Archive for August, 2005

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Tensors

Tuesday, August 2nd, 2005

Tensors are no longer quite such mysterious objects to me. Incidentally, what has confused me so much about tensors were the impression I got from physics classes and books. The fact that it’s so much easier to understand the mathematical definition— and supposedly from there figure out a physical corollary— is interestingly backwards to me. For instance, I understand div, curl, and grad in almost purely physical terms, and justify the mathematics on the basis that it gives the results needed.

The formal definition of a tensor is a multilinear mapping

\displaystyle T: \underbrace{T^\star_p \times T^\star_p \times \cdots \times T^\star_p}_k \times \underbrace{T_p \times T_p \times \cdots \times T_p}_l \rightarrow \R,

where p is a point, T_p is a vector space attached to the point (aka a tangent space ( :)), and T_p^\star is the dual space of T_p^\star. In this case, T would be called a tensor of type or rank (k,l). By definition, tensors of type (0,0) are scalars.

Compare that purely mathematical definition to the equivalent, but more confusing definition that I usually see for tensors in a physical context: A “quantity” with components T^{\mu^\prime_1 \cdots \mu^\prime_k}{}_{\nu^\prime_1 \cdots \nu^\prime_l} which transform according to the rule under a linear transformation \Lambda:

\displaystyle T^{\mu^\prime_1 \cdots \mu^\prime_k}{}_{\nu^\prime_1 \cdots \nu^\prime_l} = \Lambda^{\mu^\prime_1}{}_{\mu_1} \cdots \Lambda^{\mu_k^\prime}{}_{\mu_k} \Lambda^{\nu_1}{}_{\nu_1^\prime} \cdots \Lambda^{\nu_l}{}_{\nu_l^\prime} T^{\mu_1 \cdots \mu_k}{}_{\nu_1 \cdots \nu_l}

(where I’m using Einstein summation notation to avoid an even nastier equation.)

Clearly this form has the advantage that it would be easier for folk who don’t know what a dual space is to understand— since we engineers don’t learn that in our “engineering math” course which substitutes for linear algebra and differential equations, I could see why this form might be used in engineering/physics contexts— furthermore, it is clearly directly physically applicable (i.e. inertia is a tensor, since its components transform in the above manner).

But, those are about the only advantages it has. For one thing, this definition begs the question: why two sets of indices? Also, it is coordinate-centric, which by analogy is like teaching linear algebra and stating vectors are collections of numbers with the following properties… and matrices are collections of numbers with the following properties; i.e. inefficient and overly specific, since you lose coordinate free methods, and misleading since you tend to think in terms of matrices instead of mappings. Finally, it isn’t clearly how to determine what is a tensor without trotting through a whole bunch of calculations to see if components transform in the appropriate manner.

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Signals and Systems

Monday, August 1st, 2005

I’ve let my appointed task of typing up my Fourier class notes fall aside… good thing no one, self included, really cares or needs them. Instead, inspired by the new material we’re doing on signals and systems, I’m going to start a set of notes on that stuff. We did basically everything we’re doing now in this grad. math class in my junior level DSP course in the engineering department (minus the rigorouous proofs that e.g. the convolution of two signals is again a signal).

Today, I’m supposed to be working on my paper for the xenon project which I finished this summer, but I don’t think I’m going to. My prof. is off at a conference— so I don’t need to have it done until Friday when he gets back— and I’m at the tedious part: putting in graphs and discussing numbers. Admittedly, this is the only important part of the paper, but I don’t want to mess around with figuring out which graphs to include, and how to quantize the results today. I’ll do it tomorrow :) Really, I mean it! Instead, I think I’m going to try to take a crash course in wavelets— I read the preprint of a paper they wrote on the project I’m starting to work on this week, and my eyes glazed over at the technical sections— and after/during that, relax by studying some relativity stuff. Ordinarily, that wouldn’t be my idea of relaxing, but I found a book that is a pleasure to read: “An Introduction to General Relativity: Spacetime and Geometry” by Sean M. Carroll. What I like about it is that like the other book on (special) relativity I’ve part read: “The Geometry of SpaceTime: An Introduction to Special and General Relativity” by James J. Callahan, it can be read as a math book without proofs, as opposed to a physics book with math; i.e., no ‘practical problems’ to hurt my head trying to understand, and no units! Besides, it is a beautiful book.

So, off to the library now.

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