No BS math and physics v5.1 update

Over the last years, several readers uncovered mistakes in the No bullshit guide to math & physics, which I immediately fixed in the source. The errors were mostly minor, so they didn’t warrant a new edition, but once I reached a threshold of six errata, I decided it’s time to release a v5.1 update. With this bugfix update, I took the time to make some other minor improvements described below.

Errata

Most of the mistakes in v5 of the book were in the exercises and problems. I’m happy to announce there are no major conceptual problems, or mistakes in any of the core equations. Here are the mistakes in v5.0 (Aug 2014) of the book:

  • P1.41: Both calculations should use the radius instead of the diameter. The change in height is $h_2 – h_1 = 5.47 – 4.41 = 1.06$ cm.
  • P1.44: Answer should be 8.42m = $4\sin 40 + \frac{1}{4}(2\pi(0.5)) + 4\cos 40 + 2$.
  • P1.47: Answer should be 180 degrees – 40 degrees = 140 degrees.
  • P1.51: Question describes the water tank as $12 \times 6 \times 3$, but solution uses $12 \times 6 \times 5$. The question was changed to match the existing solution: the tank now has height $5$[m].
  • P2.9 part (3) $v_f$ should be 6 [m/s], not 10 [m/s].
  • P2.10, part 4. Distance should be 13[m] not 14[m].
  • page 202: Revolution of the Earth example: $v_t$ should be 328.32 m/s not 464.32 m/s, giving a final answer of 1181.95km/h not 1671.56 km/h.
  • 5.5 Limit formulas, page 264 at the bottom: removed formulas $\lim_{x\rightarrow0}\frac{\ln(x+a)}{x}=a$ and $\lim_{x\rightarrow0}\left(a^{1/x}-1\right)=\ln(a)$. First formula is wrong, second is not useful.
  • page 286: “Consider the point $P=(x_P,y_P)$ that lies on the circle $x^2+y^2=R$.” should be “Consider the point $P=(x_P,y_P)$ that lies on the circle $x^2+y^2=R^2$.”
  • page 401: the correct conversion formula for inch is 1[in] = 2.54[cm] and not 1[T] = 1000[kg].

 

New math exercises

Several readers complained about the math fundamentals chapter being too “rough” for complete beginners. To fix this, I did a critical rereading of the material and corrected some mistakes of continuity (there was crazy rough patch in Section 1.2 where I suddenly jump into negative and fractional exponents that—not surprisingly—some readers found confusing). I also interspersed exercises throughout Chapter 1 so readers can now test their understanding as they progress through the chapter.

 

Index

Thanks to the suggestion in a comment on amazon, I decided it the book should have an index. It turned out creating an index was a very time-consuming task, but it was very totally worth it since it allowed me to standardize certain terminology (e.g. function range vs function image) and weed out certain inconsistencies.  The exercise of cataloguing every concept used in the book will help with STRUCTURE project too.

 

New proof of the chain rule for derivatives

A mathematician friend of mine pointed out a mistake in the proof of the chain rule for derivatives in the book. The hand-wavy argument that I had improvised had possible divide by zero error: the quantity $\Delta = g(x+\delta) – g(x)$ can be zero, so it’s not OK to use it in expressions where it appears in the denominator. In other words, bullshit had creeped into the book!  We can’t have any of that in a book which claims to be bullshit free! The new proof of the chain rule for derivatives is longer and more technical, but at least it’s not wrong. Hopefully I will not alienate my readers too much by having such a technical argument in the middle of the calculus chapter, but there really wasn’t any way to make the proof simpler or more intuitive. It’s just a technical argument.

 

More calculus problems

Upon critical review, I found the section on surface and volumes of revolution was a little short. Sure all the formulas are introduced, but it would have been a stretch to assume the average reader will be able to pick up the concepts from these few pages. To remedy this, I added a few more pictures, beefed up the discussion, and added some problems. I also came up with a “append only” policy for adding problems to the book—this way the numbers won’t change between versions so if  a prof assigns P5.33 to their students, it will be the same problem in v5, v6, or whenever. The new problems added test the student’s ability to apply the volume of revolution formulas and also the infinite Riemann sum formula.

 

The diff

As with previous updates, I’ve generated a red-blue diff of all the changes between v5.0 (Aug 2014) and v5.1 (July 2016). You can check out the diff to see the details of all that changed.

 

Right now I’m waiting for the green light from my editor about the changes and the next step will be to push the updated PDFs to all distribution channels. I’ll also send out an update email to all readers.

The aims of education according to Alfred North Whitehead

Yesterday I read the fascinating essay titled The Aims of Education by Alfred North Whitehead (1861-1947). It was written 100 years ago, but every line of it rings true in the modern context. Below I’ve extracted the best quotes from the essay and added some personal comments.

The OP gives a detailed blueprint of how to structure formal education, making a distinction between “general education” (primary school and middle school) and “specialized training” (high school and college). The essay discusses learner psychology, learner user experience, curriculum customization, student assessment, and even proposes a new structure for the educational system. The essay is so full of good stuff that nearly all of it is worth quoting.

 

The danger of inert ideas

In training a child to activity of thought, above all things we must beware of what I will call “inert ideas”—that is to say, ideas that are merely received into the mind without being utilised, or tested, or thrown into fresh combinations.

In the history of education, the most striking phenomenon is that schools of learning, which at one epoch are alive with a ferment of genius, in a succeeding generation exhibit merely pedantry and routine. The reason is, that they are overladen with inert ideas. Education with inert ideas is not only useless: it is, above all things, harmful. […] Except at rare intervals of intellectual ferment, education in the past has been radically infected with inert ideas. That is the reason why uneducated clever women, who have seen much of the world, are in middle life so much the most cultured part of the community. They have been saved from this horrible burden of inert ideas. Every intellectual revolution which has ever stirred humanity into greatness has been a passionate protest against inert ideas. Then, alas, with pathetic ignorance of human psychology, it has proceeded by some educational scheme to bind humanity afresh with inert ideas of its own fashioning.

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