The Euler spiral

Students of literature read Shakespeare; students of music listen to Bach. In mathematics such a tradition is, if not entirely absent, at least fairly uncommon. This book is meant to address that situation. Although it is not intended as a history of the calculus, I have come to regard it as a gallery of the calculus. - (William Dunham)

I am reading Dunham's book. Calculus Gallery. While reading I realized that I couldn't do the integral $$\int \cos{(x^2)}\ dx ,$$ anyway to make a long story short I discovered yet another beautiful curve ( most beautiful ever? maybe ): the Euler spiral, it is parametric plot of two Fresner integrals.

An Euler spiral has the property that its curvature at any point is proportional to the distance along the spiral, measured from the origin.

Exercise ( read: question ) How can we rotate the spiral?

Peek preview of Dunham's book.

Calculus video lectures at the Worldwide Center of Mathematics

The Worldwide Center of Mathematics publishes ( modern... ) calculus textbooks ( not a bad idea if you ask James Stewart ) and has a section with freely available video lectures on calculus and multi-variable calculus as well as a section with talks on the research level. There are also talks about research which are meant for undergraduates. Sort of to give you an idea of the field these researchers operate in.

Well, the Calculus lectures can be of help if you are on MST121, MS221 or MST209. There are a lot of options nowadays if you are looking for calculus videos.

Video lectures on multivariable calculus ( and more ) ...

I read a commentary on the future of universities.

Education and knowledge have become more accessible in the internet age. Students can do a lot of their work online or at home. But does that mean that universities will become obsolete? Of course not. - Books have to be written, exams have to be prepared and marked, research has to be done and published and students will always want some form of live interaction with a professor or tutor. Even if all lectures would be given in the form of video lectures then these lectures have to produced and as is the case with textbooks, redone every few years. Things have changed and will continue to change. Let's hope for the better.

Some video lectures, I recommended:
- [ NEW ] Multivariable calculus ( Berkeley )
- Video lectures number theory
- [News] - Video Lectures Algebraic Topology ( for Undergraduates )
- video lectures complex analysis

No set of video lectures is better than a well written text-book with lots of examples and exercises. Like this magnificent book on Complex Analysis for example ( I'll start M337 october next year ):

Infinite series ( and Euler's identity revisited )

The importance of infinite series and sequences can not be underestimated in my opinion, they literally pop up everywhere. To my surprise however, there isn't a single course at the Open University ( or any other university to my knowledge ) that deals exclusively with the topic of 'Infinite Series'. Instead the theory is stuffed away somewhere else, as if it is not really important. Like in M208 for example. - I have been searching for books on the subject and there -is- a ( recent ) book on the subject. It is called 'Real Infinite Series' by D. Bonar / M. Khoury published my TMAA. Tests not in M208 but in this book are for example Raabe's Test, Rummer's Test. Cauchy's Condensation Test, Abel's Test, and Dirichlet's Test as well as Bertrand's Test. It includes an entire chapter on the harmonic series with different divergent proofs. In the appendix there is an overview of the literature on infinite series.

P.S.
This video shows the proof of $e^{i\pi}+1=0$ using infinite series.

Watched MIT 18.02 - lecture 11

In 18.02 lecture 11 Prof. Denis Aroux talks about differentials and the Chain Rule. Two of the examples used to illustrate the main topic are of particularly interesting: a new proof for the differentiation of products and quotients, and the conversion between rectangular and polar coordinates.

A main result of this lecture is $$df = f_u \frac{du}{dx} + f_v \frac{dv}{dx},$$ where $f$ is a function of two variables $u,v$ which are both dependent on $x$, and $f_u, f_v$ are partial derivatives. The quotient rule can be derived from this result as follows. Let $g(x) = \frac{u}{v}$, with $u,v$ both dependent on $x$ :
$\begin{align*}

df &=f_u \frac{du}{dx}+f_v \frac{dv}{dx} \\
&=  \frac{1}{v}\frac{du}{dx}-\frac{u}{v^2}\frac{dv}{dx} \\
&= \frac{ v \frac{du}{dx}-u \frac{dv}{dx} }{v^2}
\end{align*} $
The last expression is the quotient rule for differentiation.

