I will attempt to post notes for the coming sixty lectures on a daily basis. You can either ask questions about the topics which you don’t understand here, or email me directly, or of course come and chat in my office when I’m around.

- These notes are for the second semester of MAM1000. They are neither complete nor exact and no responsibility is held for the accuracy within. Mistakes are undoubtedly included. That being said, I hope that they can be a useful resource in addition to the course textbook (Stewart) and additional online materials.
- I am always very grateful when people find mistakes in these notes. These may be in the form of spelling, grammar, calculational errors, typos in formulae, typesetting errors and anything else which doesn’t seem to make sense. If an explanation is not clear, please contact me and I will do my best to explain it in another way. If you find errors, please email me at jonathan.shock@uct.ac.za and I shall make amendments to the notes as soon as I can.
- Many of the examples and some of the explanations, especially in the initial part of these notes come from the brilliant lectures which can be found and watched for free here. I highly recommend watching them if anything here does not make sense.
- Take a look here for a series of podcasts on the subject of mathematics. They are a lovely historical addition to the practical side of what we are doing here.
- This coming week we will start with a review of various integration techniques. You should already have studied these but they take a while, and some serious practice to become natural. Believe me, that
**with a total of a good few hours of practice they will be a breeze and you will have a good handle on not only how to get the answer, but also WHY and HOW they work – this is the difference between knowledge and understanding.** - Email me with questions, ask for more exercises, come and chat with me after the lecture or email me and find a time to come and chat. I will always be happy to clarify, explain in more detail, correct MY mistakes when I DO make them: jonathan.shock@uct.ac.za

**Tips for thinking clearly**

Of course you need to spend time, just about every day, exercising your brain to make the contents of this course feel like a breeze, but in order for this to be effective, your mind should be in the clearest, healthiest state it can be in. Here are a few things which I consider to be vital for good brain function:

- Exercise – run, swim, lift weights, play football, do something! Sweat, get your heart beat up, get your blood flowing!
- Get away from the computer, see the sky, breath fresh air as often as you can, walk on the beach, cook a meal, talk to friends in the real world! If nothing else, just spend half an hour going for a walk if you can.
- Write. Writing helps you to organise your thoughts on paper – journal, blog, diary, random streams of consciousness.
- Sleep. Remove electronics as far as possible from the place you sleep. Getting a good amount of consistent sleep will make a big difference to your daily brain function.

**Some questions for you, to help me! – please write answers in the comments if some of these questions resonate with you.
**

- Do you ever feel that you get ‘in the zone’ when you’re studying maths? How do you most effectively get there?
- What are your biggest distractions?
- Do you find that you work best on your own or with other students?
- What resources do you feel you are lacking?
- What has been the hardest thing so far about varsity?
- What has been the hardest thing so far about MAM1000?
- What has been your greatest AH-HA moment in MAM1000?

**What is this course about?**

- Is it about learning how to integrate, solving sets of linear equations and differential equations, exploring the complex plane and understanding the world of combinatorics?
- Is it about testing your understanding of mathematical methods?
- Is it about stretching your brain in new and interesting ways?
- Is it about giving you a new dimension in creativity?
- …yep…

It is all of these things and more. My hope is that by the end of the course you will not only be able to solve the problems that you’ve encountered in the course, but you will have a new mental tool set by which to tackle problems you’ve never seen before.

**Integration recap and beyond – you have already spent a week or so studying integration and integration techniques. Here is a simple question for you to ponder.
**

*Why is integration so much harder than differentiation?*

By now you should be able to differentiate just about any function we throw at you, but integrating is another story. Differentiating seems to be a very logical process, usually involving either the product rule or the chain rule and iterating. Integration seems to be a bag of random tricks! Why is this?

It’s not that hard to calculate:

(Try it, you’re unlikely to have to calculate a more messy differential than this, but you can certainly do it if you set your mind to it). How about integrating a very “simple” function?

Even though this looks like a relatively simple function you won’t be able to do this, because it isn’t given by any simple combination of so-called elementary functions.

Have you ever thought about the fact that while integrating is essentially the inverse of differentiating, it is so much harder?

You already know of another example where this is true. Multiplication and Division. They are in some sense inverse processes, but one is much harder than the other. In this case you can always do it algorithmically, but division is generally much harder.

