About Jonathan Shock

I'm a lecturer at the University of Cape Town in the department of Mathematics and Applied Mathematics. I teach mathematics both at undergraduate and at honours levels and my research interests lie in the intersection of applied mathematics and many other areas of science, from biology and neuroscience to fundamental particle physics and psychology.

Integrals with sec and tan when the power of tan is odd

We went through an example in class today which was

 

\int tan^6\theta \sec^4\theta d\theta

 

In this case we took out two powers of sec and then converted all the other \sec into $latex\ tan$, which left a function of tan times sec^2\theta d\theta. We wanted to do this because the derivative of \tan is \sec^2 and so we can do a simple substitution. If we have an odd power of \tan, we can employ a different trick. Let’s look at:

 

I=\int \tan^5\theta\sec^7\theta d\theta.

 

Here, sec is an odd power and so we can’t employ the same trick as before. Now we want to convert everything to a function of \sec and have only a factor which is the derivative of \sec left over. The derivative of \sec is \sec\tan, so let’s try and take this out:

 

I=\int \tan^5\theta\sec^7\theta d\theta=\int \tan^4\theta\sec^6\theta (\sec\theta\tan\theta)d\theta.

 

Now convert the \tan into \sec by \tan^2\theta=\sec^2\theta-1:

 

I=\int (\sec^2\theta-1)^2\sec^6\theta (\sec\theta\tan\theta)d\theta=\int (\sec^{10}\theta-2\sec^8\theta+\sec^6\theta) (\sec\theta\tan\theta)d\theta

 

where here we have just expanded out the bracket and multiplied everything out.…

Fundamental theorem of calculus example

We did an example today in class which I wanted to go through again here. The question was to calculate

 

\frac{d}{dx}\int_a^{x^4}\sec t dt

 

We spot the pattern immediately that it’s an FTC part 1 type question, but it’s not quite there yet. In the FTC part 1, the upper limit of the integral is just x, and not x^4. A question that we would be able to answer is:

 

\frac{d}{dx}\int_a^{x}\sec t dt

 

This would just be \sec x. Or, of course, we can show that in exactly the same way:

 

\frac{d}{du}\int_a^{u}\sec t dt=\sec u

 

That’s just changing the names of the variables, which is fine, right? But that’s not quite the question. So, how can we convert from x^4 to u? Well, how about a substitution? How about letting x^4=u and seeing what happens. This is actually just a chain rule. It’s like if I asked you to calculate:

 

\frac{d}{dx} g(x^4).

 

You would just say: Let x^4=u and then we have:

 

\frac{d}{dx} g(x^4)=\frac{du}{dx}\frac{d}{du}g(u)=4x^3 g'(u).…

Calculus for the ambitious, by Tom Korner, a review, by Henri Laurie

Amazon link

This is a lovely book: strong emphasis on ideas; a lively sense of humour; a sure logical touch; historical detail that is accurate, relevant, yet quirky (takes some doing!). What’s not to like?

Well, there’s this: it is not easy to decide whether to recommend the book to anybody who doesn’t already know calculus. I’ll return to that. Let me start by describing why this is such a good book.

Firstly, the light touch and the clarity, which together make it wonderfully accessible. Fans of Tom Korner, including yours truly, will be happy to hear that it it as good as his “The Pleasures of Counting” and “Fourier Analysis”, two of the best books on maths ever. Like them, it discusses applications, social context and history but always in a way that supports the maths, which remains the main focus.

Secondly, the balance between rigour and intuition is superbly judged and maintained.…

By | June 20th, 2018|Book reviews, Reviews|0 Comments

The Recamán sequence

In case you have watched the following video about the Recamán sequence.

and want to play around with it in Mathematica. Here is my code for doing so:

nums = {0};

For[i = 1, i < 66, i++,
If[nums[[-1]] – i > 0 && Position[nums, nums[[-1]] – i] === {}, nums = Append[nums, nums[[-1]] – i],
nums = Append[nums, nums[[-1]] + i]]
]

{{#[[1]], 0}, #[[2]]} & /@ Partition[Riffle[Mean[#] & /@ Partition[Riffle[nums, nums[[2 ;;]]], 2],
Abs[Differences[nums]]/2], 2];

Show[Show[
Table[Graphics[Circle[%[[i, 1]], %[[i, 2]], {(i) \[Pi], (i + 1) \[Pi]}]], {i, Length[%]}], ImageSize -> 1000], Plot[0, {x, 0, 91}],
Axes -> True]

(You may have to copy this by hand rather than copy/paste.)

This produces the following rather beautiful graphic (and answers the question posed in the video):

RecamanEvidence away my dear Watson…evidence away.

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By | June 14th, 2018|Uncategorized|0 Comments

Music by the Numbers, From Pythagoras to Schoenberg – By Eli Maor, a review

NB. I was sent this book as a review copy.

