Dork Scratchings

A proof that the net force due to pressure on a (fully or partially) submerged object is equal to the weight of the water displaced. The facts/approximations used were pressure is isotropic density is constant gravitational force is constant And the method of proof was to apply the Divergence Theorem. POSTSCRIPT I was just wondering why pressure is isotropic when I found this post and  I realised that the scratchings above actually prove it!  Imagine a blob of water submerged in water: the above proof shows that the net force on it is zero, but only if pressure is isotropic !

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...

I was never really that interested in economics until after 2008 when suddenly technical terms like structural deficit started appearing in the news.  Oddly, with anything relating to macro economics, journalists have to pretend they were born understanding all the concepts perfectly and they're not going to patronize you by explaining them.  This is in stark contrast to anything related science where they have to pretend to understand even less than they do!  Somehow this attitude has leaked out to the wider world, so that friends and colleagues down the pub - or politicians on Question time - will b******t eternally about the effect of interest rate rises, but happily or even boastfully admit to knowing nothing about how the rest of the universe works.  I, on the other hand, knew I didn't know anything about economics, but thought it was less important than all the other stuff I didn't know.  But when economics stories started to become the main content of th...

In More Mathematical Puzzles and Diversions , Martin Gardner makes a passing reference to the Ham Sandwich Theorem.  It goes like this Any 3 shapes in 3 dimensional space can be simultaneously bisected by a single plane So imagine you have two roughly cut pieces of bread and a slice of ham, then you can always cut the sandwich in half such that each half has exactly half of each piece of bread and half of the ham, no matter how roughly strewn the pieces are. According to Gardner the generalized version has been proved by Tukey and Stone: any n shapes in $\mathbb{R}^n$ can be simultaneously halved by a single $n-1$ dimensional hyperplane.  But I thought I'd have a go at proving it myself in the 3D case, just for kicks. First observe that there are at least enough degrees of freedom to make it not impossible .  A plane (other than one going through the origin) can be described by the equation $k_xx+k_yy+k_zz = 1$ for some $\boldsymbol{k} \neq \boldsymbol{0}$ so the...