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.

On the invariant measure in special relativity

I’m writing this for my string theory class. We are basing our lectures on Zwiebach – A First Course in String Theory, and starting off with special relativity. Not everybody in the class has a physics background (pure and applied mathematics students), and so there are likely to be questions which come up which show where I have to fill in some knowledge. We had a question about the invariant measure in special relativity (SR) and why there was a different sign in front of the time term compared with the space terms. I’ll do my best to explain here. Note that I am not explaining it in the precise chronological order of discoveries.

We start the picture off with relativity before SR – that is, Galilean Relativity. This simply states that the laws of motion are the same in all inertial (non-accelerating frames). That may sound straightaway like SR, but there’s a crucial ingredient missing which we will see in a bit.…

By | July 23rd, 2019|Uncategorized|0 Comments

MAM1019H fundamentals of mathematics

How clear is this post?
By | July 18th, 2019|MAM1019H|1 Comment

The South African Mathematics Olympiad problems

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By | July 14th, 2019|Uncategorized|1 Comment

The definite integral

I realise now, in all the excitement of the FTC that I hadn’t written a post about the definite integral…that’s shocking! ok, here we go…the plan for this post:

  • Look at our Riemann sums and think about taking a limit of them
  • Define the definite integral
  • Look at a couple of theorems about the definite integral
  • Do an example
  • Look at properties of definite integrals

That’s quite a lot, but we are more or less going to follow along with Stewart. Stewart just has a slightly different style to mine, so I recommend reading his for more detail, and mine for potentially a bit more intuition.

So, let’s begin…

We have seen in previous lectures/sections/semesters/lives that we can approximate the area under a curve by splitting it up into rectangular regions. Here are examples of splitting up one function into rectangles (and, in the last way trapezoids, but you don’t have to worry about this).…

By | July 12th, 2019|Courses, First year, MAM1000, Uncategorized, Undergraduate|3 Comments

The Fundamental Theory of Calculus part 2 (part ii)

OK, get ready for some Calculus-Fu!

We have now said that rather than taking pesky limits of Riemann sums to calculate areas under curves (ie. definite integrals), all we need is to find an antiderivative of the function that we are looking at.

As a reminder, to calculate the definite integral of a continuous function, we have:

 

\int_a^b f(x)dx=F(b)-F(a)

 

where F is any antiderivative of f

Remember that to calculate the area under the curve of f(x)=x^4 from, let’s say 2 to 5, we had to write:

 

\int_2^5 x^4 dx=\lim_{n\rightarrow \infty}\sum_{i=1}^n f(x_i)\Delta x=\lim_{n\rightarrow \infty} f\left(2+\frac{3i}{n}\right)\frac{3}{n}=\lim_{n\rightarrow\infty}\frac{3}{n}\left(2+\frac{3i}{n}\right)^4

 

And at that point we had barely even started because we still had to actually evaluate this sum, which is a hell of a calculation…then we have to calculate the limit. What a pain.

 

Now, we are told that all we have to do is to find any antiderivative of f(x)=x^4 and we are basically done.

 

Can we find a function which, when we take its derivative gives us x^4?…

By | July 11th, 2019|Courses, First year, MAM1000, Undergraduate|0 Comments

The Fundamental Theory of Calculus part 2 (part i)

OK, now we come onto the part of the FTC that you are going to use most. We are finally going to show the direct link between the definite integral and the antiderivative. I know that you’ve been holding your breaths until this moment. Get ready to breath a sign of relief:

The Fundamental Theorem of Calculus, Part 2 (also known as the Evaluation Theorem)

If f is continuous on [a,b] then

 

\int_a^b f(x) dx=F(b)-F(a)

 

where F is any antiderivative of f. Ie any function such that F'=f.

————-

 

This means that, very excitingly, now to calculate the area under the curve of a continuous function we no longer have to do any ghastly Riemann sums. We just have to find an antiderivative!

 

OK, let’s prove this one straight away.

 

We’ll define:

 

g(x)=\int_a^x f(t)dt

 

and we know from the FTC part 1 how to take derivatives of this. It’s just g'(x)=f(x). This says that g is an antiderivative of f.…

By | July 11th, 2019|Courses, First year, MAM1000, Uncategorized, Undergraduate|0 Comments

The Fundamental Theorem of Calculus part 1 (part iii)

So, we are now ready to prove the FTC part 1. We’re going to follow the proof in Stewart and add in some discussion as we go along to motivate what we are doing. What we are going to prove is that:

 

\frac{d}{dx} \int_a^x f(t) dt=f(x)

 

for x\in [a,b] when f is continuous on [a,b].

