certain moment!! May be it is nice to play with it. The program was doing it at once on my computer without any problem, but on other computers, you never know. ( Java has to be installed.) You can look for equipotential lines, streamlines, etc. for certain vector fields. Those vector fields, you can chose yourself. 6.6 Week WIM My intention was to keep it short, but.... The idea is to do the line-integrals and using those conservative vector fields. May be I will take a small rectangle in the plane and will ask myself and I hope, you are doing with me: If the vector field shoud be a fluid. How much fluid streams through the boundaries of that rectangle? and also: How much fluid streams along the boundaries of that rectangle? For the students, who can not wait and want the answer(s), see Answers to the questions above 14

15 ( this is a.pdf-file). If you understand these papers, you have almost done this course. Only we will ask ourselves, can you do the same in 3-dimensions, if the answer is: Yes, then the question becomes: How? This weeks will be solved: Integration of a function over a line. May be you can think about: How to integrate functions over two dimensional surfaces?. How should you solve such a problem? Next, next week, you can look, if your solution is equal to the procedure I will explain on the blackboard. You have two weeks to think about: How can I easily calculate the weight of a thin membrane, knowing the mass-density. Also: How can I calculate the fluid going through the sides of a little cube and how to calculate going along the boundary of a membrane? Looking at the solutions of these problems, you will see that is in fact nothing else then b a f(x) dx = F (b) F (a). Some integral and the boundaries are in the formula above x = a and x = b. What are you doing? Searching some strange F (.) and you put the boundary in that F (.) and you get the answer. So you integrate over some interval (1 dim.), but you are only filling in the numbers a and b, in certain sense (0 dim.)! That is what we also want to do. Integrate about something with dimension n, but we try to get some other expression, where you have to do with dimension n 1, so reducing the dimension. Another nice paper is the divergence as rate of change of a small area ( a.pdf paper). Part 14 Another nice paper is the divergence as rate of change of a small area ( a.pdf paper) WID Line-integrals are done, also I have used the potential in calculating lineintegrals, if the vectorfield F is conservative on the whole domain of definition, be careful with that. ( I always think to some circulating fluid. Going around its centerpoint. There are two situations: or you have to do a lot of work to get around that center, or you have nothing to do!) If you have a conservative vector field, you have the situation that potential energy is turned over in kinetic energy and with this benefit of kinetic energy, you can come back to point you started. So the work done by the force is equal to the work, you have to do to get back at the point you started. ( 15

16 This in the situation you go from a higher potential level to a lower potential level and back.) This means that the total energy of a particle moving under the influence of conservative forces is conserved, in the sense that a loss of potential energy is converted to an equal quantity of kinetic energy or vice versa. See, out of boof of Lang Read also the remark on page 829 of Adams! Be careful in using that remark. Always control if your domain is simply connected! ( So without holes, sources, sinks or other strange points.) 6.7 Week 6 It is the best to follow the studyguide and we are at the point to integrate functions over surfaces. Think yourself what you should do if you have to calculate the weight of a membrane. I should cut into pieces! The massdensity of such a piece times its area, should give some approximation of its weight. But how to calculate the area of such a piece of that membrane? What did we do by the calculation of the line-integrals? Can we do a similar thing? Don t be afraid of all the formulas. See page 837 of Adams, figure What is the relation between ds and da? One thing is easily seen: the area of ds is greater then the area of da, why? WIM Part Week WIM Part Week WIM Part Week WIM Part 18 16

17 6.11 Week WIM We will see on that moment. May be exercises of old exams? It can also be the case that we have to do a lot of stuff? 17

18 7 Possible Parts 7.1 Part 1.1 Part Leerstof: Appendix I 1.2 Onderwerpen: rekenen met complexe getallen poolvoorstelling van complexe getallen 1.3 Zelfstudie: Appendix I: 1, 3, 5, 7, 9, 11, 13, 15, Instructieopgaven: Appendix I: oneven opgaven van 21 t/m Part 1.2 Part Leerstof: Appendix I (vervolg), Aanvulling Complexe e-macht en complexe polynomen 2.2 Onderwerpen: complexe e-macht hoofdstelling van de algebra wortels van complexe getallen 2.3 Zelfstudie: Aanvulling: 4: 1, 3ab 2.4 Instructieopgaven: Appendix I: 48, 50, 51, 53, 55 Aanvulling: 4: 2, 3cd, 4, 7a 7.3 Part Leerstof: 5.1, 5.2, 5.3, 5.4, Onderwerpen: 18

