My brother Tim has started a blog recently (yesterday) and he’s already put up a fair amount of content, and the site looks great. The blog is titled “teebs”. Certainly his publication rate far exceeds that of vdov.net. You should all take a look here. His site is also linked from the blogroll on the right sidebar.
I used something in my course on continuum mechanics last week that was pretty interesting, and I only now realize I’ve used it all the time without ever thinking about it. It is the so-called “Du Bois-Reymond Lemma”, and it goes something like: if an integral formula equals zero and is a continuous indefinite integral, then the integrand must vanish. It’s so ridiculously simple and yet I never actually thought about it before. This is particularly useful when, for instance, deriving the equation of continuity from the mass conservation form of the Reynold’s Transport Theorem.
![\frac{d}{dt}\iiint_{V(t)}f(\mathbf{x},t)dV = \iiint_{V(t)} \left[\frac{\partial f}{\partial t} + \nabla\cdot\left(f\mathbf{v}\right)\right]dV](/latexrender/pictures/600d6f1882e5faefb37b8215ade80e1f.png "\frac{d}{dt}\iiint_{V(t)}f(\mathbf{x},t)dV = \iiint_{V(t)} \left[\frac{\partial f}{\partial t} + \nabla\cdot\left(f\mathbf{v}\right)\right]dV")
Apply to mass conservation by,
And we have,
![0 = \iiint_V \left[\frac{d\rho}{dt} + \rho\left(\nabla\cdot\mathbf{v}\right)\right]dV](/latexrender/pictures/b7003a7411beab64c14a3a78a0a2b326.png "0 = \iiint_V \left[\frac{d\rho}{dt} + \rho\left(\nabla\cdot\mathbf{v}\right)\right]dV")
Therefore the equation of continuity is simply,
Fun. Cheers.