 

Proof:

 

we define g(x)=\int_a^x f(t)dt and we want to find the derivative of g. We will do this by using the fundamental definition of the derivative, so let’s look at calculating this function at x and x+h – ie. how much does it change when we change x by a little bit?

 

g(x+h)-g(x)=\int_a^{x+h}f(t) dt-\int_a^x f(t) dt

 

But remember that the definite integral is just the area, so this difference is the area between a and x+h minus the area between a and x. Which is just the area between x and x+h. Using the properties of integrals, we can write this formally as:

 

g(x+h)-g(x)=\int_a^{x+h}f(t) dt-\int_a^x f(t) dt=\left(\int_a^{x}f(t)+\int_x^{x+h}f(t)\right)-\int_a^{x}f(t)=\int_x^{x+h}f(t)dt

 

and we can write, for h\ne 0:

 

\frac{g(x+h)-g(x)}{h}=\frac{1}{h}\int_x^{x+h}f(t)dt

 

Restated, we can think of this as the area between x and x+h divided by h.…

By | July 11th, 2019|Courses, First year, MAM1000, Uncategorized, Undergraduate|0 Comments

The Fundamental Theorem of Calculus part 1 (part ii)

OK, so up to now we can’t actually use the FTC (Fundamental Theorem of Calculus) to calculate any areas. That will come from the FTC part 2.

For now, let’s take some examples and see what the FTC is saying. I’ll restate it here:

The Fundamental Theorem of Calculus, part 1

If f is continuous on [a,b] then the function g defined by:

g(x)=\int_a^x f(t) dt,     for a\le x\le b

is continuous on [a,b] and differentiable on (a,b), and g'(x)=f(x).

——

Let’s look at some examples. We’re going to take an example that we can calculate using a Riemann sum. Let’s choose f(x)=x^2.

If we integrate this from 0 to some point x – ie. calculate the area under the curve, we get:

 

\int_0^x t^2 dt=\frac{x^3}{3}.

 

Make sure that you can indeed get this by calculating the Riemann sum.

So, what does the FTC part 1 tell us? It says that if we take the derivative of this area, with respect to the upper limit, x, then we get back f(x).…

By | July 11th, 2019|Courses, First year, MAM1000, Uncategorized, Undergraduate|0 Comments

Prime Suspects – The anatomy of integers and permutations, by Andrew Granville and Jennifer Granville, illustrated by Robert Lewis – a review

NB I was sent this book as a review copy.

What a spectacular book! I am rather blown away by it. This is a graphic novel written about two bodies discovered by cops in an American city some time around the present day, and the forensic investigation which goes into solving the case, and somehow the authors have managed to make the whole book about number theory and combinatorics.

I have to admit that when I started reading the book I was worried that it was going to have the all-too-common flaw of starting off very simple and then suddenly getting way too complicated for the average reader, but they have managed to somehow avoid that remarkably well.

It is however a book that should be read with pen and paper, or preferably computer by one’s side. As I read through and mathematical claims were made, about prime factors of the integers and about cycle groups of permutations, I coded up each one to see if I was following along, and I would recommend this to be a good way to really follow the book.…

By | July 9th, 2019|Book reviews, Reviews|1 Comment

The Fundamental Theorem of Calculus, part 1 (part i)

We’ve seen some intriguing things in this course so far, and we’ve developed some clever tricks, from how to find the gradient of just about any function we can throw at you, to proving statements to be true for an infinite number of cases.

To some extent, this is what we have looked at so far (at least in terms of calculus, and building up to calculus):

Screen Shot 2019-07-09 at 16.06.58

However, we’re about to see some magic. We’re about to see the most important thing yet on this course, and indeed one of the most important moments in all of mathematical history.

We are going to see…actually, we are going to prove, that there is a relationship between rates of change and the area under a graph. This doesn’t sound that amazing, but its consequences have essentially allowed for the development of much of modern mathematics over the last 350 years.

The link that we are going to prove will allow us to find the area under graphs of functions for which taking the Riemann sum would be really hard.…

By | July 9th, 2019|Courses, First year, MAM1000, Uncategorized, Undergraduate|1 Comment