Blog – Page 17 – Mathemafrica

Basics of Vector Representation

Not so long ago, I started reading some linear algebra, just out of interest. I was uncertain about whether or not I would understand the concepts, or if it would be worth it to go through all the trouble. I can now say that it was worth it. Honestly, it was the most frustrating, but at the same time rewarding, experience. I have come to realise that there are things that we often have to accept without knowing the beauty of the logic behind their existence, and the idea presented here is one of them. This post answers a simple question about vector notation.

You might have asked yourself at some point in your life (… or maybe you haven’t, but you should): Why is it “legal” to write a vector, $A$ , as ${ A=(a_{1},a_{2},\ldots,a_{n}) }$ , and why can we switch between different notations without finding trouble (for example, we can represent the vector in the form: ${A = \sum\ a_ia^*_i}$ ) ?…

By Siphelele Danisa| October 25th, 2017|Uncategorized|3 Comments

Permalink
Gallery
The Probability Lifesaver – by Steven J. Miller, a review

Book reviews, Reviews

The Probability Lifesaver – by Steven J. Miller, a review

NB. I was sent this book as a review copy. In addition, I lent this book to a student studying statistics, as I thought that it would be more interesting for them to let me know how much they get out of it. This is the review by Singalakha Menziwa, one of our extremely bright first year students.

http://i1.wp.com/press.princeton.edu/sites/default/files/styles/large/public/covers/9780691149547.png?resize=336%2C480&ssl=1

From Princeton University Press

All the tools you need to understand chance, the insight of statistics at base, and more complex levels. Statistics is not just about substituting into the correct formulae but requires understanding of what the numbers mean. Counting rules and Statistical inference were two of the topics I struggled with, especially the logic behind statistical inference, but this book provided great insight and explanations regarding these topics with a step by step procedure and gave enough interesting exercises. Miller’s goal when writing the book was to introduce students to the material through lots of accurately done, in depth worked examples and some fascinating coding for those who want to get more practical, to have a lot of conversations about not just why equations and theorems are true, but why they have the form they do.…

By Jonathan Shock| October 20th, 2017|Book reviews, Reviews|1 Comment

The Mathematics of Various Entertaining Subjects, Volume 2- edited by Jennifer Beineke and Jason Rosenhouse, a review

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

http://i2.wp.com/press.princeton.edu/sites/default/files/styles/large/public/covers/9780691171920_1.png?resize=316%2C480&ssl=1

From Princeton University Press

I tell my first year students that whether or not they will use their first year maths directly in the future, taking a course in mathematics is like going to a gym for your brain. Unless you are doing some good mental sweating, you are not benefiting from the study. It should be a subject in which you grow by gently (or not) applying more and more intellectual pressure to your thought patterns, and over time you will find that you can understand more complicated, or more abstract concepts than you ever thought that you could before. This translates into solving problems which may not have anything to do with maths, but require a similar pattern of logical juggling.

This book (The Mathematics of Various Entertaining Subjects, Volume 2) feels like Crossfit for the mathematics world. It’s a book filled with strength, endurance, flexibility and power exercises, each of which will stretch you in different ways.…

By Jonathan Shock| October 15th, 2017|Book reviews, Reviews, Uncategorized|3 Comments

Missing lectures after writing a challenging test – thoughts from a recent MAM1000W student

The following is by one of the current undergraduate tutors for MAM1000W, Nthabiseng Machethe, who has been providing me with extremely useful feedback and her thoughts on the course from the perspective of a recent student of it. She wrote this to me after a lot of students were disappointed with their marks from the last test.

——

This is based on my perspective as a student. I always plan to attend lectures, however as the work load increases and exhaustion kicks in, it is difficult to keep up with the plan.

It is easy to think of the things one may want (like excelling in MAM1000W) but realistically, it is hard to achieve them. In most instances, you find that students are studying a certain concept with a short term vision (passing a test), which can give one instant gratification but may not sustain in the long run (exam). Hence, one tends to quarrel about the time spent studying for the test not equating to the marks.…

By Jonathan Shock| October 1st, 2017|Courses, First year, MAM1000, Undergraduate|1 Comment

The Seduction of Curves – by Allan McRobie, a review

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

http://i0.wp.com/press.princeton.edu/sites/default/files/styles/large/public/covers/9780691175331.png?resize=384%2C480&ssl=1

From Princeton University Press

This is a beautiful book, it is a thought-provoking books and it is an informative book. It really is the intersection of mathematics, nature and art, and explores the three themes via the language of Catastrophe Theory, the theory by René Thom which aims to classify the possible folds in the solution space of natural systems and their two dimensional projections.

The book starts by introducing the alphabet of curves from the image of the human body, its curves and crevasses, its osculations and puckerings and from this alphabet it branches out to study the universe of catastrophes in the natural world.

