Abstract

Calculus and mechanics are often taught as separate subjects. It shouldn’t be like that. If you learn calculus without mechanics, it will be boring. If you learn physics without calculus, you won’t truly understand.

I think I may have found a way to solve this chicken and egg problem. It goes a little something like this:

  1. Chapter 1. You need [solving_equations,algebra,quadratic_equation] to do physics. That is all the prerequisites for first year Physics.
  2. Chapter 2. Physics laws are expressed as equations. If you know how to solve equations, then you know how to solve physics equations. In particular we will study the kinematics equations $x(t)$, $v(t)$, $a(t)$, which describe the motion of an object.
    • Start by defining kinematics concepts like time $t$, position $x(t)$, velocity $v(t)$, acceleration $a(t)$, initial position $x_i$, and initial velocity $v_i$. We can then state the UAM equations straight up: $a(t)=a$, $v(t)=v_i+at$, $x(t)=x_i+v_it+\frac{1}{2}at^2$. Example (free fall): An object on which only the force of gravity acts is said to be in free fall. Such objects experience a constant downwards acceleration of magnitude 9.81[$m/s^2$].  The classic examples are a ball thrown in the air. Using equations you can calculate the trajectory of the ball, and predict where it will land. Equations are cool and all, but where do these equations come from?In order to find out we must take a short excursion into calculus-land.
    • Calculus is the study of functions. We use calculus in to describe how quantities change over time (derivatives \(f'(t)\)) or to find the total amount of quantities that vary over time (integration \(F = \int f \;dt\)). Integrals sound fancy, but are really just a an area-under-the-curve calculation. Provide visual proofs for two important cases: if $f(t)=3$, then $F(t)=3t$. If $g(t)=t$, the integral is $G(t)=\frac{1}{2}t^2$.But why should anyone care about integrals?  What good is computing the area under a curve?
    • Integrals are the inverse operation of the derivative. In analogy with the inverse functions that we use when solving equations, the concept of an inverse operation is a useful concept in calculus: integrals are the inverse operations of derivatives.The kinematics equations of that describe the motion of objects can be derived from Newton’s law $F=ma$ and applying the integration operation twice. This is easy to see: we start by rewriting $F=ma(t)$ as $F=mx”(t)$, which means the force on an object is equal to the second derivative of $x(t)$. Recall that we just learned that integrals are the inverse operation of derivatives, so if we want to solve for $x(t)$ in $F=mx”(t)$ we can do it! First we divide both sides by m, in order to isolate the x expression on the right $F/m = x”(t)$. Then apply the integration operation twice in order to undo the two derivative operations.In particular, let us consider the case when $F=\textrm{const.}$ which implies that  then $a(t)=\textrm{const.}=a$. The equation we want to solve is $F/m = a=x”(t)$. Applying the integration operation to both sides of this equation we get $at+C=x'(t)$. By definition $x'(t)=v(t)$ so the constant $C$ can be identified as the initial velocity $v(0)=v_i$. Applying the integration operation to both sides a second time gives us $\frac{1}{2}at^2 + v_it + x_i = x(t)$. This is how the UAM equations are derived: $F=ma$ and 2x integration steps.
    • Main idea of this book: understand the math + physics is easier than just learning physics by memorizing the equations. With memorization, you would need to remember three equations of motion as separate entities. If you understand derivatives and integrals then you can remember just one equation $a(t)=a$, which is not much to remember since it is in the name UAM.
    • We have now seen kinematics in one dimension. But the real world is three dimensional so we need to learn about the math for dealing with objects in 3D.
  3. Chapter 3: Vectors.
  4. Chapter 4: Now that we know about vectors we can discuss more physics (mechanics).
    • Projectile motion. The position of the object is now a vector $\vec{r}(t)=[x(t),y(t)]$. There are two separate sets of kinematics equations. $x(t)$ is UVM (since no forces in the hz direction) while y(t) is UAM ($a_y=-9.81$ due to the force of gravity).
    • Introduce dynamics $\vec{F}=m\vec{a}$, i.e. forces cause acceleration. Forces. Force diagrams.
    • Momentum.
    • Energy.
    • Uniform circular motion.
    • Angular motion.
    • SHM.

 

The structure in Chapter 2 is the only new thing. After that, Chapter 4 is pretty much a standard course through the mechanics curriculum. So how is Chapter 2 so special, as to be worth blogging about at 2:44 in the morning?