19 sigma notatie oppervlakte bepaalde integraal, onbepaalde integraal hoofdstelling van de integraalrekening 1.3 Zelfstudie: 5.1: 1, 3, 5, 7, 9 5.3: 1, 2 5.4: 4, 5 5.5: 1, 3, 7, Instructieopgaven: 5.1: 10, : 11, : 27, : 28, 33, 35, 39, Part Leerstof: 5.6, 6.1 (tot aan Reduction Formulas ), 6.2 (alleen examples 6, 7, 8 op blz. 326/327) 2.2 Onderwerpen: substitutiemethode partiële integratie 2.3 Zelfstudie: 5.6: 1, 3, 5 6.1: 1, 2, Instructieopgaven: 5.6: 6, 7, 12, 24, : 5, 6, 10, 13, : 30 19

20 7.5 Part Leerstof: 6.3, 6.5 (tot aan Improper Integrals of Type II ) 1.2 Onderwerpen: integralen van rationale functies oneigenlijke integralen 1.3 Zelfstudie: 6.3: 4, 5, 9, 13, : 1, 2, 3, Instructieopgaven: 6.3: 20, 22, : 10, 18, 20, Part 3.2 Part Leerstof: 2.10, Onderwerpen: eerste orde differentiaalvergelijking beginwaarde-probleem scheiding van variabelen eerste orde lineaire differentiaalvergelijking 2.3 Zelfstudie: 2.10: 27, : 1, 3, 4, 13, Instructieopgaven: 2.10: 37, : 6, 7, 11, 12, 15, 16 20

21 7.7 Part 4.1 Part Leerstof: 3.7, Aanvulling Tweede Orde Differentiaalvergelijkingen 1.2 Onderwerpen: tweede orde lineaire differentiaalvergelijking met constante coëfficiënten homogene, particuliere oplossing 1.3 Zelfstudie: 3.7: 1, 3, 5 Aanvulling: 1, Instructieopgaven: 3.7: 7, 8, 9, 15 Aanvulling: 3, 4, 5, Part 4.2 Part Leerstof: 12.1, Onderwerpen: functies van meer variabelen hoogtelijn, hoogtekaart partiële afgeleiden raakvlak 2.3 Zelfstudie: 12.1: 1, 2, 3, 19, : 2, 7, 17, 19 (in opgaven 17 en 19 alleen raakvlak) 2.4 Instructieopgaven: 12.1: 14, 15, : 4, 6, 13, 16, 20, 21, 23 (in opgaven 13, 16, 20 en 21 alleen raakvlak) 21

22 7.9 Part 5.1 Part Leerstof: 12.4 (tot aan The Laplace and Wave Equations ), 12.5 (tot aan Homogeneous Functions ) 1.2 Onderwerpen: hogere orde partiële afgeleiden kettingregels 1.3 Zelfstudie: 12.4: 1, 2, : 1, 6, Instructieopgaven: 12.4: 4, 5, : 9, 15, 17, Part 5.2 Part Leerstof: 12.6 (tot aan Definition 5 ), 12.7 ( Example 5 en Rates Perceived by a Moving Observer niet) 2.2 Onderwerpen: linearisatie gradiënt richtingsafgeleide 2.3 Zelfstudie: 12.6: 1, 2, : 1, 2, 3 (alleen (a) en (b)), Instructieopgaven: 12.6: 5, : 5, 6 (alleen (a) en (b)), 11, 13, 17 22