As a fan/devotee/obsessive of atmospheric optics, the fold catastrophe which occurs in the production of the rainbow was bound to appeal to me. As Rene Descartes said in 1673:

A single ray of light has a pathetic repertoire, limited to bending and bouncing (into water, glass or air, and from mirrors).

…

By Jonathan Shock| October 1st, 2017|Book reviews, Reviews|1 Comment

2017 2/3rds numbers game

This is the fourth year that I’ve played the 2/3rds numbers game with my first year maths class. I’m always interested to see how, knowing previous results will affect this year’s results. Of course I am sure that a great deal depends on exactly how I explain the game, and so I imagine that this is the largest confounding factor in this ‘study’.
If you don’t know about the 2/3rds numbers game, take a look at the post here.

Here are the histograms from the last three years:

This year I told the class the mean results from the previous years to see if it would make a difference (as it seemed to last year). This year, the results are somewhat lower:

The winner was thus the person who got closest to 2/3 of 24.4=16.3. This year one person guessed 16, and one person guessed 16.2. Because everyone was asked to write down an integer, unfortunately I can’t claim that 16.2 is the winner, but they will get a second prize.…

By Jonathan Shock| September 22nd, 2017|Courses, First year, MAM1000, Undergraduate|0 Comments

Some more volume visualisations

Here is an animation which may help you imaging a shape which has a circular base, with parallel slices perpendicular to the base being equilateral triangles:

The same thing, where the slices are squares.

And here is the region in the (x,y) plane between $y=\sqrt{x}$ , the x-axis and the line x=1. rotated about the y-axis. Here a thin shell is drawn in the volume, then pulled out. Then it is replaced, then the volume is filled with shells, and each of them is pulled out of the volume vertically. This is to give you an idea about how to visualise the method of cylindrical shells.

…

By Jonathan Shock| September 14th, 2017|Courses, First year, MAM1000, Undergraduate|1 Comment

Permalink
Gallery
Guidelines for visualising and calculating volumes of revolution

Courses, First year, MAM1000, Uncategorized, Undergraduate

Guidelines for visualising and calculating volumes of revolution

I have seen some people try to blindly use the formulae for volumes of revolution by cylindrical cross-sections and by cylindrical shells, and I thought that I would write a guide as to how I would recommend tackling such problems, as generally just using the formulae will lead you down blind alleys.

I’ve created an example, with an animation, which I hope will help to master this technique.

So, here is a relatively fool-proof strategy:

Draw the region which you are going to have to rotate around some axis. This will generally be a matter of:
- Drawing the curves that you have been given
- Finding where they intersect
Draw the line about which you are supposed to rotate the region
Draw the reflection of the region about the line of rotation: This gives you a slice through the volume that will be formed
Now you have to decide which method to use:
- Take a slice through the volume perpendicular to the axis of rotation.

…

By Jonathan Shock| September 13th, 2017|Courses, First year, MAM1000, Uncategorized, Undergraduate|2 Comments

Using integration to calculate the volume of a solid with a known cross-sectional area.

Hi there again, I have not written a post in while, here goes my second post.

I would like us to discuss one of the important applications of integration. We have seen how integration can be used to solve the area problem, in this post we are going to see how we can use a similar idea to solve the volume problem. I suggest that we start by looking at the solids whose volume we know very well. You should be able to calculate the volumes of the cylinders below (yes, they are all cylinders.)

Cylinders are nice, we only need to multiply the cross-sectional area by the height/length to find the volume. This is because they have two identical flat ends and the same cross-section from one end to the other. Unfortunately, not all the solid figures that we come across everyday are cylinders. The figures below are not cylinders.…

By Mashudu Mokhithi| September 7th, 2017|Background, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|1 Comment

Introduction to trigonometric substitution

I have decided to start writing some posts here, and this is my first post. I would like to introduce trig substitution by presenting an example that you have seen before. Trig substitution is one of the techniques of integration, it’s like u substitution, except that you use a trig function only.

Let’s get into the example already!

$\int_{-1}^{1} \sqrt{1-x^2} dx$

If you equate the integrand to y (and get $x^2+y^2=1$ , $y\geq 0$ ), you should be able to see that this is the area of the upper half of a unit circle. The answer to this definite integral is therefore the area of the upper half of the unit circle (yes, the definite integral of f(x) from a to b gives you the net area between f(x) and the x-axis from x=a to x=b), is $\frac{\pi}{2}$ .

We relied on the geometrical interpretation of the integral to solve the definite integral, but can we also show this algebraically?…

By Mashudu Mokhithi| August 27th, 2017|English, First year, MAM1000|3 Comments

Previous 15 16 17 18 19 Next