I will tell you in point form, because it is kind of late indeed:

  • It connects nicely with the Precalculus chapter. You just learned how to solve equations for 50 pages, and now I am telling you that you can do physics with this equation solving skill. Yey! Math is useful.
  • Then we introduce a bit of basic kinematics concepts $x$, $v$, $a$ and the equations of motion. But then we say where did these equations come from (this is kind of a weak point?). To tell you, we must learn Calculus.
  • Bam—drette là—we do a mini course on calculus in 5 pages. Integrals with pictures and FTC. Sure it is complicated but the analogy to f and f-inverse should make it go through.
  • Then show derivation of $x(t)$ via int( int($F/m$) ). I can use the exact integral formula since students just saw those formulas as pictures 4 pages ago so they can’t say “i don’t know integrals”.
  • Having this early exposure to integrals also helps with the work and potential energy section later on in Chapter 4.
  • Basically the 5-page mini introduction to integral calculus is sufficient to do calculus-based mechanics course. The sin/cos derivative info and the chain rule required for deriving the SHM is presented in a just-in-time manner (i.e. in the last chapter).

But who am-I to say what is and isn’t a good way to teach. Only the students can tell me.

 

Update May 2021: I improved the formatting of the equations in the blog post and added links to the current version of the MATH & PHYS book preview.

December launch

I realize that it is a little late to launch an “educational product” one week before the finals, but it must be done. Besides, I think next week will be a sweet spot for exam prep.


Get an A

This will be the new marketing line. Six characters — if this were twitter, I would have 134 characters left ;).

Mechanics explained in seven pages

I finally managed to complete the Mechanics Tutorial which I have been working on for the past couple of weeks. I am pretty proud of the result. The material from the entire Mechanics class is covered in just seven pages. This should illustrate to my future clients that they don’t need to read a 500 page book to learn mechanics.

All we can do now is wait and see if people will get in touch with me.

September launch?

I have to add a question mark there. The book, well the book is not finished. I could get through the corrections in a couple of days, plus one day to finish up linear algebra. It is possible to do some kind of book launch. Let’s not do a half-assed launch though, and print a book with typos and mistakes in it. I have to do something with the resource that is September and the start of classes. I owe it to my company and to myself to test the hypothesis that mathematics and physics can be taught well in just 400 pages.

So what do we do with the next four days then?

  1. Flyer. Math lesson + physics lesson + pitch: “come preview the book for free on minireference.com”.
  2. Doku-wiki website. (must setup nginx rewrite rules + nice-up the homepage)

This is all I have time for. We are basically going to redo the launch from November 2011, but focus on the web aspect. By analyzing doku-wiki’s access logs I will be able to see whether there is engagement with the audience.

First day of classes

The book is not quite done yet. This sucks, as I wanted to get the book content ready (especially the new chapter on Physics) so that I can crete the flyers from that content. Linear algebra is not done either. And my ticket to Singapore is for the 13th of September. Hmm…

Makes me wonder what the best strategy for the coming week is. The main goal is to get the book into student’s hands this semester and let them read and learn from it. I want to judge the interest in the book (will it sell?), but also judge whether the teaching in it is effective (will they pass?). I suppose testing the quality of the content  is more important than testing how well it will sell. I mean print has been my  favourite monetization strategy since the beginning, but it is unlikely to be the winning strategy. Realistically speaking, I think I am much more likely to see sales on Kindle, iBooks and Kobo rather than in lulu-print. Plus I recently learned that they want a 20% cut from the book profits. WTF? I am pretty sure this 20% thing wasn’t there before.

So in the week that I have left, these are the priorities:

  1. Website (Preview PDF, give-me-your-email box)
  2. Flyers & advertisement. (depends on 1)
  3. Work on corrections & Chapter 2.

So tomorrow is going to be a Django day I guess 😉

Cory Doctorow on writing for a living

There is an interesting talk about who controls computers http://boingboing.net/2012/08/23/civilwar.html via HN.

At around 51:30, there is an off-topic question from the audience, which leads to an great answer:

Q: What can you say about making a living writing things. Will you advise it?

A: If you want to make a living writing things I would advise you to stop trying, because that is a bit like saying “I want to make a living buying lottery tickets”. Sounds like a If you don’t have a plan B for earning a living, you have the wrong career. Writing is a very very high-risk entrepreneurial venture that almost everyone who tries it fails at. Some people have succeeded using CC and some fraction without using CC, but they are rounding errors against all the people who try to earn a living with writing.

