I would have asked in math, but I was hoping the context of lattice boltzmann may make my question clearer. Given f is the number density of particles, v velocity, and u equilibrium velocity.(adsbygoogle = window.adsbygoogle || []).push({});

In a book(http://www.ndsu.edu/fileadmin/physics.ndsu.edu/Wagner/LBbook.pdf equation 3.14), he derives mass conservation/continuity equation in terms of distribution function f:

NOTE: He's using Einstein index notation.

[tex]

\begin{align}

\partial_{t}\int f d v +

\partial_\alpha\int f v_\alpha d v +

F \int \partial_v f d v & =

\frac{1}{\tau}\int (f^0 - f)d v \\

\partial_t n + \partial_\alpha(n u_\alpha) & = 0

\end{align}

[/tex]

where I'm using equilbrium distribution f0 in place of f:

[tex]

f^0(v)=\frac{n}{(2\pi\theta)^{3/2}}\exp{-\frac{(v-u)^2}{2\theta}}

[/tex]

I'm reading along and get the first 2 terms of the lhs and rhs to be n, n u_alpha, and 0 respectively, but the third term on the lhs looks like it evaluates to zero. My question is, it looks like he's integrating a derivative of f wrt to a vector. I'm assuming this is a directional derivative and looked it up in wikipedia. However, when I throw it into mathematica (using the equilibrium distrubtion f0) and triple integrate, I don't get zero. Anybody shed some light on how he gets this? (If you can show me the integral in mathematica - so I can see that it works - that would be a bonus)

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# Question about integral used in Lattice Boltzmann text

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