23 7.11 Part 6.1 Part Leerstof: 13.1 (tot aan Example 9; Theorem 3 alleen voor het geval n = 2, zie ook Remark op p. 712) 1.2 Onderwerpen: extremen van functies van twee variabelen kritieke punten classificeren van kritieke punten 1.3 Zelfstudie: 13.1: 1, 2, Opgaven voor de oefeningen: 13.1: 4, 5, 7, 19, Part Leerstof: 13.4 (tot aan Applications of the Least Squares Method to Integrals) 2.2 Onderwerpen: kleinste kwadraten methode lineaire regressie 2.3 Zelfstudie: 13.4: 1, 2, Opgaven voor de oefeningen: 13.4: 6, 7, 8, 9, Part Leerstof: 13.3 (tot aan Problems with More than One Constraint) 1.2 Onderwerpen: extremen van een functie onder nevenvoorwaarden multiplicatoren-methode van Lagrange 23

24 1.3 Zelfstudie: 13.3: 1, 2, 3b,c 1.4 Opgaven voor de oefeningen: 13.3: 4, 6, 8, 9, Part 7.2 Part Leerstof: 14.1, Onderwerpen: tweevoudige integralen veranderen van integratie-volgorde 2.3 Zelfstudie: 14.1: 13, : 1, 4, 11, Opgaven voor de oefeningen: 14.1: 15, : 9, 15, 16, 19, 21, Part 8.1 Part Leerstof: 14.4 (tot aan Change of Variables in Double Integrals 1.2 Onderwerp: dubbele integralen in poolcoördinaten 1.3 Zelfstudie: 14.4: 1, 2, 3, Opgaven voor de oefeningen: 14.4: 8, 9, 11, 12, 19 24

25 7.16 Part 8.2 Part Leerstof: Onderwerp: drievoudige integralen 2.3 Zelfstudie: 14.5: 1, 2, Opgaven voor de oefeningen: 14.5: 11, 14, 17, 20, Part 9.1 Part Leerstof: 14.6 (vanaf Cylindrical Coordinates) 1.2 Onderwerpen: drievoudige integralen in cilinder-coördinaten drievoudige integralen in bolcoördinaten 1.3 Zelfstudie: 14.6: 1, 2, 3, 5, 7, 9, 11, Opgaven voor de oefeningen: 14.6: 13, 14, 17, 20, 24, 25, 26, Part 9.2 Part Leerstof: 15.1 (tot aan Vector Fields in Polar Coordinates), 15.2 (tot aan Sources, Sinks and Dipoles) 2.2 Onderwerpen: vectorvelden veldlijnen van een vectorveld 25

26 conservatieve vectorvelden 2.3 Zelfstudie: 15.2: 1, 2, 3, Opgaven voor de oefeningen: 15.2: 7, 9, 10 (in 9 en 10 hoeven geen veldlijnen bepaald te worden) 7.19 Part 10.1 Part Leerstof: Onderwerpen: lijnintegralen van vectorvelden kringintegralen enkelvoudig samenhangende gebieden 1.3 Zelfstudie: 15.4: 1, 2, Opgaven voor de oefeningen: 15.4: 4, 5, 8, 9, 12, Part 10.2 Part Leerstof: Onderwerpen: georienteerd oppervlak normaal op een oppervlak flux van een vectorveld 2.3 Zelfstudie: 15.6: 1, Opgaven voor de oefeningen: 15.6: 4, 5, 7, 8, 10, 11, 12, 17 26

27 7.21 Part 11.1 Part Leerstof: Onderwerpen: divergence of a vectorfield Guass s theorem 2.3 Zelfstudie: 16.1: 1, Opgaven voor de oefeningen: 16.1: 3, 9, : 2, 9, 16.4: 5, 9, 11, Part 11.2 Part Leerstof: Onderwerpen: theorem of Green 2.3 Zelfstudie: 16.3: 2.4 Opgaven voor de oefeningen: 16.3: 1, 3, 5, Part 12.1 Part Leerstof: Onderwerpen: rotation of a vectorfield all kind of arithmetic rules 27

28 theorem of Stokes 2.3 Zelfstudie: 16.1, 16.2, Opgaven voor de oefeningen: 16.1: 3, 9, : 4, 5, 7, : 3, 5, 7, Part 12.2 Part Leerstof: Onderwerpen: theorem of Green 2.3 Zelfstudie: 16.3: 2.4 Opgaven voor de oefeningen: 16.3: 1, 3, 5, 7 28