Here are some thoughts about why there is such a fundamental difference in the two operations when they are seemingly so similar, at first glance. Don’t worry too much about the details (the answers below go into some pretty complex ideas)- this is just a thought to ponder on while you are scratching your head over a seemingly trivial integral.

Do not worry if these explanations seem pretty deep – they are, this is just an exercise in pondering some interesting ideas about what you are studying.

Take a look at some nice thoughts on this from mathoverflow.

and From Roger Penrose in The Road to Reality: “*… there is a striking contrast between the operations of differentiation and integration, in this calculus, with regard to which is the “easy” one and which is the “difficult” one. When it is a matter of applying the operations to explicit formulae involving known functions, it is differentiation which is “easy” and integration “difficult”, and in many cases the latter may not be possible to carry out at all in an explicit way. On the other hand, when functions are not given in terms of formulae, but are provided in the form of tabulated lists of numerical data, then it is integration which is “easy” and differentiation “difficult”, and the latter may not, strictly speaking, be possible at all in the ordinary way. Numerical techniques are generally concerned with approximations, but there is also a close analogue of this aspect of things in the exact theory, and again it is integration which can be performed in circumstances where differentiation cannot.*”

In the next section we will review some techniques of integration and also discover a new method – integration by parts.

DanielJuly 20, 2015 at 12:10 pmI usually get in the zone when I need to show my work to someone besides myself (eg: exams or tests, working in tuts, studying in groups of strangers). It usually forces me to think and not make myself look like a fool. We had a math tutorial today with whiteboards and we worked in groups of about 3 or 4 and just worked through that weeks tutorial. I feel like I learnt more in that tutorial then I did in any other math tutorial. And the time flew.

By far the most distracting thing is my phone or the internet in general. I don’t study well with my friends.

Jonathan ShockJuly 20, 2015 at 1:55 pmYep – it’s amazing how much the internet can be a huge distraction!!

The whiteboard workshops really are very good, aren’t they

KennethJuly 21, 2015 at 9:34 amDo you ever feel that you get ‘in the zone’ when you’re studying maths? How do you most effectively get there?

I sit with problems until something clicks in my mind. Sometimes I can sit for hours attempting different approaches until I get it.

What are your biggest distractions?

Definitely my cellphone, or anything that isn’t maths related (because sometimes I feel like I’m trying to move a mountain when I do maths…)

Do you find that you work best on your own or with other students?

I work best in groups because often I won’t have a key concept or I need a different view on the question at hand.

What resources do you feel you are lacking?

I feel like I lack organisational skills or how to optimise study time. (Although I have more than enough resources I think I lack guidance).

What has been the hardest thing so far about varsity?

Consistently working hard while feeling like I’m drowning in work.

What has been the hardest thing so far about MAM1000?

Honestly, and this is going to sound melodramatic, it’s passing the tests.

What has been your greatest AH-HA moment in MAM1000?

MAM1000W requires intuition. High school maths didn’t. For instance, I came across a question on Newton’s Method, it required me to know what cos(1) and cos(2) were. I didn’t realise that cos(pi/2) is approx equal to cos(3/2) which is basically 0. Therefore cos(1) -> +ve and cos(4) -> -ve. It sounds silly but I didn’t know how to solve that without a calculator.

By the way, do the Wednesday tutorials also have the whiteboard styled workshops? I ask because last semester we had a rather free tut and it didn’t really promote much interaction between people.

Jonathan ShockJuly 21, 2015 at 9:56 amCheers Kenneth, this is great feedback!

J

mackkJuly 21, 2015 at 5:59 pmhalf of the time i get in the zone….normally when i grasp the material fast!

biggest distraction? SocialMedia,,,

work best on my own, but sometimes with fellow students

hardest thing about varsity is keeping up with the materialin all the courses, and meeting new people

MAM1k; preparing for test

aha moment? not yet

Jonathan ShockJuly 21, 2015 at 6:02 pmHey Mackk, yep, I think that social media is a serious problem. I would advise setting aside time when all access to social media is going to be turned off. Set yourself an hour at a time and then allow yourself a five minute splurge while taking a walk around if you want.

Make sure that when you’re working through the notes, you’re not just reading, but actually writing out all the steps – it makes a big difference!

Mathemafrica - UCT MAM1000 lecture notes subject listOctober 3, 2015 at 9:03 am[…] MAM1000 lecture notes part 1, part i – Preamble and introduction […]