Music by the numbers leads us on a journey, as stated in the title, from Pythagoras to Schoenberg. In many ways the endpoint is stated early on, giving us clues that a revolution in mathematical thinking about musical scales will be encountered in the early twentieth century. Indeed the journey through musical practice, mathematics, physics and the biology of hearing is woven rather beautifully together, giving the account of our step by step explorations of tonal systems and their links to the physics of vibration. The development of calculus and the triumph of Fourier take as from the somewhat numerological and empiric realms of musical experimentation to the age of a true understanding of timbre – the way different instruments express harmonics and their overtones in different admixtures. A lot of emphasis is placed on the development of scales based on subtly different frequency ratios, which were developed over the years (particularly within European music, non-European music being given only very brief comment) to balance the physical, mathematical and aesthetic qualities of the various possible tunings of instruments.…

By | June 2nd, 2018|Book reviews, Reviews|1 Comment

Kovalevskaia research grants for female mathematicians in Southern African region

KovaleskaiaAward2018

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By | May 28th, 2018|Advertising|0 Comments

What can be computed? A practical guide to the theory of computation – by John MacCormick, a review

NB. I was sent this book as a review copy.

It’s not often that a textbook comes along that is compelling enough that you want to read it from cover to cover. It’s also not often that the seed of inspiration of a textbook is quoted as being Douglas Hofstadter’s Pulitzer prize-winning book Godel, Escher Bach. However, in the case of “What can be computed”, both of these things are true.

I am not a computer scientist, but I have spent some time thinking about computability, Turing machines, automata, regular expressions and the like, but to read this book you don’t even need to have dipped your toes into such waters. This is a textbook of truly outstanding clarity, which feels much more like a popular science book in terms of the journey that it takes you on. If it weren’t for the fact that it is a rigorous guide to the theory of computability and computational complexity, complete with a lot of well thought through exercises, formal definitions and huge numbers of examples, you might be fooled by the easy-reading nature of it into thinking that this book couldn’t take you that far.…

By | May 27th, 2018|Book reviews, Reviews|1 Comment

On Gravity, a brief tour of a weighty subject – By Tony Zee, a review

NB. I was sent this book as a review copy.

In the era when our eyes are being opened to the Universe in the gravitational spectrum via the recent gravitational wave observations, this book is exactly what is needed to communicate to the general public the beauty and depth of Einstein’s theory of gravity, as well as the interplay between gravity and quantum mechanics which takes place at the event horizon of a black hole.

Starting with the observations of merging black holes black in 2016, Tony Zee takes the reader on a clear and swift journey through the ideas of the incredible weakness of gravity, the basics of field theory, relativity, curved space-times, quantum weirdness, black holes and Hawking radiation, back, full circle to the consequences of General Relativity including the existence of gravitational waves and the detectors now observing them and those which will give us a far clearer picture of the gravitational universe in the near future.…

By | May 27th, 2018|Book reviews, Reviews|1 Comment

Advice for MAM1000W students from former MAM1000W students – part 5

While I resisted Mam1000W every single day, I even complained about how it isn’t useful to myself. Little did I know when it all finally clicked towards the end that even though I wasn’t going to be using math in my life directly, the methodology of thinking and applying helps me to this day.

Surviving Mam1000W isn’t really a miraculous thing. While everyone tends to make it seem like it’s impossible, it is challenging (Not hard) and I said that because I have seen first-hand that practice makes it better each time. Getting to know the principles by actually doing the tuts which is the most important element of the course in my opinion will make sure that even though you feel like you aren’t learning anything when the time comes (usually 2nd semester) it will all click on how you actually are linking the information together.

Another important aspect is playing the numbers game.

By | May 14th, 2018|Courses, First year, MAM1000, Undergraduate|1 Comment

Advice for MAM1000W students from former MAM1000W students – part 4

In high school, as I believe was the case for many students, there wasn’t much incentive to work very hard regularly on math – concepts were easy to grasp first hand in class. That’s the kind of attitude I brought towards MAM1000W last year (2017). Unfortunately things didn’t turn out as anticipated…by as early as April I had already started playing “catch-up” for I hadn’t been putting in any practice on the staff done in class. Tests were nightmares. With every course demanding its share of my attention, I found myself crying the other day alone in my room, asking myself, “what went wrong?”. Well, the answer was pretty simple – EVERYTHING.

Eventually, I figured a way to potentially get back on my feet – I became a very good friend of my WebAssign voluntary quizzes. In combination with past papers (WHICH I HIGHLY RECOMMEND) and the daily uploaded ‘practice questions and solutions’, I was able to gain back some bit of confidence.

By | May 9th, 2018|Uncategorized|0 Comments