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Author: Silvanus P. Thompson and Martin Gardner
Amazon info

Who could resist a book co-authored by Martin Gardner and somebody named Silvanus? The archaic name provides a bit of a hint into the genesis of this book. It was written in 1916 by the aforementioned Silvanus with an update in 1991 by Gardner.

It is an excellent book, but the title is somewhat misleading. It may be easy for most "schoolboys" of Silvanus' era - as he will often quote that "as any schoolboy knows" and then allude to some slightly obscure bit of mathematics. While the book does not make calculus easy, it is indeed a very practical guide to the calculus. In this book you will not find proofs of the mean value theorem, or an in-depth derivation of the fundmental theorem of calculus, you won't even find the term "limit"! In the area of differential calculus, what you will find is the meaning of the derivative, rules for differentiating, "dodges" (e.g. useful little tricks), and many real life exercises (if, in real life, you are an engineer). Similarly, when covering integral calculus, the focus is on how to integrate (or, if you will, find the anti-derivative).

The book is written in early 20th century language (which Gardner updates as necessary) and I was struck by the purity of the grammar. Tenses are used correctly, prepositions are never left hanging - and it all reads so wonderfully.

I read the book because I couldn't remember how to prove that the volume of the cone was 1/3*pi*r^2*h - and that was one of the exercises in the book. So I sat down and worked it out (technically I did it in my head while lying in bed) - and the book gave me the confidence to push through and solve it. It is rather elementary, something any schoolgirl would know.

Recommended to all math types.

Posted by Richard Berger at 09:50 PM | Permalink | Comments (0) | TrackBacks (0)

As every reader of this blog undoubtedly knows, the volume of a cone with height H and radius R is 1/3*pi*R^2*H. As a reader of this blog, of course I also knew this - but I found myself unable to prove it. Until a few nights ago. Inspired by the classic book "Calculus Made Easy" by Silvanus P. Thompson and updated by Martin Gardner, which I am currently reading, I attempted to rediscover the simple proof for myself.

I jot it down here so that twenty years from now, I need only to recover this blog entry to remind myself of the proof.

Imagine a triangle with (x, y) coordinates (0,0), (H, 0), and (H, R). Now, imagine this triangle rotating around the X-axis. You will see that this forms a cone with the specified dimensions. Our goal is to determine the volume of this cone. Imagine a little strip of that triangle, with an infinitely small width and height y. When this is rotated around the X-axis, it will form a circle, with the area pi*y^2, as the radius is y. To find the area of the cone, we need to add all these little circles together as x goes from 0 to H - e.g. a little tiny circle at the top of the cone to the full big radius-R circle at the bottom of the cone. (Since my cone is "lying down" it is really from the left to the right - but I think you get the picture).

So, we have to take the integral of 1/2*pi*y^2 from x=0 to x=H. First we must figure out the value of y, in terms of x. At any point x along the x axis, the triangle formed by (0,0), (x, 0), (x, y) must be geometrically similar (e.g. proportional) to the triangle (0, 0), (H, 0), (H, R), as the two triangles have the same three angles, they must be similar. Thus H/R = x/y. Or rewritten as y/x = R/H or y = R*x/H. So, we know have to integrate:
pi*(R*x/H)^2 which is the same as itegrating
pi*R^2/H^2 * the integral of x^2 taken from 0 to H.

The integral of x^2 is x^3/3, taking the value at H, we have H^3/3 and at 0, we have 0. So the integral is H^3/3. Now multiplying by the constants from earlier, we have to solve
pi*R^2/H^2 * H^3/3 which yields
pi*R^2*H/3 or 1/3*pi*R^2*H.

Apologies for horrible mathematical formatting!

Posted by Richard Berger at 10:08 PM | Permalink | Comments (0) | TrackBacks (0)

Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions
Author: Lisa Randall
Amazon info

An excellent addition to the physics reading library! This book is deeper than Brian Greene's Elegant Universe, but quite accessible. Like most books in this genre, Warped Passages covers the 20th century revolutions in physics - the emergence of relativity and quantum mechanics - and the lack of coexistence between the two. Randall then moves on to one of her specialities as a "model builder" - the standard model. At this point she veers into more complex territory (but it is always marked as optional and there are always metaphorical introductions). But it is this more complex territory that is the most interesting and challenging - including the Higgs particle/field and detailed discussions of symmetry breaking that must occur for particles to have mass. From there we move to some of the big questions - like why is gravity so weak. And, Randall covers string theory in a fair amount of detail, as well as branes and M-theory.

But just as you think the book is about to end, it is really just beginning as the first 300 pages or so are merely introduction for Randall's extensively cited theories of warped space, namely RS1 - the theory that there universe might consist of two branes, separated by a very small amount, but such that all the standard particles are on one brane (the Weakbrane) and gravity resides on the other brane (the Gravitybrane). These branes are separated along a fifth dimension of spacetime (e.g. a fourth physical dimension), but the key is that because this dimension is warped or curved, the gravitational forces drop off exponentially with distance - thus a small physical separation leads to a huge difference in the power of gravity. Randall then extends this to RS2, which allows for an infinitely big extra dimension of space, but one that wouldn't be noticed in our world (again, due to the warped nature of this extra dimension).

Beyond the intellectual achievement of this book, I also saw Randall as uniquely capable of bridging the philosophical gap between string theorists (this string theory stuff is so beautiful it just has to be write) and particle physicists (if we can't test it, then it is a philsophy, not a science). Randall seems to avoid being an idealogue (although she is a particle physicist/model person, not a string theorist) and is interested in finding ways to bring the two groups together.

And this book also struck me on a personal note as the author went to the same math camp that I attended (I think she was there two years later).

My only criticism of the book is that there are times when the writing is a bit repetitive - e.g. mentioning the same point several times in the space of a paragraph of two.

Highly recommended to all those interested in recent advances in the world of physics - no math is required, but you will have to think deeply at times.

Posted by Richard Berger at 10:16 AM | Permalink | Comments (0) | TrackBacks (0)