Ouch! I guess he is talking specifically about writing fiction. Textbooks are OK. I mean somebody has to teach people science. So I will keep going despite this advice.

▷▷   M A T H
P H Y S I C S

I am thinking of de-emphasizing  the “minireference” brand for the upcoming release. It just doesn’t tell that the product is an actual textbook-level coverage. Maybe we could have TUTORIALS in the title? “MATH and PHYSICS textbook”. Ok back to work now. 4 days left till launch.

Reading analytics for eBooks

We are coming into very interesting times for publishing books because the new ePub 3 format that is coming up will have the possibility to track reading behaviour. From the readers’ perspective, this isn’t necessarily good because it amounts to a massive invasion of privacy, but from the perspective of someone trying new ways to teach mathematics like me, this would be a major tool.

“So much of the time, it’s an editor and agent and publisher telling you, ‘This is what readers want,’ but this is hands-on reader data,” says Ms. Fenske

I love the above quote as it succinctly describes what is going on. We don’t need the publishers anymore. I mean yes, the average writer might still need some support along the way, say to write some js tracking code, and install piwik, but in general the hacking-author can go a long way on his/her own now.

But J. says I should delegate more, so let us look at what platforms are out there. For $20 per month, Hyptype will let you see what is going on in your book. I love the idea. This is perhaps the first good startup idea I have heard about in a long time. The value proposition is loud and clear. GA for eBooks. Bam! I will definitely have to get in touch with them.

What other platforms are out there? The WSJ article lists several other companies doing stuff in that market. First there is big A, with their kindle dev program which is a way to create interactive reading stuff on the kindle. The startup Coliloquy is using the kindle dev API to offer choose-your-own-adventure kind of books. Very cool, though a little Kindle specific. It seems that every one is getting  in on this stuff.

Like I said, interesting times are upon us.

Realism

I have been working every day on the linear algebra chapter in order to push them forward as much as possible before September 1st. I can’t say that it is going too well. The old cadence of one section per day seems to hold true regardless of how much effort I put into the process. I guess there is some  natural daily capacity limit for the human mind (at least mine) for the production of thoughtful written word.

This is not good in terms of having Minireference ready on time for the first day of classes. Let’s do a quick assessment right now:

  • Math: 90% done.
  • Physics and Easy Calculus digression: 20% done (will require 4 days)
  • Vectors: 95% done
  • Mechanics: 80% done
  • Calculus: 90% done
  • Electricity and Magnetism: 70% done (couple of days)
  • Waves and Optics: 20% done (~ a week of work)
  • Linear Algebra: 60% done (~ a week to go)

From the above estimates, it means that I am about 3 weeks away from a finished product. But wait! Recall that any time estimate ought to be doubled if you want to have a realistic time estimate for the real completion date. What does that leave us with? Aug 19th –2 montsh–> Oct 19th. Not to mention that I have to give a talk in Singapore in the meantime. Hmm… Not good. Shall we aim for a launch date mid-term?

Perhaps more important to the whole entreprise and this September’s market test is the support material and not the actual book content. These include:

  • Book cover design (Charlotte?)
  • minireferece.com website (design, pitch text, PDF sample)
  • One-page condensed content flyers for Calc & Mech
  • lulu.com (print) / ejunkie (pdf) purchase page tests
  • facebook page for Minireference Co. ?
  • Who can sell for me at McGill while I am in Bulgaria?
  • Concordia sales?

Wow! That is quite a bit of things to do before September 1st. It is kind of depressing to think about all that needs to be done. I have to go into another energy level if I am to get through this. Well it is not like I have never overcome obstacles before. I just have to stay together and minimize the down-days. Keep your eyes on the product. The world belongs to those that ship!

How to learn PHYSICS from the web

An interesting discussion over on HN was going down today about online resources for learning Physics. Some good links came from that. The calculus notes here: http://tutorial.math.lamar.edu/ and a very comprehensive collection of physics links: http://www.staff.science.uu.nl/~hooft101/theorist.html it is really worth taking a look at the resources linked to from there.

I will close on a personal recommendation from 1914. Calculus Made Easy by Silvanus P. Thompson http://www.gutenberg.org/files/33283/33283-pdf.pdf