## An excellent addition to your mathematics educationGo Top

If you want to broaden your mathematical education there is an excellent online course run by Keith Devlin aimed at first-year university students called "Introduction to Mathematical Thinking" on Coursera. Check out one of Keith Devlin's videos in the Calculus extras below...from the course synopsis:

The goal of the course is to help you develop a valuable mental ability — a powerful way of thinking that our ancestors have developed over three thousand years.

Mathematical thinking is not the same as doing mathematics — at least not as mathematics is typically presented in our school system. School maths typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school maths is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box — a valuable ability in today's world. This course helps develop that crucial way of thinking.

The course starts on the **3 ^{rd} February** and is completely free — highly recommended!

## 1S "Algebra"Go Top

As I explained in the lectures, Algebra is in inverted commas as the course is much more about Geometry, Logic and Counting Methods than Algebra! Somehow we are stuck with the historical labels...

### Extra resources

- I am a parallelogram...apply the parallelogram property (
*update:*& also remember the magnitude of the cross product of two vectors) with this catchy tune stuck in your head — just be careful not to sing it in the exam! - Unfolding a four-dimensional cube...in 1S we stop at the third dimension, this video shows some of the quirks when you try to visualise higher dimensions. I blabbed on a bit about cubism and the concept of a four-dimensional creature being able to view all the sides of a three-dimensional object at the same time (much like us and two-dimensional objects). Also we talked about the
*Manifeste Dimensioniste*: writing is one-dimensional art, painting two-dimensional, sculpture three-dimensional and higher dimensions being (at that time) unexplored. One attempt to rectify this deficit by Dali was with this painting...compare it to the unfolded four-dimensional cube of the above youtube link. Someone mentioned that this painting is in Kelvingrove Art Gallery...unfortunately it's not this one (it's in the Metropolitan Museum of Art, New York City) but the painting Christ of St. John of the Cross. - For a further higher dimensional fix check out this video of Carl Sagen...or this more modern version.
- If parallelograms weren't your thing, maybe triangles are?
- From the Creator's Project channel on YouTube, check out this wonderful illusion impossible without the vector mathematics we have been learning!
- As a bit of an aside, here's a short video explaining the mathematics in one episode of Futurama in which they present a group theory theorem that resolves the mess that the Professor has created (group theory is the branch of mathematics that examines symmetry in a broad sense).
- Episode 5 of the 1980s program
*Mechanical Universe*covering vectors and all tied together with an amusing tale about drunken sailors lost at sea on a boat called 'Irish Coffee'. We watched segments starting at 16m 19s (definition of dot and cross products plus animations...see also this animation of the cross product) and 18m 43s (expression in component form). - Some amusing addition...can you see where the error is in their sums?

## 1S CalculusGo Top

The slides below follow very closely the (2012-13) lectures notes on moodle, the main difference being the presentation. There are no `answers' on the slides, that is, there are neither solutions to examples nor proofs to theorems (as these are shown on the board during lectures). Therefore, you may find the slides useful in revising material — consulting the lecture notes on moodle when needed.

- Slides 1 (Covers Sections 1.1 - 1.5 of the lecture notes on moodle)
- Slides 2 (Covers Sections 1.6 - 1.7 of the lecture notes)
- Slides 3 (Covers Sections 1.8 - 1.11 of the lecture notes)
- Proof of the Fundamental Theorem of Calculus
- Slides 4 (Covers Sections 1.12 - 1.13 of the lecture notes)
- Slides 5 (Covers Sections 1.14 - 1.15 of the lecture notes)
- Slides 6 (Covers Sections 1.16 - 1.17 of the lecture notes)
- Slides 7 (Covers Chapter 2 of the lecture notes)
- Slides 8 (Covers Chapter 3 of the lecture notes)
- Slides 9 (Covers Chapter 4 of the lecture notes)
- Slides 10 (Covers Chapter 5 of the lecture notes)

### Extra Resources

An excellent introductory calculus course focusing on nurturing intuition.

A TedEd video on logarithms and their role in navigation (I think I pointed out the link between logs and the British Colonialists during the lecture).

An animated explanation of Zeno's Paradox...from TedEd.

A Wolfram Demonstration illustrating the approximation of sine, cosine and exponential functions by (truncated) MacLaurin Series.

An animated gif of simple harmonic motion [Wikipedia].

Another segment from Mechanical Universe, this time introducing Simple Harmonic Motion and the associated ODE.

An old and fun (cheesy but clear) video on Calculus.

~~The segment we watched in lectures starts at 16:41~~. I've uploaded a copy of the segment here as it's been removed from YouTube. It's part of a series of 52 half-hour videos on Physics produced in the early 1980s called Mechanical Universe. Look out for the moments in the clip~~(starting at 16:41)~~when, as the width of the rectangles approaches zero, the sum of these areas explodes into an integral.- A timeline of the main characters involved in the development of the calculus.
The video below is Keith Devlin giving a lecture (part of a series - check the others out too) on Calculus as a technology. Highly recommended.

- An interesting student essay on the history of calculus and computer algebra systems (e.g. Maple, Mathematica, Magma, etc.).
- An essay "Why Do We Study Calculus?" by Eric Schechter.

## HelpGo Top

Feel free to drop in to my office (Room **427**) for additional help outside of the tutorials and lectures. My office hours are:

**Wednesday: 1430 -- 1600,****Friday: 1200 -- 1330.**

If you can't make it to see me at these times then email me at: **Andrew{dot}Wilson{at}glasgow{dot}ac{dot}uk** and we can arrange a more suitable time.

This semester I'm also offering Skype sessions for those that find it hard to get to my office (or it's just being typical West Coast weather!). Add me to your Skype contacts, my username is "**Andrew.Wilson.427**". I'll be online during my office hours and sporadically at other times. I've got my webcam pointed at my whiteboard, so I'm hoping that it'll prove to be a good way to communicate mathematics...