responses

So, I'm reading over other people's thoughts before I go to class, and I thought I could respond to some of them! W00ts!

Michael mentions that "A question about 1.6 and partial derivatives: in 1.6 a partial derivative was defined, and shown to transform like a tensor. But if I'm remembering correctly, in G.R. we saw a different 'partial derivative' (the "box operator"?) that was developed specifically because the old "standard" partial derivative did NOT transform in a covariant fashion. Is this simply because of some extra consideration arising in a non-Euclidean space, or am I getting these two cases mixed up or confused in some sad way?"

I think the reason that we needed a 'new' gradient operator is because in GR (or even in the special-relativity treatment of E&M) we want our vectors to be invariant not just under the standard rotations that we know and love in 3-space, but also under Lorentz transforms (essentially, 'rotations' in 4-space). In special relativity, as applied to E&M, needed the "box" operator because we needed the minus sign on the t component to make the physics and math work out.

And on a miscellaneous note, in response to " In composing this blog, I've been wishing that I could use cut-and-paste in the "Compose" window, but it doesn't seem to work. Is this a bug or a feature? Or is it just a quirk of my browser (Safari)? Thanks for any insight or advice!"

It's probably a bug. In Firefox under Linux, I can copy/paste fine into the text box, but it does behave strangely in other respects and isn't just a text box, it's got all sorts of fancy javascript behind it... yep, according to http://help.blogger.com/bin/answer.py?hl=en&answer=42247 , Safari isn't fully supported yet. Perhaps, if you want to copy/paste, temporarily switch over to the "Edit HTML" mode first (that looks like it just gives a standard text box) instead of the default "Compose" mode?

Sections 2.1-2.3

Ok, so I'm trying to get an early start on this semester, and hence I'm making a blog post about the reading due this wednesday :) I think I'm getting how we're defining the stress tensor; it seems to me to be pretty close qualitatively to the
Maxwell stress tensor from E&M; as then, it seems like the most concise way to put it is that the {ij} component of the stress tensor is the force in the i direction on a unit area normal to the j direction. I'm not sure I'm visualizing the "make a cut in the medium" explanation though. Am I getting this right, or do should I think about the problem setup some more?

Also, in response to Steven Ning wanting to get more fancy LaTeX than the online image converter offers - they cite the script they use as being available at http://www.nought.de/tex2im.html ; it's a bash script which uses "latex" and "convert" to make the images. If you have a unix system with bash available, it seems like it should be a pretty simple matter to stick '\include{amsmath}' into the script and use it, though I haven't tried it myself yet.

Reading for 1/28, 1.1-1.5

Whoa, expressing the standard dot products and cross products (and, later, div and curl) in terms of tensor rank-reductions is really cool!

I'm not sure what to make of the fact that the gradient of a scalar doesn't transform the same way as a 'normal' vector. What is the distinction, what does contravariance actually mean? I don't get it, and the text doesn't go into detail.

Frist psot.

Hey all!

I'm Max, and I'm currently an HMC senior majoring in both Physics and Computer Science. I'm taking the class because fluid dynamics is super-cool! And there's a pretty high chance I'll be using a lot of this stuff in grad school. I'd like to come out of this class with a much better understanding of tensors and how to use them - I never felt like I completely got them when taking GR, or when we used the Maxwell stress tensor in Big E&M, so I'm looking forward to a more in-depth treatment of them. My favorite two equations are the Lorentz transformations. These two go together; I couldn't pick just one, and I didn't want to convert them to a single equation that's less understandable.








A fun fact about me is that I've gotten my computer to beat expert minesweeper in 41 seconds without cheating :)