DU BOIS LEMMA
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.
Ha! Nice to know the name of that little trick. Now I can sound smart when I’m talking to mantle dynamics people. Yay.
1) I’d call this the “fundamental theorem of calculus” applied to a homogeneous equation or a conversion from the variational (in functional analysis) or integral form to the differential form. But I never learned the named organic reactions either.
2) This lemma is a good named lemma to know. It also appears in electromagnetism (to convert between the forms of Maxwell’s equations) and in classical mechanics (in the derivation of the Euler-Lagrange equations).
3) The integral can be definite.
4) What’s going on with your domain of integration? Is it really time-dependent?