Dork Scratchings

I heard somewhere or another the claim that soap film forms a "surface of zero mean curvature".  I wasn't sure exactly what that meant until I read this book which gave me the tools to understand what that means, and prove it.  It turns out to be a simple consequence of Gauss's Divergence theorem.  In tensor notation Gauss's Divergence theorem states $$ \int_{\Omega}\nabla_iT^idV = \int_{\partial{\Omega}}T_iN^idA $$ where $\Omega$ is some volume of space $\nabla_i$ is the covariant derivative along the $i$th coordinate $z^i$ $\delta{\Omega}$ is the surface of the volume $T^i$ is any single index tensor defined over the whole volume $N^i$ is the unit vector normal to the surface in contravariant form  This works in any number of dimensions, so if you take surface embedded in 3 dimensional euclidean space $\vec{z} = \vec{z}(s^1,s^2)$ and cut it, then the theorem tells us $$ \int_{S}\nabla_{\alpha}T^{\alpha}dA = \int_{\partial{S}}T_{\alpha}n^{\alph...

A non-euclidean surface with coordinate system Metrics are important in both the calculus of surfaces and in General Relativity.  These objects are essentially NxN matrices whose values are defined by a) the coordinate system used to parametize the space, and b) the particular point $\boldsymbol{x} = (x^i)$.  In general, theorems can be categorized as either intrinsic or extrinsic , where extrinsic means that information about how the space is embedded in a higher dimensional space is used, and intrinsic means that nothing other than the metric and it's partial derivatives were used.  The word intrinsic comes from the idea that a being which "lives" in the space can calculate the metric. So how does such a being find the metric?   2 & 3 dimensional spaces In three dimensions it's pretty straightforward: Impose a coordinate system $x^i = x^1, x^2, x^3$ on the space Define the covariant basis $\boldsymbol{e_i} = \frac{\partial{\boldsymbol{R}}...

I said I was going to use this blog to save my HandWrite Pro scratchings.  Well here is the first.  A statement of Gauss' Divergence Theorem in both traditional euclidean coordinates, and in general coordinates. ​

I ended up buying the book for the Tensor Calculus course.  Almost all the material is available for free in the youtube series, but I wanted to own a trophy.  I was also partly motivated by a sense of guilt - having gatecrashed a complete university course for free I felt I should at least leave the host a bottle of wine.... Having said that, I have delved into it a couple of times since.  The first time was because I was left with a slight sense of incompleteness by the course after it stated but did not prove Gauss' Divergence Theorem in general coordinates.  (The proof is in one of the final chapters.)  The second time was much later. I was preparing to learn GR and I needed a refresher in TC to boost my confidence.

After a few abortive attempts to teach myself General Relativity I eventually realized that the best approach would be to master (as best as possible) Tensor Calculus on its own, and then tackle GR.  That sounds easy doesn't it!  Well, no, as it turns out.  There's certainly lots of material online, and there are books on the subject at the Cambridge University Press Bookshop, but in general they are all very bad at explaining the motivation behind the maths, or they get hung up on relating back to the building blocks of pure maths, thereby obscuring the vision. Eventually I found Pavel Grinfeld's excellent online course.  48 lectures (yes!) that take you by the hand and really do explain what Tensor Calculus is about, and how to do it.  I watched these during my lunch hours over the course of a couple of months.  I'm sure all the other cafe goers thought the man sitting in the corner with his headphones in was a complete dork, but I didn't care - I...