This lecture inspired me to some experimentation ( play ) with Mathematica's PolarPlot function. A polar coordinate is in fact a function of two variables $x,y$ which are both dependent on $r$ and $\theta$ with $x=r \cos(\theta)$, $y=r \sin(\theta)$. By applying the theory above one suddenly gets control over geometric objects like this:

Click to enlarge

Finally the concept of a gradient was mentioned which is merely a vector of partial derivatives. Gradients are the topic of lecture 12. I designed some problems and exercises ( and other experiments ) for functions in polar coordinates. I am delighted I feel more able in that regard.

More on MST209 in relation to MIT video lectures

MIT has a video lecture series on multivariable calculus: 18.02, which is in fact a prerequisite for 18.03. I looked further into the topics of MST209 and I now think that 18.02 is a much better preparation for MST209 than 18.03.

It's more or less like this:
MST209 = 18.02 + ( part of ) 18.03
MST209 + MST326 = 18.02 + 18.03

18.02 has lectures on
Lecture 15: Partial Differential Equations
Lecture 16: Double Integrals
Lecture 19: Vector Fields
Lecture 21: Gradient Fields
Lecture 25: Triple Integrals
Lecture 27: Vector Fields in 3D
Lecture 30: Line Integrals
which are topics in MST209.

18.02 Multivariable Calculus

Khan Academy

I was listening to one of those great talks by Lew Rockwell. He talked about some 530 million dollar school building in Los Angeles built to house 4,000 (!) high school students and which looked like a prison. Probably because it is a prison for these kids, he added. Lew mentioned that teaching like that is not at all like it should be with the ( technological ) possibilities we have today. Or if you like, with our ( read: the US's ) financial situation. He pointed out that it can be done differently. That it in fact -is- done differently by The Khan Academy, a not-for-profit 501(c)(3) with the mission of providing a world-class education to anyone, anywhere.

For me too? I thought, immediately. And yes, he recently added video lectures for a course in Differential Equations to the approximately 1800 video's already stored on their website. http://www.khanacademy.org/ Besides differential equations they also have an impressive set of calculus video's.

Since I already started the 18.03 lecture program ( competition breeds innovation ! ) I probably continue to do so but I'll be making comparisons.

Welcome

Thanks for visiting this blog. If you are studying math then you will find this text on "Tips for Success in Undergraduate Math Courses" interesting.

Philosophy of Learning

I found this on a webpage from the MIT OpenCourseWare Calculus course.

Philosophy of Learning

1. Amount learned is proportional to time put in.
2. Best way to learn is to figure out ideas yourself or teach them to someone else.
3. Second best is to do so with hints from others like your friends or us.
4. Third best is to get the ideas from reading; but pause in your reading to think about them.
5. Fourth best: unacceptable: don't get them at all.
6. The object of a lecture is not so much to inform you of important facts, but rather to stimulate you to try to learn about some concept.
7. The object of the course is to empower you to use the concepts of calculus in any context.

I would like to make some comments on these points.
1. Of course. I nevertheless disagree.
- Finding the right time to study is very important. Study when you feel energetic, hungry to learn, wanting to know and agressive enough to tackle any hard problem thrown at you.
- Better study one hour each day than seven hours every Saturday. The brain somehow needs backup time to process new concepts learned.
2. Very true! The trick is to find 'things' to find out for yourself which add the knowledge required for the course you are taking. About the 'teaching', I guess he means that you can verify if you have mastered a subject by explaining it in your own words.
3, 4, 5. Yeah...
6. Lectures. Personally, I don't like lectures. They cost you a LOT of time. You either understand what's being told at a lecture ( and that's because you already mastered the subject ) or you simply don't understand what's being told which makes it all a big time-waster if not worse.
7. Yeah...