Recent content by joda80

That makes sense … thanks for helping out!

Hi All, I think I have confused myself ... perhaps you can tell me where my reasoning is wrong. The idea is that in general coordinates the partial derivative of a vector, \frac{\partial A^i}{\partial x^j}, is not a tensor because an additional term arises (which is the motivation for...

Thank you Chet, for your detailed responses. I have to re-read your exposition a few more times, but it seems you are pointing to the same direction as I was heading (regarding the transformation of the basis vectors of the deformation gradient). Cheers, Johannes

Thanks! I think there's an error in the way I defined the permutation symbol. It should be {+1, -1, 0}. If defined in terms of the basis vectors as done above, we actually wouldn't need the sqrt-factor in the definition of the curl. But my main confusion has been settled...

Hi Chet, Thank you for your detailed response! So my thoughts about the transformation behavior: When changing the coordinate basis of a "conventional" vector gradient \partial A^i / \partial x^k, with x^i = x^i(\xi^j) the tensor behavior is destroyed because not only the derivative...

Hi All, I'm trying to figure out how the components of the curl transform upon changing the coordinate system. In general coordinates, the contravariant components of the curl (if applied to the velocity field; then the curl is known as vorticity) are defined as \omega^k =...

Hi Chet, Thanks for your comments. I certainly am interested in more discussion. Were you hinting at two-point tensors? It would seem that whenever we map a tangent vector from one manifold to another, we would have the problem that we need to express the vector on each manifold w.r.t...

@Micromass: Thanks, now it works! @Newton: Indeed, this is what I meant. What you say makes sense (also consistent with Wald's General Relativity, p. 437), and I already suspected that the statement in Fliessbach's textbook might be wrong, but wanted to make sure. If you're not bored yet...

Hi All, I have a question about transformation matrices (sorry about the typo in the title). The background is that I've spent some time learning differential geometry in the context of continuum mechanics and general relativity, but I'm unable to connect some of the concepts. So I have this...

Great, thanks! My first correct statement in this thread! :)

I updated the URL in my previous post, sorry it didn't work. I'm not sure if we mean the same thing regarding gluing the surfaces together (though we might -- I never stumbled upon the wegde sum before). This was just my way of saying that the above pattern (see URL) was supposed to repeat...

Thanks ... I knew I was going to get in trouble for using the word "global" ;) Point taken! Just as a check, though: Let's assume a 2D surface that looks like this: http://www.bu.edu/tech/files/images/surface.jpg (from www.bu.edu) And let's also assume that this image repeats itself...

Just to follow up: I think I found out what my misunderstanding was. Somehow I thought that the coordinate systems on a manifold were always the Riemann normal coordinates (i.e., valid only in a small neighborhood of a given point). Accordingly, I thought all coordinate systems were describing...

Thanks, guys! I think I understand (the metric tensor and the curvature are properties of the manifold, not of the tangent spaces; accordingly, the coordinates chosen to describe the tensors merely determine their components, not the tensors themselves, which of course is the purpose of...

Indeed, I'm working with a couple of pure math books aside from physics books ... though mainly with Wald's "General Relativity", which seems to be rather clean/rigorous regarding the mathematical treatment. I think my problem is mostly of mathematical nature. If you're no bored yet, I'll give...