Hello friends! This will one day be my personal web site, though it is currently under construction.
My favourite theorem is the Reynolds transport theorem, which says that for any (possibly vector-valued) function \(\Psi (\vec x, t)\) and a simple volume \(\Omega (t)\) immersed in a fluid, the following holds true:
\[\frac{\mathrm d}{\mathrm d t} \, \int\limits_{\Omega (t)} \Psi \, \rho \, \mathrm d V = \int\limits_{\Omega (t)} \frac{\mathrm D \Psi}{\mathrm D t} \, \rho \, \mathrm d V + \oint\limits_{\partial \Omega (t)} \Psi \, \rho \left( {\vec v}_{\text b} - \vec u \right) \cdot \mathrm d \vec \Sigma \, ,\]
where \(\rho\) is the fluid’s density, \(\vec u\) its velocity, \(\mathrm D / \mathrm D t\) the material derivative, \(\vec{v}_{\text b}\) is the velocity of \(\partial \Omega\) and \(\partial \Omega\), as usual, denotes the volume’s boundary.
In fact, I like it so much I’ve even written a sketch proof of it here.