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?

• 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 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:

$\frac{d}{dx} \left(\sin \left(x+\cos \left(\frac{1}{\tan (x)+\sqrt{x}+\exp\left(x^{3/2}\right)}\right)\right)\sqrt{x^{15}-\tan ^{-1}\left(\frac{x}{\sin(x)}\right)}\right)$

(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?

$\int \sin(x+\cos x)dx$

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.

 How clear is this post?