Learn Physics or Get Bored Trying

Deviating standards: epilogue

A subtle variant of the “inconsistencies” in the last post can arise where a concept is introduced and used in a simplified form, which can catch you unawares in more general settings.  For example (and while I’m trying to keep these posts agnostic of any particular field, it’s hard to describe this without example; but don’t worry if the details aren’t clear, it’s the idea that matters):

A vector space has associated with it a scalar field.  The most obvious one is the field of real numbers; they are also commonly associated with complex numbers.  Given a vector space over either of these fields, you can turn this into an inner product space by defining the inner product of two vectors u and v as x = \langle u, v \rangle, where x is a member of the scalar field.  The inner product must satisfy a number of properties; one of these dictates a form of commutation: \langle u, v \rangle = \overline{\langle v, u \rangle}, where \overline{x} denotes complex conjugation—a process that leaves real numbers unchanged.  You will therefore, very occasionally, see an author avoid mentioning complex numbers if they are making only brief use of an inner product space with real scalars, by quoting this property as \langle u, v \rangle = \langle v, u \rangle.  If not aware of this, you may miss the fact that conjugation is required in the complex environment.

Again, the only real protection to this is to make sure you read any introductory chapters where terms ought to be properly defined.  If in doubt, and the text makes assumptions of prior knowledge, Wikipedia is generally a good place to look for definitions, as it will tend to quote the most general form.


June 16, 2010 Posted by | Self-study | Leave a comment

Deviating standards

One of the purposes of this blog was to discuss the additional challenges presented by self-study, over structured courses such as those offered by Universities and the like. I touched upon the lack of an obvious “road-map” in the Background page, and I’ll expand on that later. First I’d like to talk about a problem that is not unique to self-study, but that is exacerbated by going it alone.

Simply put, once you get past the “general physics” books, aimed at late high-school to early undergraduate (such as the rather fine “Fundamentals of Physics” by Halliday, Resnick and Walker), you will be flicking between books specialising in particular subjects, and no two of them will use the same notation or terminology.

Starting with conceptual differences, there are three main issues:

  1. Different names being used for the same concept,
  2. The same name being used for different concepts,
  3. Different definitions for equivalent concepts.

In some cases you’ll see two or more of these at the same time (you could say that the general problem is a linear combination of these basis difficulties, but you might strain a friendship or two if you do).

Some examples I’ve met so far:

  1. Where most books I’ve encountered talk of Hermitian operators (or matrices), the book I’m currently reading on Fourier Analysis calls these symmetric operators. This sort of thing is not too tricky to negotiate, unless compounded with a different, but equivalent, definition for the concept in question (more of which later).
  2. Whereas most books in the subjects I’m currently studying will use adjoint to mean “conjugate transpose” of a given matrix, I did spend some time brushing up on my differential equations using a book that defined the adjoint of a matrix A as A^{\dagger} = \det(A) A^{-1}. The fact that three chapters later he switched to using it as the conjugate transpose without thinking to warn us was, I think, just to piss me off. (Entertainingly, this was not a very new book; in the chapter on improving the efficiency of approximation algorithms we were warned that computer time can cost up to $1200 per hour.) Fortunately, in this case, a previous book had warned of the clashing terminology and confirmed that these were entirely unrelated usages, or I might have spent an evening scratching my head.
  3. In a nicely circular conclusion to this tale of three, in the book above that used “symmetric” where other books use “Hermitian”, the defintion of a symmetric operator was an operator A for which inner products \langle u, Av \rangle and \langle Au, v \rangle were equivalent, while another book defined an operator to be Hermitian if it was self-adjoint, i.e. A^{\dagger} = A. As you may have guessed (or already know), these definitions are equivalent—as long as you use the right adjoint!

Once you get beyond conceptual differences, you of course have notation to deal with. Will the author use primes for derivatives (f'(x))? Or Leibniz notation (df/dx)? Perhaps subscripts (f_x), or (for time derivatives of a vector) dot notation (\dot{\vec{x}}, \dot{\mathbf{x}} or \dot{\underline{x}})? Often these will be used interchangably within any single book, according to what seems to the author to be most natural for the given problem, but you can be sure that at some point you’re going to fail to recognise a friendly equation in unfriendly clothes.

So how does one deal with this sort of silliness? Well there are a few things you can do to help:

  • Never skip the opening chapters of a new book. Pretty much every text will start with a review of topics, and while it can be dull going over the same ground time after time, this is also where the author lays his/her notational cards on the table. The use of different but equivalent conditions for a certain concept will mean that you may follow the text but get hopelessly lost in a proof if you aren’t prepared for it. And if nothing else, you may get treated to an elegant or more enlightening derivation of something.
  • The more sources you have with overlapping interests, the better you’ll get at recognizing the well trodden paths; in the example above of the use of “symmetric” in place of “Hermitian”, we’d just introduced a particular concept (eigenvalues), and I was 90% sure we’d be moving immediately to Hermitian operators, simply because it’s the natural next step.
  • Make use of sample chapters (for e-books) or “look inside” buttons on web-stores—or you could even (gasp!) try a book shop (try between “iPods For Dummies” and “Book-reading For Dummies”, if it’s anything like my nearest Waterstones), if you’re particuarly keen to follow a particular style.
  • Finally, and a theme that will be picked up later, sometimes if you’re struggling with a particular section of a book, take a step back, and just keep reading without focusing on any proofs. Sometimes you will find that a few pages ahead you will realize that the author is actually doing something you recognize but in a different form, and with this understanding it all slots in place.

Of course, any further tips are always welcome!

June 15, 2010 Posted by | Self-study | | Leave a comment