Reading 1/28/2008: 1.1-1.5

Sections 1.1,1.2: Okay, so these are basically what we did in class. Fairly simple stuff to those who have seen tensors. The mathematician in me is still freaked out by the index stuff, and the physicist is wondering what happened to superscript/subscript Einstein notation used in GR. But otherwise, it's pretty intuitive. That said, it does seem the notation is incredibly cumbersome in certain cases. For example, equation 1.31

||a_ij||² = ||a_ij|| ||a_ij|| = ||a_ij a_jk|| = ||&delta_ij|| = 1

seems vastly more clear when written as

|A|² = |A| |A| = |A| |A|^T = |A| |A|^-1 = 1

As the middle step in index notation does not seem easily justified, whereas we know that orthonormal matrices have transpose inverses. It's a minor point, though.

Section 1.3: The subscript issue was nicely resolved here, though I wish there were a more rigorous definition of contravariant vs. covariant. I'll need to go dig that up on Wikipedia. It seems to be an issue that doesn't come up as prominently (if at all) in the mathematical notation. Contravariant vectors transform 'like' coordinates and covariant transform 'like' the gradient, but what exactly does that mean? I realize this won't be necessary for the class, but it's intriguing. Another point (definition-wise) I'd be curious about here is exactly what it means to be an isotropic tensor - I'm presuming this means that its an eigenvector of the rotation matrix with eigenvalue one.

Section 1.4: Pseudovectors are intriguing. I haven't seen them in a mathematical context and am wondering if they have a common analogue, perhaps under a different name. Mathematicians, of course, aren't looking at things from the same perspective - cross products are not as prominent and the central focus isn't how tensors transform under transformations. That said, what is the fundamental difference? And where does the name tensor density come from - in particular, why density? I expected this to be something like a tensor field, but obviously it is not.

Section 1.5: This is pretty much the standard tensor stuff. It's actually a little strange to see contraction without the need to raise or lower indices, but it makes sense given the Cartesian focus. The transition between tensors and tensor densities is definitely new, however, though I'd still like to know what is fundamentally going on - at least in the mathematical realm - with densities. Somehow, that usually makes the physics make more sense.

So we're still doing fundamentals. Tensors are cool little creatures, and it should be good to see how they work in this context, as opposed to GR or Diff. Geo.